Physics Textbook: Mechanics, Heat, Sound & Light

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EX P ER I M EN T A L
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PHY S I C S
S O U N D AN D L I G H T
BY
W ILSO N M A D
ly F ll w
f T i i t y C ll g
C m b i dg
f Ph y i
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Ri
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H A RO L D A
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PRE FAC E
'
book is i tended as text book for use i connection with
a course of experimental lectures on mechanics properties of
matter heat sound and light N previous knowledge of physics
is as sumed but nevertheless the bo ok is primarily i tended
for a first year college course and the maj ority f the students
attendi ng such a course have studied element ry physics at
school Th writing of such a book does not offer much scope
for originality ; the aim of th writer should be to present
fundamental principles clearly and accu ately Th e chief difficulty
is to decide what to include d what to leave out I have
endeavoured to leave out everythi g t of fu damental importance
It is important for the student to learn some facts and to get to
understa nd some methods and fu damental pri ciples ; if he learns
nothing about certai phen mena harm is do e and he can make
up the deficie cy in his knowledge at a later date if necessary
Th kind of text book which contains a little about everything does
more harm than g d
Care has been taken not to discuss questions which ca not be
t r eated adequately in an elementary way and to avoid stating
formulae without proving them A few expe rime ts are rather f lly
described in nearly e v e y chapter ; these have been selected from
the ma y which might ha v e been merely mentioned
In Part I Chapters VI VI I d parts of I X may be omitted
at the first reading I Part II Chapters X d XI may also be
omitted by students whose time is limited
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3 5 8 3 43
PRE FAC E
.
book is inte ded as text book for use i connection with
a cours e of experimental lectures on mechanics properties of
matter heat sound and light N previous k owledge of physics
as sumed but nevertheless the bo ok is p imarily intended
i
for a fi st year college course and the ma jority f the students
attendi ng such a course have studied elementary physics at
school Th e writi ng of such a book does t offer much scope
for originality ; the aim of th writer should be to present
fundamental principles clearly and accurately Th e chief di fficulty
is to decide what to include d what to leave out I have
endeavoured to leave out e v ery thi g t of fu damental importance
It is important for the student to learn some facts and to get to
understand some methods and fu dame n tal principles ; if he lear s
nothi g about certain phen mena no harm is do e d he can make
up the deficie cy in his knowledge at a later date if necessary
Th kind of text book which contains a little about everything does
more harm tha g od
Care has been taken
t t discuss questions which ca not be
treated adequately in an elementary way and to avoid stati g
formulae without proving them A few experime ts are rather fully
described in nearly e v ery chapter ; these have been selected from
the ma y which might ha v e been merely mentioned
In Part I Chapters VI VI I d parts of I X may be omitted
at the first reading In Part II Chapters X d XI may also be
omitted by students whose time is limited
HI S
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3 5 8 3 43
P REF AC E
vi
I am indebted to Messrs J J Griffi and S o s Ltd for
permission to rep oduce Fi g 5 7 Pt I ( Oe tli g bala ce ) from
thei catalogue ; to the Cavendish Laboratory f leave to draw
Ltd for Fi g 8 2 P t I ;
Fi g 6 2 P t I ; to M essrs G C s
to Messrs W G Py e d C f Fi g 22 P t II and to M Edward
A n ld for permissio to reproduce Fi g 5 4 P t I V from S chuster s
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I wish to exp ess my thanks to M T G Bedford the Editor
of the Cambridge Physical S e ies for many valuable suggestions
and correcti s in readi g the proofs d f the preparatio of
Fi g 24 P t III
My tha ks are als due to the sta ff of the U niver ity Press
for their excellent work
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C ONT E N T S
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MEC HA I CS AN D P ROPERTI ES OF MATTER
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HEA T A F O RM O F EN ERG Y
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viii
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C ON T N TS
III
PART
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M OTI ON
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VI I
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VI I I
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SO
U ND
NOTES
FEREN CE OF S O U N D
P ERP EN D I C UL A R V I B RATI O N S
A
P OS I TI ON
.
OF
I NS
REF L EX I ON , REF R
VI
ND
OF
CTI O N ,
I N TER
N CE
VI B RATI O N
V I B RATI ON
OF
S TRI N GS
OF
AI R
OP EN
IN
C L OS ED P I P ES
AN D
P ART I V
L I G HT
S O U RC ES
REF L EX I ON
S P H ERI C
L I G HT
OF
AL
P H OTO
.
AN D
REF R
A
M
CTI ON
ETR
Y
AT
P
L A N E S U RF A CES
M I RRORS
L EN S ES
DI S P ERS I O N
.
CO L O U R
OP TI C
TH E
AL
I N S TR
V EL OCI TY
I N TERF EREN CE
PO
L A RI Z A TI ON
EN ERG Y
IN
D
EX
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OF
U M ENTS
L I G HT
OF
AN D
AN D
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DI FF R
A
CTI O N
D O UB L E REF R
IN
A
V BL RA D A
I SI
E
CTI ON
I
.
TI O N S
AN D
I NTRODU CTI ON
physical sciences relate to the study of phenomena in
systems of bodies compo s ed of lifeless or inanimate matter Th e
name p h y s i c s denotes a restricted bra ch of physic l science
dea lin g usually with only th e implest kinds of phenomena which
happen under artificially arranged circumstances These C rcum
stances are designed with the Obj ect of simplifyi g as far as
pos sible the actions taking place so tha t the real nature of the
phenomena Observed may be the more easily discovered Physics
then m a y be said t relate to the succession of events in com
paratively S imple material systems artificially contrived for special
purposes
Th e o ther great branch of physical science chemistry deals
chi efly with the composition and S pecial properties of particular
substances and their methods of preparation in a state of pu rity
All man s knowledge of things is the result of experi ence
To
study physical science is to acquire for ne s own use the c
cumulated experience and wisdom of mankind in dealing with
nature and trying to turn the mani fold properties of m atter to
the good of the race
Experience S hows that natural phenomena are subj ect to
defin i te rules or laws which are invariable Events do not occur
i
a haphaz rd manner but follow ea ch other in regular order
An y given event is determined by the state of things preceding it
acco rdi ng to definite laws If a given state of things at any time
or place is followed by a certain event then at any other time and
place a precisely S imilar state o f things will be followed by an
identical event O this foundation science rests and without it
no reliable knowledge would be pos sible It is sometimes said
that like causes produce like e ffects d this is true if li ke c u
means causes which di ffer only with respect to the time and place
It is also true that in ma y cases a small change in a system
produ c es only a small change in the events taking place in it :
THE
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NTRO DU CTI ON
but there are other cases in which this is not true for example
a small moveme t f a gun j ust before it is fired
Clerk Maxwell stated the general maxim of physical science i
the following w ds
Th e di fference betwee
one event d another does t
depend on the mere differe ce of the times or the places at which
they occur but only di fferences in the nature configuration or
motion of the bodies concer ed
O
belief in the truth of this maxim is based on experience
N O exceptions t it supported by reliable evidence are kno wn
To describe phenomena exactly it is necessary t use words
with definite meanings Many words are us ed in ordinary con
versation in a loose manner sometimes with one mean i g and
sometimes with another I scientific wo k an attempt is made
to formulate precise definitio s of the m eani gs of words d
whe a word has bee n given a meaning it hould never be used
with any other O e of the chief advantages of a sci e ntific
trai ing ought to be the ability to use language having only one
possible meaning
T measure a quantity it is necessary to have a unit in terms
of which the size of the qua tity can be expressed Th e S ize of
any quantity is a unit of the same kind as the qua tity itsel f
multiplied by a number Th e choice of suitable units of exactly
fixed magnitudes is one of the most important aids to scientific
prog ess
Th e discovery of new phenomena a d the exact measurement
of all the quantities co ce ned is one of the chief Obj ects of
scientific investigations It is also desired to find out the general
laws regulating phenomena to analyse complicated phenomena
into S impler components and to formulate mental pictures of the
innermost structure of things S that the succession of events may
be explai ed and the behaviour of matter in given circumstances
predicted
When matter is found under some circumstances to act i n
accorda ce with a certain rule or law this rule may be suggested
as a u iversally true law of nature Th e consequences to be
expected if the rule is always obeyed are the worked out as
completely as possible and compared with the results of observation
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I NTROD U CTI O N
3
When the results in an immense number of cases have been found
to agree with those predicted and when no cases of want of agree
ment are know the rule comes to be regarded as an established
law of nature
When an event is shown to have occ u rred in accordance with
previously established laws f nature S O that it could have been
co mpletely predi cted by assuming the laws to be obeyed then it is
sometimes sa id to have been explained in terms of the natural
S uch s o called explanations are of course not
laws in question
complete Th e event h a s merely b een given its proper place i
a c lass of S imilar events determined by the same laws A complete
expla ation would require the laws themselves to be explained
N o such thing a s a really complete explanation of any event can
be gi v en
If it could be shown that all phenomena were due to the
m tion of a single continuous mediu m fillin g all space and that
othing else existed i the uni v erse the laws of motion of the
medium would still remain to be explained
Many of the properties of matter depend on the structure of
parts far too small to be Observed di ectly To explain such
properties a mental picture of the small parts may be i m agined
and they may be supposed to obey certain laws Th e truth of
such an hypothesis or theory can be tested by compari g the
properties of matter in bu lk with the properties to be expected
Even if the expected properties
ac cordi g to the theory in question
ag ree perfectly with those Observed we can never be sure that the
assumptions made are really true in fact Other assumptions
might lead to identical properties
S uch theories of the nature of matter enable new phenomena
to be predicted and when such predictions are found true the
truth of the theory becomes more pro b able When new phenomena
are di scovered an attempt is generally immediately made to
formulate a more or less complete theory to explain them This
theory then serves as a guide in devising new experiments with
the obj ect of elucidating the real nature of the phenomena Thus
the theory serves a useful p u rpose even if it ultimately proves to
be w ong
When a theory enables a great body of facts to be explained
when it has successfu lly predicted new facts and when no facts
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4
I N TRO
DU CTI O N
inconsistent with it are known it is usually regarded as probably
more or less completely true F example the atomic theory
according to which each eleme ta y substance i s made up of an
immense number of minute parts all exactly equal in size and
structure explains successfully such a vast a ray of well established
facts that it is unive sally accepted as true
When two different theories are capable of explai ing a set f
facts then to decide between them their consequences are worked
out until some practical case is found for which they predict
different results Th e matter is then t ested by experiment and i f
the consequences predicted by one of the t heories are in agreemen t
with the results obtained the other theory must be abandoned
or modified S uch an experiment t decide between rival theorie s
is called a c c i a l e p e i m e t
In studying physics it is important to distinguish between
simple but inexact experiments design ed merely to illustrate well
established principles and the exact investigations by which such
principles have been established F example though Atwood s
machine enables the laws of motion to be illustra ted our belief in
their universal truth does not rest on such a crude basis but is
founded on the results of many series of investigatio s of high
prec s o
It is convenient to divide physics into several parts each
dealing with closely related phenomena
M ec h i c s is the study of the motion of matter
Under the heading Ge e l P op e ti e of M tte are included
gravitation elasticity ca pillarity v iscosity and other misce llaneous
phe omena
S o d is the part of physics which deals with the vibrations
of elastic bodies which produce sensations in our ears
He a t is the branch of physics dealing with all phenome a
dependi g on W hether bodies are hot or cold
Li g h t relates to the physical pr cess which produces the
sensation of sight d is closely related to another great division
of physics Elec t i c i ty and M g ti s m
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P AR T I
M EC HAN I CS AN D P RO PERTI ES OF MATTER
C HAP T ER I
S PACE AN D TI M E
ME C HAN I CS
is the study of the modes f motion of matter
Th e motion of matter involves the ideas of position time and
uantity
of
matter
or
mass
q
Th e position of a body
be described by stating the length
o f the straight line between it and another body whose position i s
known and the direction of this line Thus if we are told that
a Village is 20 miles fi o m the town where we live and due West
of it we know the position of the Village Th e position of a body
can only be described relatively to the positions f othe bodies Th e
position of a S ingle particle by itself in space could not be specified
because there would be nothing from which its distance could be
meas ured ; for all parts of space are exactly S imilar With two
particles in space the distance between them could be measured
and would give the position of either relative to the other When
a pe rson gets to know the relative position s of the chief obj ects in
his neighbourhood s ch as the buildings and streets in the town
where he lives and the di fferent th i ngs in his house he has
acquired by experience the ideas of pos ition and of space or
volume Th e idea of spa ce is acquired by experience ; it cann ot
b e explained
A y particular body occupies a certain por tion of
s pac e
Th idea of space carries with it the ideas of length breadth
and height or more generally of the poss ibility of movi ng in three
di rection s pe rpendicular to each other Only three st aight lines
c a
be drawn through a point so that each one i s perpendicular to
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6
PROPE RTI ES
M C HAN CS AND
P
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A
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[
E
O F M ATT R
the other two O this account space is said to have thre e
dimensions
Dista ces are measured in terms of a unit of length
Th e
sta dard yard is a metal bar having two arrow transverse li es
ruled
it which is carefully preserved in London Engla d
Th e dista ce between the centres of these two lines when the b
is surrou ded by melting ice is the English unit of le gth th e
yard Copies of this standard very nearly equal to it are made
and are used as seco dary standards with which ordinary yard
measures for general use may be compared Th e Bureau of
possesses such 0 0 pies whose
S tandards at Washington
lengths have been accurately compared with the original standard
Th e French standard of length called the metre is preserved
at S evres and consists of a similar bar On e thirty S ixth part of
a yard is one inch and the metre is equal to 3 93 7 0 inches On e
inch is equal to 25 400 centimetres
To find the length of any body it is necessary to determin e
how ma ny times a unit of length is contained in its length
If L denotes a unit of length such as one yard one metre o
one centimetre then any length is equal to a number times
the unit or L F example consider a length of 1 00 cms or
inches
Units of area and volume are derived from the un ts of length
Th e unit of area is the a ea of a square with sides of unit length
Th e unit of volume is the volume of a cube with S ides of uni t
f
length I f L d enotes a unit of length then the correspond i n g
units of area and volume may be denoted by L and L respectively
for the unit of area varies as the square of the unit of length and
the unit of volume as the cube
Th e area of a rectangle with sides of lengths L and m L i s
equal to ( n L) ( m L) which may be written n m L and denote s
Th e arithmetical operation indicated by th e
n m units of area
expression m L is the mul tiplication of the number n by th e
1 the expression m L reduces to L and
number m If
m
denote s one u n it of area ; the arithmetical operation indicated by
L being the multipli c ation of the number one by the nu m ber one
In the same way the arithmetical operation indicated b y L i s
I x l x l
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P ROPE RTI E S
M EC HAN C S AN D
P
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A
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[
O F M ATT ER
u its of length time and mass which we have denoted by
L T and M are usually regarded as the fundame tal units
Whe ever possible other quantities are expressed in terms of
units derived from the e fundamental units Thus the unit of
volume which may be denoted by L is de ived fr m the unit of
length A another example consider density Th density of a
body is its mass per unit volume Th e unit of density is taken to
be a density of u it m ass per unit volume A body of mass M
and v olume m L has a density equal to M/ L which may be
Th e
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denotes
units
density
h
arith
e
/
%
metical operatio indicated by M/m L is the di v ision of the
number
by the number m If = m = 1 the expression
M / L reduces to M/L and denotes one unit of density Th e
arithmetical operation indicated by M/L is the division of the
number one by the number e or one unit Of mass divided by
one unit of volume gives one unit of density An y density is
equal to p ML where p denotes a number Th e ge eral
expression for any unit derived from the fund ame tal units can
w
be written L M T and of course is numerically e qual to one It
is not alwa ys possible to derive units from the fundame tal units ;
for example the unit of temperature is not so derived
S uppose we have an equatio expressing a relation between
physical quantities say p = g p must be a qua tity of the same
nature as q for it is impossib le for example to have a volume
equal to an area or a density Consider the equation m = vd
where m denotes the mass of a body d its density and 7) its
volume m means m times the unit of mass M 2 means times
the unit of volu me L d means d times the unit of density
—
ML
Th u s the equation may be written
w
ritte
n
*3
o
n
n
n
n
.
3
ni
3
n
n
.
,
3
.
3
on
.
“
3
n
.
n
,
y
,
z
,
.
n
,
.
n
n
.
,
,
,
.
,
,
.
)
.
1)
3
.
3
.
mM
(
vL
3
)
where m i) and at now denote m ere umbers S ince m = vd
this reduces to M = L ML and so to M = M which shows that
both sides of the equation represent a mass Th e equation might
be written
m grams
2
cubic
e
m
s
grams
per
c
c
d
)
)
(
(
if the fundamental units adopted were the gram and centimetre
n
,
3
-
.
3
.
)
.
.
.
.
CH 1
.
]
S
P AC E
I E
9
AN D T M
If both sides of an equat o cannot be shown to represent
quantities f the same kind in this way then the equatio must
be false
When a given unit varies as the th power of one f the
fundamental
units
it
is
said
to
be
of
dimensio
s
i i
as regards that unit Thus the expres ion ML
for the unit f density i dicates that the dimensions of the nit
of de s ity in terms of the fu damental units are one as regards
mass and — 3 as regards length
i
o
n
n
,
,
.
n
D m
o
n
,
en s o n s
—3
.
s
.
n
o
n
n
u
n
,
,
.
'
Ma tte r
a n
F
N
RE ERE CE
d Mo ti o n
,
J C
.
l
er k
-
Ma x we
ll
.
C HAP T ER II
M OTI O
N
W HEN the position of a body with respect to surr undi g
bodies remains uncha ged it is aid to be at rest Whe its
position continually changes it is said to be in motion If a body
moves from a position A to a other position B then the straight
line AB is called the displaceme t of the body A displacement
has magn itude and direction Q uantities like volumes and masse s
which have m agn itude but not direction are called s c la s whil e
quantities which have magnitude
also direction are called
n
o
S
n
n
.
.
n
,
n
.
.
.
a
r
,
vector can be represented by a straight line such
that its length represents the size of the vector and its direction
the direction of the vector
If a body is displaced from A to B and from B to a third
point 0 then AO the resultant displacement is called the vector
S imply the sum of the two displacements AB and B C
t AB ( Fi g 1 ) can be resolved into two more
component parts such as A O CD and DB ; for the resultant of
displacements f om A to C C to D and D to B is the same as
a displacement from A to B
ve c to r s
An y
.
.
,
,
,
.
,
or
.
,
r
,
.
C H 11
.
I N
]
11
MOT O
If a body moves along at a n i fo m r a te so that it c overs e qua l
distances in equa l times it is said to be moving with a unifor m
v elocity Its veloc ity i s meas ured by the di s tance it de scribes in
un it time In a ti m e t it will des cribe a dis tance 3 given by the
equ ation 3 tv where 1) denote s its velocity Th e u nit of velocity
is the velocity of a body w hi ch tra vels unit length in un i t t i m e
and is denoted by L T or L T Velocity i s a vector A uni form
v elocity c a n be repre sented by a str aight line contain ing as many
u
r
,
,
.
.
.
,
“
.
.
units of length a s the velocity co ntains u n i ts of v elocity and drawn
in the same direction as the velocity Velocity like position and
displacement is relative to surrounding bodies If a body were
alone in otherwise empty space it would not be possible to tell
whether it wa s at rest or i n motion
?
When the obj ects near a body are thems elves moving rela t ve
to ea c h other the velocity of the body rel tive to one obj ect
wi ll not be the same a s i ts velocity r el tive t o a n other Fo
e x ampl e consi der the velocity of a man on the deck of a ste a m
.
.
.
i
a
,
a
,
.
r
I
E
12
M C HAN C S AN D
P RO PE RTI E S
E
O F M ATT R
A
P
R
T
I
[
boat S uppose the man walks alo g the deck at 4 miles an hour
relative to the boat L t the boat be moving thr ugh the water
at 20 miles an hour d let the water be flo w
ing over the earth
at say 6 miles an hour If all three Velocities are in the same
di ection say from North t S outh i t is easy to see that the
V elocity of the man relative to the water is 20 4 24 miles an
hour and relative to the earth 20 4 + 6 3 0 miles
hour If
they are not in the same direction the velocity of the man
relative to the earth can be found in the same way as the resultant
of two or more displacements; for the velocities are equal to the
displacements in unit time To do this draw A B ( Fi g 2) to
represent in magnitude and direction the velocity of the water
relative to the earth Draw B C representing the velocity of th
boat relative to the water a d OD representing the velocity of the
man relative to the boat Then AD represents the velocity of the
’
man relative to the earth Also A C represents the velocity of the
boat relative to the earth and B D represents the velocity of the
man relative to the water Corresponding to the four bodies earth
water boat and man we have the four points A B C and D and
a line from any one point to any other represents the relative
velocity of the corresponding pair of bodies S uch a diagram as
this is called a velocity diagram
n
.
e
.
o
’
an
,
,
.
,
r
o
,
,
an
.
,
.
.
e
.
n
.
.
.
,
,
,
,
,
,
,
.
.
Fi g 3
.
.
All other
kinds of vector quantities besides displacements and
velocities can be represented by properly drawn straight lines and
,
CH
.
II ]
I
13
MOT O N
can be combined together in th e same way Thus if AB and CD
represent
two
vectors
of
the
same
kind
their
resultant
can
i
F
3
( g )
be got by drawi g from D a line D E equal and parallel to AB
Then CE represents in magnitude and direction the resultant or
vector sum f CD and DE or of CD and AB If AB ( Fi g 4)
represents any vector then A C and CB represent component parts
of the vector AB If the angle at C is a right angle then A C is
.
,
.
n
.
O
.
.
,
.
,
Fi g
.
4
.
called the resolved part f AB in the direction of A C L t the
angle BA C 0 then A C AB cos 0 T get the resolved part of
any vecto r along a direction making an angle 9 with it we multiply
by c o s 0
Th e v ector represented by AB is equal and opposite to the
v ec tor represented by BA so that AB
BA
When a body moves with a changing velocity it is said to have
acceleration
In
such
cases
the
space
described
by
A
l ti
the body in any i nterval of time divi ded by the
time gives the average value of the velocity during this time
Th e equation 3
tv is st i ll true if 1) now denotes the a verage
velocity during the time t If the time interval t is taken small
enough the changes i n the velocity during it will become negli g ible
and then the average velocity will be equal to the actual velocity
during the extremely S hort interval Th e velocity at any instant
is the space described during
extremely short interval of time
containing the instant divided by the length of the interval Th e
interval taken must be so short that o appreciable change in the
velocity takes pl e d u rin g it It may be one thousandth or one
o
e
.
.
o
.
:
,
.
.
c c e e ra
o n
.
.
.
,
.
an
.
n
ac
-
.
I
E
14
PROPE RTI E S
M C HAN CS AN D
E
OF M ATT R
P
I
A
R
T
[
ten millionth of a second or as much smaller a s we please to
imagine it
When the veloci ty of a body moving along a straight line
changes at a constant rate the body is said to have a uniform
acceleration Th e acceleration is then equal to the change of
velocity in unit time If the velocity at the be gi nning of an
i n terval of time t is equal to and at the end equal to then
the uniform acceleration is g ven by the equation
-
.
.
.
11 1
a
71 2
112 ,
,
i
’
U
unit of acceleration is unit change of velocity in unit
time it is therefo e denoted by L T
Th e average velocity of a body movi g with a unifor m accelera
tion along a straight line is (v 2 ) so that the space described
in the interval t is given by
Th e
‘
r
,
2
.
n
5
2
If the velocity diminishes the acceleration must be taken to be of
opposite S ign to the velocity In this case the velocity becomes
zero after a time t give by
0 = v + a t or t
,
.
’
n
'
'
1
Thus
if
200
v,
cms per sec and
20
We have
we get
a
.
.
a
secs
.
t,
2s
01
”2
“
T
Multiplyin
g the corresponding S ides Of these equations together
2a s
we get 11 —
This S hows that the change in the square
of the velocity is proportional to the S pace described
If the body starts from rest we have i) 0 so that
2
2
cl
2
.
.
,
and
An
SO
v2
a
s
§
2
v2
,
t,
2
a t,
2a s
.
cceleration like a displacement or a velocity is a vector and
can be represented by a properly drawn straight line ; and the
a
,
,
N I CS
E
1 6
M C HA
AN D
P ROPE RTI E S
T
P
A
R
I
[
E
O F MATT R
direction from the body to the centre of the circle Th e accelera
tion is perpe dicular to the velocity ; and therefore it does not alter
the magnitude of the velocity but it changes its direction
.
n
.
,
Fi g 6
.
a point
.
be moving with u iform velocity
round a circle BRA Q Drop a perpe dicular P N
m
B
A
on
the
diameter
O
of
the
circle
A
s P goes
fififgg
round suppose that N moves along BA so as to
keep always on a line through P perpe dicular to BA Then as
P moves alo g BRA N moves alo g B OA and as P goes along
Thus as P goes round and round
A Q B N goes back along A OB
the circle N moves backwards and forwards along the diameter
between A and B Th e motion of N is what is called a s i m p le
While P goes once around N is said to make
m o i c m o ti o
h
one complete vibratio or oscillation
Th e greatest distance N goes from O which is equal to the
radius r of the circle is called the amplitude of the S imple harmonic
motion Th e number of complete vibrations performed i uni t
time is called the frequency of the vibration L t the time f
one complete vibration be T and let the frequency be n so that
Le t
F
i
( g
P
.
7)
n
n
.
fi
a
m
c
.
n
n
.
n
,
,
.
,
,
.
ar
n
n
,
.
n
.
,
,
n
.
.
,
e
o
,
CH
II ]
.
I N
17
MOT O
acceleration of P
is equal to v / d is directed from P to 0 We may take P 0 to
represent th i s acceleration in magnitude as well as in di rection
Th i s acceleration represented by P O c a
b e regar ded a s the
of two components rep e sented by P N and N O Of
r es u l t
th es e P N is pe rpendicular to AB and S O does not affect the
motion of N Th e other represented by N O is equal to th e
No w P O represents an acceleration equal to
cce le ation of
n
=l T
/
.
We have
fi
r
Tu o r 2m m
27 r r
Th e
v
.
an
.
.
n
r
.
.
r
a
Fi g
.
7
.
fig
2
x
so
that
N
represents
one
equal
to
x N0
H
ence
0
/
;
i
we find that N has an acceleration directed towards O and e qual
Th e
to its distance from O multiplied by v / Le t
acce leration of N is then equal to “ N O or to
We have
ON
’
v r
”
,
fi
r
fi
.
u.
,
T
21 r
27
.
.
W»
Thus
it appears that when a point moves along a straight line
i n such a way that its acceleration is eq ual t p times its distan ce
from a fixed point in the li ne and is direc ted towards that poin t
wP
2
o
.
,
.
.
E
18
N I CS
M CH A
AND
P ROPE RTI E S
E
O F M ATT R
P
I
A
T
R
[
then it oscillates backwards and forwards and the time of one
complete oscillation is equal to
If y is a constant then
the number of vibrations per second is a constant independent of
the amplitude of the oscillations At 0 the middle point of the
swings the velocity of N is equal to that Of P and so is equal to
,
.
,
.
,
27 m m
.
F
N
J Cl
RE ERE CE
M a tter
an
d Mo ti o n
,
.
er k
-
Ma x we
ll
C HAP T ER III
THE
LAWS
M OTI O
OF
N
MATTER
OF
is found that the w
a
a
body
moves
is
modified
by
the
y
presence of other bodies There is said to be an action between
the bodies which changes thei r modes of moti o n S uch actions
take place between bodies which are in contact and a lS between
bodi es separated from each other Fo example the cars on a
railway are set in motion by the engi e and the motion f the
engine is modified by the presence of the cars Again bodies
when unsupported fall towards the earth ; there is an action
between the earth and the bodies near it A mag net a n d a piece
of iron move together when near ea h other Th e motion of a
falling body is changed when it meets the g round ; and so on In
all such cases there is a mutual action between two or more bodies
which mo difi es their motion
A action which causes bodies to come together is ca lled an
attraction W hile one which te ds to separa te them is called a
repulsion Thus we say that there is an attraction between the
earth and other bodies We may consider the e ffect of such
actions on one alone of the bodies concerned Th e change in the
mode of motion of the body considered is said to be due to a force
ac ting on the body which force is said to be produced by the other
body or bodies Force then may be defined as that which changes
the motion of a body Th e mutual action between two bodies
changes the motion of both so that eac h exerts a force on the
other Th e two forces are merely two di fferent as pects of the one
mutual action
If a horse pulls a boat along by means of a rope there is an
action between the horse and the boat which is tran mitted by the
rope Th e rope i s said to be in a state of tension Consider the
two parts of the rope on either side of any cross section P ( Fi g 8 )
IT
'
.
.
O
r
.
,
n
O
,
.
,
.
c
.
.
.
n
n
,
.
.
.
,
.
.
,
.
.
,
s
.
.
.
2— 2
N I CS
E
20
M C HA
of it
from
AND
A
R
T
P
I
[
E
O F M ATT R
and B A exerts a force on B in the direction
exerts a force on A in the direction from A to
B
These two forces are merely two aspects or views of the one
tension in the rope at P
.
Call these A
B to A a n d B
P ROPE RTI E S
.
,
.
.
If a body is pushed along by a rigid rod there is an action
transmitted along the rod which is called a pressure and is of the
opposite kind to a tension Consider as with the rope two parts
Call them
of the rod on either side of a cross section Q ( Fi g
,
.
,
,
»
.
and B A pushes B in the direction from
pushes A in th e direction from B to A Th e
are opposite to those in a tension
A
.
.
A
to B Also B
in a pressure
.
.
imple experiment illustrating that when there is an action
between two bodies they are acted on by forces in opposite
A
S
CH
.
III ]
L AW S
THE
I N
OF M OT O
21
O F M ATTER
dire tions is the following A bar AB ( Fi g 1 0) abo ut 3 feet long
has a tube fixed to it at right angles at C This tube is supported
by a fixed horizontal rod p assi n g thro u gh it so that the bar ca
turn about a hori zontal axis at 0 On e end of the bar carries a
pulle y P and a S liding weight N can be fixed in any position near
its o ther end A string is passed over the pulley from which a
mas s M is suspended Th e string is passed over the tube at
Th e sliding weight is
C and then do wn and is held at D
adj uste d so that the bar is balanced and wi ll remain at rest in
a horizontal pos ition If now the string at D is suddenly pulled
down this causes a te n s i o ri in it which draws P and M together
M moves up and P moves down sho wing that the force on M is
upwards and that on P downwards
If we bear in mi nd that a f r c e is only one aspect of an action
between two or more bodi es we may go on to c ons ider the
beha v iour of a single body when acted on by a force A fo rc e
changes the motion of a body consequently if the motion of
a body remains unchanged we must conclude that there is no
force acting on it This idea was express ed by N ewton in h i s
firs t law of motion which may be sta ted as follows
c
.
.
’
.
n
.
,
.
.
.
,
.
.
,
,
.
o
,
.
,
.
I Every body perseveres in its state of rest or of moving
uni formly in a straight li ne except in so fa as it is made to
change that state by external forces
A tra i n runn i n g on a straight level track goes on with nearly
uniform velocity after the steam is shut o ff and scarc ely slows up
appreciably un til the brak e s are applied What little slowin g up
there is without the brakes c a n be explai ned as bein g due to the
res i s tan e f the i and friction on the track Anything which
di minishes the resistance to the m otion of a body enables it to
co tinue longer motion Th e rotation of the earth goes on so
far as we kno w at a practically uniform rate because there are no
appreciable forc es tending to stop it
If the velocity of a body is changing that is if it h a s an
acceleration we say a force i acting on it and producing the
change Fo example it is found that any body fall i ng f e ly has
m
an ac cele ation of approximately 9 8 0
v r ti c a llv downwards
La w
.
r
.
,
.
a r
O
c
.
In
n
.
,
.
,
s
,
r
.
,
r e
,
e
s
r
se e
.
2
e
.
N I CS
E
22
M CHA
R
T
P
A
I
[
E
’‘
AND P ROP ERI I ES OF M ATT R
‘
We say then that there is a
acting on i t in this direction
This fo e is called its weight
It is one aspect of the mutual
action between the body and the earth In order to deal with
forces scientifically it is necessary to define what we mean by
the magnitude of a force and to adopt units in terms of which
forces can be measured A a preliminary to this we have to
co sider a number of experimental m ethods and results on which
the plan adopted for the measurement of forces is based
It has already been stated that the quantity of matter in a
body is called its mass but nothing has been said about how the
quantities of matter in two bodies can be compared A u it of
mass is a certain carefully preserved piece of platinum S uch a
unit is of no value unless the masses of other bodies
be
expressed i n terms of it To make this possible we have to adopt
some property of bodies as the measure of their mass es Th e
practical method wh i ch is nearly always used for the comparison
of quantities of matter consists in comparing their weights by
Weighing with a balance and a set o f weights
When a body is weighed on a balance the weights are adj usted
till they balance the body Th e weights are then attracted by
the earth with a force equal to the force with which the earth
attracts the body If the body is weighed in another place the
same weights are found to balance it but this does not how that
the force with which the earth attracts the body is the same as at
the first place ; it merely S hows that the forces on the body at the
two places are in the same ratio as the forces on the weights It
is found as a matter of fact that the force with which the earth
attracts bodies does vary considerably from place to place It is
less near the equator than near the poles and greater at sea level
than at the top of a mountain
A b alance consists essentially of a rigid bar called the beam to
which three knife edges are fixed These knife edges are per
e
d
i
c
l
to
the
length
of
the
bar
hen
the
bar
is
horizontal
W
p
the middle knife edge faces downwards and the end edges upwards
Th e middle edge S hould be exactly half way betwee the other
two Th e middle edge rests on a horizontal plane and two pans
are hung from planes which rest on the end edges Th balance
is symmetrical about a vertical plane through the middle knife
fo r c e
rc
'
.
.
.
S
.
n
.
,
n
.
.
c an
.
.
.
,
.
.
S
,
.
.
,
.
,
,
.
n
u ar
.
.
n
.
.
e
I
E
24
M CHAN CS AND
P ROPE RTI E S
A
R
T
P
I
[
E
OF M ATT R
or cold It is fo u d that the sum of the weights of two or
more bodies is equal to the weight required to balance them
when they are all put together in one pan of the bala ce
Also if a portion of a body is removed then the we i ght of the
body is diminished by an amount equal to the weight of the
part removed It has also be en found as the result of ma y
very careful experiments that the total we ght of the matter
in a system as determined by weighing with a balance and
weights remains invariable so long as no matter either enters or
leaves the system This is true whatever processes chemical
physical or biological take place i n the system That the total
weight of the matter in a system as determined by weighing
with a balance and a set of weights remains constant is one
of the most fi m ly established laws of nature Th e accuracy
with which weighings on a balance can be carried out is greater
than that of any other measurement It is possible to weigh a
body of say 500 grams weight to one part in a million without
very great difficulty
S ince it is found that the weight of any quantity of matter
as measured with a balance and a set of wei ghts remains
invariable it i atural to adopt the weight of a body as found
with a balance and a set Of weights as the measure of its mass or
of the q u antity of matter in it We may therefore provisionally
adopt the convention which is in accordance with universa l
practice that the quantities of matter in bodies are to be reckoned
proportional to their weights as determined with a balance and
a set of weights We shall see later that the adoption of this
convention can be further j ustified
Th e unit of weight employed when the weight of a body is
found with a balance and a set of weights is the weight of the
unit mass at the place where the weighing is done Th e numeri cal
value of the mass f a bo dy is therefore equal to the numerica l
value of its weight as found with a balance and a set of weights
Thus for example if we weigh a body with a balance and a set of
weights and find that it weighs 20 po unds then its mass i s
20 pounds or if it weighs 1 25 grams then its mass is 1 25 grams
A balance n d a set of weights therefore enable the mass of a
body to be found in terms of the unit of mass
n
.
n
.
,
n
.
i
.
,
,
.
,
r
.
.
.
s
,
n
.
.
.
.
O
,
.
,
:
,
,
a
.
CH
.
III ]
L AW S
TH E
I
E
25
O F MOT O N O F M ATT R
'
We may now go on to consider the motion of bodies under
the action of fo r ces If a body moves with a n ac es l ti o a
It can be shown expe iment ally
force s said to b e a c ti ri g on it
that the ac eleration imparted to a body by a f rce i p o
portional to the force T do this we may use a set of weights to
F
o
w
i
v
e
us
forces
having
known
ratios
example
the
force
ith
g
which the earth attracts a five gram weight is five times that
with which it attracts a e gram weight at the same place
A convenient ins tr u ment for m aking experiments on the
motion of bodi es due to forces is k own as Atwood s machine
This m a chine i S O ly s uitable for rather
roughly illustrating the l ws f motion
Exact methods o f verifying them will
be described i later chapters Atwood s
machine consists of a light wheel W
mo
nte
on
a
horizont
a
l
axle
Fi
1
1
d
( g
)
u
A with a thread TT carrying two masse s
M and
passed over it If the two
masses are equal their weights balance
each other and they will remain at rest
in any position If they are set in
motion up or dow they move with
u n iform velocity i accordance with the
first law of motion S uppose now that
a small additional mass say one gram
is put on the top of the mass M Th e
weight of this additional mass will cause
M to move down and M up at an equal
rate If the times they take to move
known distances starting from rest are
Fi g 1 1
m easured the d i stances are found to
be propo rtional to the squares of the times This S hows that
the mass es move with a uniform acceleration for we have
t
a
where
is
the
d
i
stance
described
in
a
time
t
by
8
s
s
a body starting from rest and mo v ing with a uni form acceleration
’
Th e acceleration is equal to 2s /t and so can b e calculated
from the observed distance and times Thus it appears that
the weight of one gram which is a co nstant force a cting on the
e ra
_
.
1
s
o
c
r
.
,
on
.
’
n
.
n
_
a
o
.
’
n
.
.
1
.
,
.
,
n
.
,
,
2.
2
1
.
.
,
.
,
z
,
a
r
o
.
n
,
r
.
_
n
_
.
,
s
.
.
N I CS
E
26
M CHA
PRO PE RTI ES
AND
A
T
P
R
I
[
E
OF MATT R
machine gives it a uniform acceleration If now we
take away one gram from M and add it to M the m o vm g forc e
will be the weight of three grams and the total m ss moved
will be unchanged It will then be found that the masses
move with three times their previous acceleration If we take
another gram from M a d put it on M we get five times the
acceleration due to the one gram weight Thus it appears
that the acceleration imparted to the machine is proportional
to the fo e drivi g it This is found to be true for all bodies
Fo any body then the ratio force
acceleration has a definite value
Instead of changing the force while the mass is kept co stant
we can chan ge the mass while the force is kept constant
Th e masses c
be found with a balance and a set of weights
S uppose we find the acceleration of the Atwood s machine with
masses of 9 9 and 1 01 g ams Th e mass moved then is 200 grams
and the driving force is the weight of two grams If we ext
find the acceleration with masses of 1 9 9 and 201 grams it will be
only one half that previously obtained With masses of 499 d
5 01 grams it will be only one fifth
Thus when the driving
force is kept constant the acceleration is inversely proportional to
the mass moved If we find the acceleration with masses of 1 9 8
and 202 grams or with 49 5 and 5 05 grams it will be the same as
with 9 9 and 1 01 grams This shows th at the force required to
give a body a certain acceleration is proportional to the mass of
the body These laws which can be roughly proved true with
Atwood s machine have been verified by man y exact experiments
some of which will be described in later chapters
Th e facts that the ratio f
has
a
definite
value
e
c e le a ti
/
for any body and that the force required to impart to a body a
given acceleration is proportional to its mass S how that
Atwo o dfs
;
2,
,
a
.
.
n
,
2
.
n
rc
.
.
r
.
n
.
an
.
’
r
.
.
n
an
-
.
-
.
.
,
.
.
’
‘
,
.
orc
a
c
r
on
CL
where f denotes the force acting on a body m the mass of the body
the acceleration imparted t the body by the force j ; and c is
a constant
By means of this equation we c a define the unit of force
Th e u n i ts of acceleration and Of mass have already been defined so
that the value of the unit of force is determined by the value of
,
,
a
o
.
n
.
,
CH
.
III ]
I N
THE L AWS
O F M OT O
'
E
27
O F MATT R
con s ta n t 0 Thus if m = 1 a d a 1 we get f c
of force u sed in scientific work is chose n so as to make 0
th e
n
.
m
a
.
or f
ma
unit
so that
Th e
1,
.
If then m = 1 and = 1 we get f 1 Th e unit force therefore is
the force which g ives unit mass unit acceleration This uni t of
force may be denoted by MLT where M L and T denote as
before the fundamental units of m ass length and time It is
c lled the dynam ical unit of force
If the uni t of mass is the g am the un it of length the
centimetre and the unit of time the second the c orrespond i ng unit
of force is called a dy e A dyne is a force which gives a mass f
one gram an acceleration of one cm per sec per se c
Th e property of matter meas u red by the ratio f/a is sometime s
called inertia We have seen that it is found to be proportional
to mass as measured by a balan ce and a set of weights It is that
property in virtue of which force is required to change the motion
of matter We know little more about force and inertia than the
experimental fact that f /a is constant for any piece of matter
Th e ideas Of force and inertia cannot b e explained ; they are
fu ndamental in mechani c s like the ideas of S pace and time and are
a cquired b
A
ex
erience
nyone
who
had
to
deal
with
large
h
a
s
y
p
masses in motion such a s motor cars railway trains or heavy fly
wheels knows the di ffie u lty of stopping or starting them quickly
and h as acquired the ideas of force and inertia by experience In
the scientific study of mechanics we ende vour to make our ideas
as precise as possible and to develop methods of accurately
measuring such quantities as space time force and mass but we
ne ed not attempt to explain their real atur e ; we are in fact
unable to do so
a body movin g along a straight line with a u iform
Fo
acceleration we have
i
a
.
.
'
Q
,
,
.
,
a
.
r
,
,
,
,
n
o
.
.
.
.
.
.
.
.
.
,
.
,
,
,
.
a
,
.
,
n
.
n
r
a
t
where
an
is its velocity at the beginning of an interval of time t
its velocity at the e d S ubstituting this in the equation
d
m a we get
v,
n
f
.
m v2
m v,
:
t
E
28
N I CS
M CH A
P ROPE RTI E S
AN D
E
O F M ATT R
P
I
R
T
A
[
product of the mass of a body and its velocity is called its
m om e t m
Thus th e force acting on the body is equal to the
increase in the momentum of the body i unit time or to the
rate of i crease of the momentum with time N ewton expressed
the relations between force and momentum in his second law of
m otion which may be stated as f llows
Th e change of momentum of a body is proportional
L w II
to the force acting on it and to the time during which the force
acts and is in the same direction as the force
= t which is
This agrees with the equation m (
f
equivale t to f m a
Th e mutual action between two bodies A and B can b e lo o k e d
upon as made up of the force which A exerts on B and the force
which B exerts on A These two forces are the different aspects
They are sometimes referred to as the
o f the mutual action
a ction and reaction
N ewton s third law of motion may be stated as follows
Th action and reaction between two bodies are
L w III
equal and in opposite directions
It is important to remember that the action and reaction act
on di fferent bodies Th e action is the fo c e exerted by A on B
and the reac tion the force exerted in the opposite direction by B
on A Consider the case of a horse pulling a b e t by means of a
rope Le t P and Q ( Fi g 1 2) be two c o s s sections of the rope
Th e
u
n
.
n
n
.
.
o
a
.
.
a,
n
.
‘
.
.
.
’
a
e
.
.
r
.
a
.
.
M
r
.
a
.
'
P
m
F
a s
Q
F
Fi g 1 2
.
Consider
.
the forces acting o the portion o f the rope between P
At P the rope on the left is pulling P Q towards the
d Q
horse with a force F say At Q the rope to the right of Q is
pulling P Q towards the boat with a fo c e F say Th e resultant
force on the portion of the rope between P and Q is therefore
Le t the mass of the rope
F — F in the direction of the horse
between P n d Q be m Then we have
an
n
.
.
,
’
r
'
'
’
.
a
.
‘
F
F
’
ma,
,
.
III ]
CH
.
h e re
left
w
.
TH E
LAWS
I N
E
O F MOT O
is the ac celeration with whi ch the rope is moving to the
N o w suppose that P an d Q are taken clos e together
a
’
F
P
F
Q
Fi g 1 3
.
Fi
1
( g 3)
.
29
OF MATT R
.
so that there is no mass between them
.
Then
0
m
so
that
'
F
’
When P and Q coincide F and F beco me th e two forces at P Q
which make up the te n sion in the rope at that cross section Th e
eq uation F F 0 j ust obtained shows that F and F are equal
and they are in opposite di rections F and F re the action and
r eaction at the cross section P Q of th e rope
When one body ac ts on another the action and reaction across
any surface s epa r at i ng them may be seen to be nec e ssarily equal
and opposite in the same way as at the section of the rope j ust
.
’
’
’
.
a
.
In the case of a mutual action between bodi es whi ch are not
in c ontac t such as the attraction between the earth and a body
nea r it or the attr action between a piece of iron and a magnet the
acti o n is transm itted through a m e dium called the ether which
fills all spac e Th e ether enables the magnet and the piece of
ir on to ac t on each other i n so me way not well under sto od so that
it ta kes the plac e of the rope in the ca s e of the horse d boat
Th e a c ti o n a n d reaction at any cross section in the ether are equal
and op po si te j u st as in the rope
When a horse pulls a cart along a level roa d the backwar d
for ce exerted by the cart on the hors e is equal and opposi te to the
forwar d force exer ted by the horse on the car t Th e forc es acting
on the cart are this forward force due to the horse and reta d ing
forces due to friction If the forwa d pull is greater than the
retardi ng forc es the cart moves forward with an acceleration Th e
force s a cting on the horse are the b c kwa d pull of the cart and
for ward forces due to the r eaction of the ro d when the horse
pres ses on it with its hoofs If the forwa rd forces due to the
,
,
.
,
an
.
i
.
.
r
r
.
.
,
a
r
a
.
E
30
N I CS
M C HA
PROPE RTI E S
AND
A
R
T I
P
[
E
O F M ATT R
exertions of the horse are greater than the backward pull of the
cart the horse moves forward with an acceleration
In any system of bodies the mutual actions between the b odies
These
i the system are made up of equal and opposite forces
internal forces therefore balance each other so that they have no
e ffect on the motion of the sys tem as a whole They merely
modify the arrangement of its parts Forces acting on bodies in
the syste m due to mutual actions between bodies outside the
system and bodies in it are called external forces and modify the
m otion of the system as a whole
In the case of the horse and cart on a level road the action
between them is made up of two internal forces and S does not
affect the motion of the horse and cart considered as a whole
Th e exter al forces are the frictional resistances retarding the cart
and the forward forces due to the reaction of the road on the
horse s hoofs Wh e the forward forces are greater than the
retarding friction the horse and cart move forward with an
acceleration If the forward forces due to the exertions of the
horse are equal to the backward forces due to friction the horse and
cart either remain at rest or if moving continue to move along
with a constan t velocity
Consider two bodies A and B and suppose that there is a
mutual action between them a repulsion say There will be a
force on B in the direction from A to B which will increase the
mome tum of B in that direction Th e rate of increase of the
momentum of B is equal to the force There will also be an
equal and Opposite force on A which will increase the momentum
of A in the direction from B to A and the rate of increase Of the
momentum of A in this direction will be equal to the rate of
increase of the momentum of B in the Opposite direction If we
regard momentum in the direction from A to B as positive and
that in the opposite direction as negative then A gains positive
momentum and B gains negati v e momentum at an equal rate
Thus th e total algebraical cha n ge of momentum is zero and the
momentum gained by A is equal to that lost by B A mutual
actio between two bodies may therefore be said to consist of a
tra sfere ce of momentum from one to the other Th e total
momentum remains constant A repulsion produces a flow of
.
,
n
.
.
.
.
O
‘
.
n
’
.
n
.
,
,
,
.
,
.
,
n
.
.
.
,
.
.
n
n
n
.
.
N I CS
E
32
M CHA
AN D
PRO PE RTI E S
P
I
R
A
T
[
E
O F M ATT R
where e is a constant less than one Th e value of e depends on the
nature of the bodi es Th constant e is called the coe fficient f
restitution
Th e most important kind of force is the force with which the
earth attracts bodies This force is called weight AS we have
seen the weight of a body is usually stated in terms of the weigh t
of unit mass at the same place It is the numerically equal to
the mass of the body Th e weight of unit mass at any place is
often used as a co n venient unit of force but as a unit of force i t
.
e
.
o
.
.
.
,
n
.
.
,
Fi g 1 4
.
.
has the ser ous drawback tha t it is not of exactly the same value
in di fferen t places In dealing with the weights of bodies we are
usually concerned with their weights as a measure of the quantities
of matter in them In s u ch cases the weight is expressed i terms
of the weight of unit mass When the weight of a body is
employed to set other bodies in motion or to bala ce forces we are
concerned with it as a force and then it i s best to express it in
terms of a dy namical unit of force such as th e dyne which always
has the same value It is important to bear in mind these two
i
.
n
.
.
n
,
,
,
.
,
C H II I
.
]
L AWS
T HE
OF
I
E
33
M OT O N O F MATT R
ways f using weight either as a measur e of mass or as a force
Weight of co u rse is a force in all cases
If a number of balls made of di fie n t subs tances say cork
wood iron and lead are put i n a box and the box is turned upside
down a n d then opened so that they all f ll out together it is found
that all reach the ground at the same time Very light bodies
like fe athers or pieces of paper fall more S lowly ; but this can be
If a feather n d a
S hown to be due to the resistance of the air
piece of lead are dropped together in a vacuum they fall at the
same rate
Th e way in which a falling body moves can be examined i
the following way
A glass plate A B ( Fi g 1 4 ) is hung vertically by a thread from
a hook H Th e plate has bee coated with lamp black by holding
o
.
.
f
re
,
,
,
,
a
,
.
a
.
,
.
n
.
n
.
Fi g 1 5
.
.
it in a smoky fla m e A tu n ing fork F is s pported so that the tip
of a light pointer S attached to one prong touches the plate near
the bottom Th e tuni ng fork is set vibrating and then the thread
is burned with a match so that the plate falls As it moves past
the vibrating fork the pointer ca ried by the fork t aces a wavy
li ne in the lamp black on the plate Th e appearance of s h a
lin e is S hown i Fi g 1 5 If the fork makes say 200 vibrations i
a second then the distance between the crests of the waves will be
the distance tra v elled by the plate i uccessive m ths of a second
If we make a mark on the plate at eve y ten waves and then
meas re the distances of these marks from the beginning of the
wavy lin e we S hall have the di stances travelled in 516
20 9 6
w P
3
u
.
.
,
.
r
r
uc
.
n
.
.
,
n
,
,
n
s
.
r
u
,
.
.
3
,
3
”
P ROPE RTI E S
E
34
M C HAN I CS AND
E
O F M ATT R
AR
T
P
I
[
seconds and so on Th e following table gives results which might
have been obtained in this way
2 /t
Di t
Ti m ( t)
( )
.
,
.
s an c e
e
00 5
s ec
1 23
.
c
m
2
3
8
98 4
.
49 0
980
1103
98 1
1 9 60
980
980
third column gives the values of 2 /t which are
practically co stant d equal to 9 8 0 Thus it is found that the
distances fallen by the plate are proportional to the squares of the
times from the start We have seen that when a body starts
from rest and moves with a uniform acceleration the space
a time t is given by s = g r Hence a = 2 /t
3 described i
Thus the experiment shows that the plate falls with a uniform
m
acceleration early equal to 9 8 0 63 i S ince all bodies fall
equal distances from rest in equal times when allowed to fall
freely it follows that all bodies fall with a uniform acceleration
em s
This acceleration is usually denoted by g
nearly equal to 9 8 0 e
and is called the acceleration of gravity More exact methods of
fi ding its val e and proving more accurately that it has the same
value for all substances will be described in a later chapter It is
found that the value of g is not exactly the same at different
places Fo example it is less at the top of a mountain than at
sea level ; but at any par ticular place it is equal for all substances
It was stated earlier i this chapter that the adoption of the
weight of a body as measured with a balance and a set of weights
as a measure of its mass would be further j stified Theoretically
it may be obj ected that this way of measuring the mass of a body
depends too much thi gs accide tally present like the earth
and the balance It ought to be possible to f rmulate a definition
of the mass of a body in terms of properties of the body and
independent of the presence of ther bodies This can be do e by
means of the property of inertia which is measured by the ratio of
force to acceleratio or f + a We define the ma s m of a body
as proportional to its inertia so that
Th e
2
s
n
an
,
.
.
a
n
a
s
.
2
.
“
n
.
8
0
.
,
.
s
e
?
’
.
n
u
.
.
r
,
.
n
,
,
u
on
.
n
n
-
,
o
.
o
n
.
,
n
.
,
s
CH
.
III ]
THE
L AWS
I
E
O F M OT O N O F M ATT R
35
where c is a constant We then fix the S ize of the u n it f force by
putting c 1 and so get f m as before
If w denotes the weight of a body expressed in dynamical
its f force then since it gives the body an accele ation 9 we
have w m g Bu t at any particular place 9 is th e same for all
bodies hence w is proportional to m If then the weight is
express ed in terms of the weight of unit mass it is numerically
equal to the mass It follows that the weight as found with a
balance and a s t of weights is numerically equal to the mass
when the mass is defined as proportiona l to the inertia All the
experimental results descri bed earlier in this chapter proving
weight as measured with a balance and a set of weights that is
when expressed in terms of the weight of unit mass to be
in variable apply therefore to inertia ; so that we have the sa m e
reasons for a dopting inertia as the measure of the quantity of
matter as we had for adopti g the weight measured with a
balance and a set f weights Inertia has the advantage that it is
a property of the body independent of the presence of other bodies
Thi s i s a purely theoretical advantage and in practice masses are
almost always dete r mined by weighing with a balance and a set of
weights S ince 9 is the same for all bodi es at the same plac e the
b la ce method gives th e same results as would be obta ined if the
masses were c ompared by finding their accelerations due to know
forces Th e fact that the total mass of any sys tem remains
invariable so long as no matter enters or leaves it is often referred
to as the pri nciple of the con s ervation of matter or of m ass
Th e equation w = m g gives the weight i ) of a mass m i n
If m is in grams and g in e m s and s e e
dynamical units of force
then w is in dyn e s Th e weight of a g ram is therefore g dynes or
nearly 9 8 0 dynes If m is expressed i pounds and g in fee t
and seconds we get the we i g h t i terms of the force which gives a
one pound mass an acceleration of one fe e t per sec per s e e This
feet
u n i t of f rce is called a p d l S ince 9 is about 3 2SEE the weight
o
.
a
un
o
.
r
,
z
“
.
.
,
,
.
e
'
.
,
,
,
,
n
o
.
.
.
a
,
n
n
.
,
,
.
t
.
s
.
,
.
n
.
n
,
.
o
oun
a
.
—
2
.
of a pound is equal to about 3 2 poundals S ince the value of g is
di fferent at di fferent places it follows that the numbe r of dynamical
units of force i the weight of a body is di fferent at di fferent
places Th e value of g at sea level and latitude A is nearly equal
.
n
.
3— 2
to
I
E
36
M CH AN C S AN D
9 78
Sin
1
(
2
P RO PE RTI ES
At
A)
9 78
g
em s
sec
A
T
P
R
I
[
E
OF M ATT R
the equator this gi v es
.
”
.
and at the north pole where A
,
98 3 1 9
9
In practical work a force equal to the weight of a unit of mass
is often used as a convenient unit of force Fo example a pound
weight is the unit of force used by English and American engineers
If a force f gives a body whose weight is w units of f rce an
acceleration a then since its weight would give the body an
acceleration 9 we have
r
.
,
.
o
,
w
( I,
9
This
equation is true whatever unit of force is used S O long as both
and
are
expressed
in
terms
of
the
same
unit
If
is
given
in
w
f
dynes we get f in dy es and if i t) is given i pou ds weight we get
1
0
in
pounds
weight
example
if
a
body
whose
weight
is
F
o
96
f
,
i t)
.
n
r
.
,
dy es moves with an acceleration
n
1 96 0
20
x
dynes
.
n
n
,
10
em s
see
.
2
the force acting on it is
.
If the weight of the body is taken to be
10
x 9 8 9 00 2041 grams
weight If a force of 1 0 pounds weight acts on a body whose
weight is 1 00 pounds the acceleration produced is
10
cms
10
x 98 0 9 8
x 32
3 2
1 00
1 00
sec
2
grams weight we get the force equal to
2
.
,
.
?
?
equation f i t)
is a very con v enient form of the
fundame tal equation and may ofte be used in worki g out
practical problems
and 9 must both be expressed i terms of
the same unit and f d also Fo example if a kilogram weight
Th e
n
n
.
n
n
a
an
.
r
,
CH
.
III ]
TH E
L AWS
I N
O F M OT O
mo v es with an acceleration of 1 6
gi v en by
f
16
1 000
500
32
E
O F M ATT R
37
then the force acting on it is
grams weight
.
If a body weighi g 1 0 pounds moves with an acceleration of
490
em
the force on t g ven by f 1 0 x
49 0
5 po unds weight
n
s
se e
.
I
?
IS
i
98 0
.
If the force is wanted in poundals then the weight of
is put equal to 3 20 poun dals so that
f
3 20
49 0
x
98 0
1 60
poundals
.
10
pounds
C HAP TER
F ORCE
AN D
IV
M OTI O
N
two forces act on a particle of matter each imparts to it the
acceleration given by the equation f m Th e
o o f
Co
F
resulting motion c n be got by adding toge ther the
two motions due to each force acting alone Le t P ( Fi g 1 6 ) be a
particle of mass m and let a fo c e f act on it in the direction P A Le t
IF
m p
s
° r °e s
,
i ti
n
a
o
°
.
a
.
.
r
.
Fi g 1 6
.
.
be the distances it would travel along P A in times
2 and 3 starting from rest at P if acted on by f alone
These distances are given by the equation 3
t
and
are
§
therefore proportional to 1 4 9 respectively Le t another force f
act on the particle in the direction P B and let P R P R and P R
d 3 due
be the distances it would describe in the intervals 2
to the action of this fo e If both forces act together in the time
the particle will move a distance P S in the direction P A
ve
and a distance P R in the direction P B It will therefore
at K the opposite corner of the parallelogram
In the
P S, P S2 P SS
‘
T,
,
,
7
7
.
2
a
'
,
.
,
,
7
rc
,
3
,
an
1
.
1
,
,
,
.
a rrI
I
E
40
PRO PE RTI E S
M C HAN CS AND
If
PQ
i
F
( g
.
1 7)
a
PS
,
PR
2
PR
2
PR
2
a
6
Sin
b
2
b
2
sin
P
S
(
2
b
2
2
9
sin
a
2
0
2
E
O F M ATT R
and the a gle Q P S
2a h cos 9
cos
S
R
P
S
(
2a h os 9
cos 0
6 cos 0 b
n
P
I
R
T
A
[
9
,
then
,
2
c
2
2
2
.
resultant of three forces acti g on a particle can be found
by finding the resultant of two of them and then the resultant of
this and the third Thus suppose P A P B and P O represent three
Th e
n
.
,
CH
.
IV ]
I
F ORC E
41
AN D M O T O N
Draw A D equal
forces acting on a p rticle at P ( Fi g
d
parallel to P C d DR equal and parallel to P B Then P R
repr e sents the resultant of the three forces In the same way the
resultant of any number of forces a cting on a particle can be found
If the resultan t of any number of forces acting on a particle is
zero then the fo ces are said to be in equilibrium Fo example if
with three forc e s P A P B and P C ( Fi g 1 9 ) we draw A D equal
and parallel to P C and DR equal and parallel to P B and R falls
on P then the three f rces j ust balance each other and have no
r es ultant
In this case th e S ides of the triangle P AD taken in
order represent the three forces in magnitude and di rection
a
,
an
.
an
.
.
.
r
,
r
.
,
.
,
o
,
.
.
F i g 20
.
.
a triangle representi ng three for ces in
e quilibrium a cting on a particle at P in directions P D P E and P F
which e parallel respectively to CA A B and BC Produce
A B to B
and B C to C
CA to A
We ha v e
A AB DPE B BC EPF and ( MA F PD
Th e sides A B BC and CA of the tria gle A B C are proportional
to the S ines of the angles BOA CA B and A B C These S ines are
equal to the S i nes of the angles C CA A A B and B B C C
sequently since the sides of the triangle AB C represent in
m agni tude and di rection the three forces acting at P it follows that
Le t AB C ( Fi g 20) b e
,
,
ar
.
,
’
’
'
,
’
:
,
,
.
n
,
,
.
,
’
’
.
,
,
,
on
E
42
A I
M C H N CS AN D P RO
PE RTI E S
OF
P
I
A
R
T
[
E
M ATT R
the forces in the directio s P D P E and P F are proportional to
the S ines of the angles EP F DP F and EP D ; that is each force
is proportional to the si e of the angle between the other two
This can be verified with the apparatus S hown in Fi g 21
TT
is a horizontal circular table with its circumference graduated in
degrees Rou d it three pulleys A B C can be clamped in any
positions Threads passing over the pulleys
w
e i h ts W W W
y
g
and are tied together at 0 By ad justing the weights and moving
the pulleys round the table 0 can be made to rest over the centre
of the table Th e angles between the threads can then be read Off
n
,
,
n
.
.
n
.
,
,
c a rr
.
.
, ,
,
,
,
,
.
,
.
F i g 21
.
on
.
circumference Th e S ines of the angles COB
A GO and A OB are found to be proportional to the weights W W
and W S ince any two sides of a triangle are together greater
than the third S ide it follows that any two of three forces in
equilibrium must b e together not le S S than the third
If two forces act on a particle in directions at right angles
to each other then the distance the particle moves in the direction
of either force is the same as if the other were n et acting Le t P Q
Fi
22
P
T
represent
a
force
ac
ing
at
ake
a
line
i
n
any
direction
t
( g
)
P A and drop a perpendicular Q N from Q on to P A
Th e force P Q
may be regarded as the resultant of the two forces represented by
th e g r a du a te d
f
.
,
,,
,
.
,
.
,
.
.
.
.
,
CE
I
F O RC E
W]
.
43
AN D M O T O N
and N Q Th e force P N is called the resolved part f P Q
along the di rection P A Th e perpendicular component N Q does
not a ffect the motion in the direction of P A If the a gle Q P A
is denoted by 0 then we have P N P Q cos 9 and N Q P Q sin 0
PN
o
.
.
n
.
.
,
F i g 22
.
.
an example consider the case of a body P rest i ng on a
smooth inclined plane AB C ( Fi g
Le t P Q drawn vertically downwards represent the weight of
the body D aw P K parallel to AB and drop a pe r pendicular Q N
on it from Q Th e compone t of the weight represent e d by N Q
does not tend to move the body along the plane and is balanced b y
As
.
r
.
.
n
F i g 23
.
.
eq ual and opposite reaction of the plane on the body Th e
If m denotes the mass of the
c omponent P N acts along the plane
body then its weight is m g dynami cal un i ts of force and the com
th
e
.
.
,
ponent represen ted by P N is equal to
mg
2g
.
Th e
angle
ABC
I
E
44
M C HAN CS AN D
P ROPE RTI E S
E
P
A
R
T
1
[
M ATT R
OF
is equal to the angle P Q N Denoting this angle by 9 we have for
the compone t along the plane m g s i n f) If there were no friction
the body would therefore move down the plane with an acceleration
given by the equation m g i 9 m or a g i 9 Actually
a
there will be frictio between the body d the plane This friction
exerts a retarding force proportional to the force exerted by the
body on the plan e which is represented by Q N and so is equal to
L t the friction then be a m g cos 9 where p is a constant
m g cos
Th e acceleration down the plane is then given by
.
,
'
n
.
S n
S n
a
an
n
.
.
e
.
g
(
s in
9
—
p
.
c os
.
6)
If the friction is g reater than m g S i n 0 the body will remai n at
rest o the plane Th e greatest possible value of 0 for which this
happen is given by
c
sin 9
= tan 0
a
,
n
.
an
.
is called the coe fficient of friction It depends on the state and
nature of the surfaces in contact It is diminished usually by
coating the surfaces with oil
We have seen in Chapter II that a point m vi g round a circle
of radius with a uniform velocity v has an c
F
b d
m
directed towards the centre of the circle
c el
ti
and equal to v / I f a particle of mass m moves round a circle a
fo e e q l to m o / directed towards the centre is therefore required
to keep it on the circular path If the body makes revolutions
per second we have v = 2
so that the force required is equal
This force may be measured approximately with the
to 4 m
apparatus S hown in Fi g 24
A frame F is mounted on a vertical axle and c
be made to
rotate by mea s of a belt and pulley as S hown Th e frame carries
a horizontal rod BB on which a mass M can S lide freely Two
cords attached to this mass pass round pulleys at P attached
to the rod and then go up along the axis of rotation to a spring
balance S Th e mass M is balanced by an equal mass M which
can be fixed at any distance fro m the centre of the rod
When the apparatus is rotating the mass M moves towards B
until the tension in the stri gs is sufficient to keep it from moving
.
a
l
.
.
.
o
o n
a
o
g in
a c
ird
o rc e
m
°v
’
z
'
rc
.
r
y
e
I
.
2
ua
on
era
r
a
,
r
n
.
'
2
-
71
n
7rr n
2
r
.
.
.
an
n
.
'
.
,
’
.
,
.
,
n
FO RC E
]
CH I v
.
I
45
AN D M OT O N
f ther Th e te sion in the strings is indicated by the balance
With thi s apparatus the force
the mass M the radius of its
circular path d the number of revolutions pe second c a be easily
f und d the force can be compared with the the retical value
ur
n
.
.
on
,
r
an
o
an
n
o
,
F i g 24
.
.
te sio i the st ings it should be obser v ed pulls
the mass M i n toward the centre and also pulls the ce tre outwards
with equal f rce These two equal and opposite forces are an
example of acti
d reaction
47 r m n
fl
2
r
.
Th e
n
n
n
r
n
s
an
o
an
.
on
,
,
an
.
E
I
M CHAN C S AND
46
P RO PE RTI ES
OF
P
I
A
R
T
[
E
M ATT R
When a wheel is rotat i ng the rim is kept on its circular path by
a tension in the S pokes of the wheel If the spokes are not strong
enough the wheel m y break in pieces and then its parts fly out
alo g tangents to their ci rcular pat h s
f a small ball P ( Fi g 25 ) is hung up from a fixed point 0 by
a fine thread it can be made to move round a
circular path by starting it properly Le t OA be
a vertical li e through 0 and P C a perpendicular
from the ball on to OA S uppose the ball m oves
round OA in a circle of radius CP Th e forces
acting o the ball are its weight and the tension
in the string Th e resultant of these two forces
must be a force along P C equal to m o / where m
is the mass of the ball 1) its velocity d
CP
Th weight is parallel to OC and the tension acts
along P O d their resultant must act along P C
Th e three ides P O OC and CP of the triangle
P OC are therefore proportional respectively to the
tension the weight and a force equal and opposite
to the resultant force on the ball for these three
fo ces would keep the ball i n equilibrium at P
Th e weight of the ball in dynam cal units of force
is m g We have therefore
,
.
a
,
n
.
.
,
.
n
.
.
n
.
2
r,
an
,
r
.
e
‘
,
an
.
S
,
,
,
r
.
i
.
PC
00
where
l= PO
.
me
g
/
r
r
x/ l
mg
2
r
2
Hence
r
i
g
/l
x
If the ball makes
so that
n
2
r
2
revolutions per second
g
477
3
71
2
we
have
1)
27r r n
z
/
Nl
It can easily be verified by experiment that such a suspended ball
moving round circles makes the number of revolutions per second
given by this formula This shows that the expression m e / or
4 m
for the force required to keep the ball on its circular path
is correct
2
.
7r
2
rn
2
.
r
M E C H AN I CS
48
AN D
PROPE RTI E S
E
O F M ATT R
T I
R
A
P
[
position of equilibrium If the mass is pulled down and th e
let go it therefore oscillates up and down in a simple harmon c
n
.
i
,
T
which is independent of its amplitude
motion of p
Th e spring oscillates up and do wn with the mass and it can be
S hown tha t one third of the mass of the spring S hould be added
t the mass of the body when calculati g the period by mea s
of th formula j ust found
.
,
-
n
n
o
e
.
N o w consider
what is called a simple pendulum This consists
of a small heavy S phere hung from a fixed point
m
by a fine thread If the sphere is pulled to one
ide and let go i t scillates backwards and forwards
Le t 0 ( Fi g 27 ) be the fixed point and P the small sphere
Le t CP = l
Draw a vertical line 0 0 through O
f mass m
and a circle P CQ with centre 0 Then P oscillates along P CQ
and C is its equilibrium position Th e tension in the thread
has no component along the a of the circle at P so that it does
S i m p le
P e n du
.
m
.
o
S
.
.
o
.
.
.
.
re
,
]
.
I
F ORC E
CH I v
AN D M OT O N
49
not a ffect the moti of P along the arc Draw a tangent at P
meeti g 0 0 produced at A Th weight o f P acts parallel to
0 0 so that its compo ent along P A is equal to
on
.
n
e
.
n
mg
cos P AO or
Fi g 27
.
Le t
the length of the
are
OP
P 60
so
mg
P OO
.
.
be denoted by w
.
Then
a:
l
)
that the acceleratio of P along the
n
g
Sin
a re
is equal to
si n
If is very small we may put
a:
on
so that the acceleration of P along the arc is equal to
w
g
7
,
and
so is proportional to the displacement of P from C and in the
oppos ite direction It appe ars therefore that P will move with
.
4
I
E
50
P RO PE RTI ES
M C HAN C S AN D
P
I
A
T
R
[
E
O F M ATT R
a simple h a m o n i c motion alo g the a r e provided its amplitude
of V ibration is very small Th e time of a complete oscillation
is given by
n
r
.
T = 27r
mg
l
It is easy to verify this expression for the time of oscillation of
a S imple pendulum experimentally and to S how that it is i
dependent o f the amplitude provided this is small If l and T
are measured the value of g can be calculated
Another example of S imple harmonic motion is the motion
of the prongs of a vibrating tuning fork Th e restori ng force is
due to the elasticity of the prongs and is proportional to their
displacements It is found that the number of vibrations per
second made by a tuning fork is indepe dent of the amplitude
of vibration as it S houl d be for a S imple harmonic motion
Th e fact that in cases like those j ust considered where a body
is acted on by a restori g force proportional to its displacement
from a fixed point the period is found to be independent of the
amplitude as it S hould be theoretically is the best proof we have
that the acceleration with which a body moves is proportional to
the force acting on it for the periods of such vibrations can be
measured with great accuracy
n
,
.
,
.
.
.
n
.
,
,
n
,
,
,
,
.
REFEREN CES
Mec h a n i c s, Co x
Exp er i m en ta l Me c h a n
.
i c s,
S ir R
.
B ll
a
.
C HAP T ER
WORK
W HE N a
V
N
AN D E ERG
‘
Y
acts o n a body and the body moves in the
di
r ection of the f rce the force is said to do work
W
Th e work done is equal to the product of the
If 11) denotes the
force i to the distance through which it acts
=
the
force
and
the
distance
then
w
s
s
work done f
f
If f = 1 and s = 1 then w = 1 so that the u n i t of work is
t h e work done by a unit force when it acts through unit distance
If the unit force is a dyne and the unit distance a centimetre
then the unit of work is called an erg Th e unit of work may be
denoted by
ML T x L or MU T
where M L and T respectively denote the fundamental units of
mas s length d time
Le t A B ( Fi g 28 ) represent a force f acting at B and suppose
fOrc e
o
o rk
,
.
.
n
.
.
,
,
,
.
,
.
‘
2
-
2
,
,
an
,
.
,
.
F i g 28
.
,
.
is di splaced to B Draw B N perpendicular to AB produced
Th e w rk done by the force f is f x BN not f x B B
If BB
were perpendicular to AB then no work would be done by f
Th e most familiar example o f work is that done in ra i sing
weights to a higher level Th e weight of a body acts vertically
down wards so the work done in raising a body is equal to its
weight multiplied by the vertical height through which it is
’
B
’
.
.
’
o
’
.
.
,
4— 2
N I CS
E
52
M CHA
AND
P RO PE RTI E S
T I
R
A
P
[
E
O F M ATT R
raised Fo example if a body of mass m is pushed up an
inclined plane of length l and height h the work done agai st
weight is m g h and S is independent of l If the inclinati of
the pla e to the horizontal is 9 then the component of the weight
down the plane is m g S i 9 so that the work done in pushing the
body up is m g S i 9 x l B t l s i 9 h so that
m g sin 9 x Z m g h
In calculating work we may therefore either multiply the force by
the component of the displacement in the direction of the force
or multiply the displacement by the component of the force
the direction of the displacement
Forces are sometimes expressed in terms of the weight of unit
mass instead of i terms of the force required to give unit mass
u it acceleration Thus we may say a force is equal to the
weight of ten pounds This w y of measuring forces is often
c nvenient It is not suitable for very exact work because as
we have seen the weight of unit mass varies slightly from place
to place If the force is measured i terms of the weight of
a pound as unit and the distance through which it acts is
expressed in feet then the work is said to be i fo o t p ds
amount of work equal to the work do e i n
A foot pound is
raising one pound a foot high Th e foot pound is a co venien t
unit of work often used by engineers
AS another example of work consider the work done in
tretching a spiral S pring Le t l be the length of the spring
when not stretched and let l be the length when pulled out by
a force f Th e force required to stretch a spring is proportional
to the i crease of length of the spri n g so that the force required
increases uniformly from zero when the length is I to f when
it is l Th e average value of the force over the distance l l
is therefore f so that we have
w % f ( l Z)
where w denotes the work done This result may be obtained
in another way as follows In Fi g 29 let the horizontal distance
1
ON measured along OA from 0 represent the extension l
f the spring and the vertical distance N P above the line OA
r
.
,
n
,
O
on
.
n
,
n
n
,
n
u
.
:
.
In
.
n
n
.
a
.
o
.
,
,
n
.
n
,
-
an
-
ou n
.
n
n
-
.
.
s
o
.
'
.
n
,
,
’
’
o
.
,
’
O ’
.
.
.
,
o
,
,
,
0
CH v
.
W ORK
]
ENE RG Y
AN D
53
the corresponding force required to stretch the spring from the
length l to l
If a series of such points P P P P etc are marked they
will all fall on a straight line through 0 because the force required
0
,
.
,
,
,
,
,
.
,
,
N,
F i g 29
.
N4
.
is proportional to the extension Take two such poin ts P and
P ve y near togethe and consider the work done in stretching
the spring from N to N Th e force at N is N P and at N
it is N P Th work is therefore between N N x N P and
Th e area
also lies between N N x N P
N N
N P
Th e di fference between these limits be comes
d N N x N P
negligible if N N is made very small S O that we see that the
work is represented by the area
Th e area between
OB and OA can be divided into a great m any very narrow vertical
strips like N P P N the area of each one of which can be shown
in the same way as for N P P N to represent work done in
s tretching the spri ng Thus we see that the area N P P N
represe ts the work done in stretching the S pring from N to N
If OA represents the extension l l due to a force 7 represented
by AB then the work done during this extension is represented
by the area OBA which is equal to AAB OA or to ( l l ) f
A diagram like the one j ust considered which sh ws the
relation between force and displacement and on which area
r epresents work is called an i di c to
di g a m and is o ften made
se of in engineering work
S uppose on such a diagram the
relation between force and di splacement is represented by a closed
cur v e like ABCD ( Fi g
When the displacement incre ases
,
.
r,
r
,
,
,
,
,
an
,
,
x
,
,
,
.
,
e
.
,
,
,
,
,
,
,
,
2
,
.
2
,
,
,
,
.
,
2
,
,
,
,
,
,
2
,
.
n
,
,
’
4.
”
o
,
’
,
,
n
,
u
.
.
a
r
a
r
,
o
,
,
E
54
M C HA
N I CS
PROPERTI E S
AN D
A
R
T
P
I
[
E
O F M ATT R
’
’
’
from OA to OC the work done is equal to the area A A B CC
’
but w h en the displacement diminishes from OC back to OA the
’
,
’
Fi g 3 0
.
.
’
work done is negative and equal to the area A AD CC Th e area
AB CD therefore represents the total work done duri g a dis
"
A
lacement
from
to
and
then
back
to
A
O
p
external force does work on a system of bodies the
When
arrangement
of
the
bodies
in
the
system
is
E
changed I t is found that to bring the system
back exactly to its original state the system must be allowed
to do an amount of work equal to the work done on it by the
external force Thus it appears that doing a definite amount of
work on a system increases the system s power of doing work by
an equal amount Th e power of doing work posse s se d by a
system is called its en e g y Thus when work is done on any
system by external forces the energy of the system is increased
by an amount equal to the work done Th e external system
whose action provides the external forces loses an amount of
energy equal to that gained by the other system Th e total
amount of energy therefore remains uncha ged In any system
of bodi es on which there are no external forces acti g the total
energy remains constant This is a firmly established law of
nature and is known as the principle of the conservation of
energy
Fo example if a heavy body is moved up to a higher level
work is done on the system consisting of the body and the earth
If the body is allowed to move back to its original position it
does an amount of work equal to that done in raising it Th e
’
.
n
’
’
an
n erg
y
.
.
,
.
’
.
r
.
.
.
n
.
n
.
.
r
,
,
.
,
.
N I CS
E
56
M C HA
PRO PE RTI E S
AN D
P
I
A
R
T
[
E
O F M ATT R
work which the body can do before being brought to rest F
s uppose that a body of mass m m oving along a straight line
with velocity v is acted on by a force i the opposite di ection
to that in which it is moving After a time t its velocity will
be reduced to zero We have 3 Ac t so that the work done in
the time t is given by
or
.
n
r
.
.
w—
f
S —
Wb
7)
X
'
i
v
t
é
1
“
2
far as we know all the di fferent forms of energy such as
heat and electrical energy can be regarded as partly potential
and partly kinetic Th e energy of a system of material particles
motion is partly potential and partly kinetic Th e potential
i
e ergy at any instant is the same as it would be if the particles
were all at rest in their positions at the instant considered and
the kinetic energy is equal to
So
.
n
.
n
A
3 mm
1
where
7
J
m
v
,
,
i
my );
?
2
etc denote the masses of the particles and
If P denotes the potential energy
v v v etc their velocities
of a system and K its kinetic energy then S O lo g as no exter al
actions take place we have P K = constant This equation is
the mathematical expression of the principle of the conservation
of energy
When work is bei g done the rate of doing it or the work
done
per
unit
time
is
called
the
If
an
o
w
e
p
P w
amount of work w is done in a time t at a
, ,
,
m,, m, , m,
,
,
,
.
.
.
,
n
,
n
.
,
.
n
r
o
.
er.
uniform rate then
w
p
-
t
where
p
denotes the power
.
unit of power is one unit of w rk done in u it time Th e
unit of power m y be denoted by M U T where M L and T as
us al denote the units of mass length and time
Th e unit of power most frequently used by engineers is called
one horse power and is equal to
foot pounds of work
per minute or 5 50 foot pounds per seco d A other unit of
power used in electrical eng ineering is called one Watt and is
equal to ten million ergs per second O e horse power is equal
to early 7 46 Watts A kilowatt is one thousand Watts and
is equal to 1 3 4 horse power
Th e
o
n
—3
a
,
u
,
.
,
-
-
n
-
.
n
.
.
n
.
n
-
,
-
.
CH v
.
W ORK
]
A ND
E NE RG Y
57
If the point at which a force f acts i s moving with a v elocity v
in the direction of the force the work done by the force in unit
t i me i f v which is therefore equal to the power developed by the
action of the force Cons ider the case of water flow i ng along
a pipe S uppose the water is under a pressure P that is let P
it
denote the force with which the water presses o ea c h
ea
with which it is in contact Consider a cross section AB ( Fi g 3 1 )
of the pipe Th e water to the left of AB is exerting a force
on the water to the right of AB which is equal to P if a denotes
the area of the cross section AB S uppose that at AB the water
is movi ng from left to right with a velocity
Then the work
done by the water to the left of AB on that to the right in
unit time is equal to P w because 1) is the distance the water
moves in unit time Th e power represe ted by the stream of
,
s
,
.
.
,
,
n
ar
un
.
.
.
a,
'
.
a
,
n
.
Fi 0O
.
31
.
water that is the work done per second at the cross se c tion AB
is therefore equal to P v Bu t v is the volume V of water
flowing past AB in uni t time Th e power of a stream of water
is therefore equal to P V or its pressure multiplied by the volume
flowing in unit time F example the power of 1 000 cubic feet
of water pe minute at a pressure of 3 3 0 pounds weight on the
sq uare foot i s equal to
foot pounds per minute or t
ten horse power In this calculation of the power we have
neglected the kinetic energy f the stream of water
There are a number of mech a nical appliances usually referred
to
as
machines
in
which
a
force
acti
g
at
one
M
hi
point causes the machine to exe t a greater or less
force at another point If the machi e moves then the f
acting on the machine and the f e exerted by the machine both
do work Le t f denote the force acti g on the machi e the
di stance through which it acts f the f rce exerted by the machine
and s the dis tance through which it acts The the work done
,
,
,
a
a
.
.
,
or
.
,
r
o
-
-
.
o
.
n
a c
n es
.
r
n
.
o rc e
,
o rc
n
.
n
’
,
o
’
.
n
,
3
I
E
58
P RO PE RTI E S
M CHAN CS AN D
P
I
T
AR
[
ER
O F M ATT
on the machine is f s and the work don e by the machine is
If
no energy is lost or stored up the machine it follows from the
principle of the conservatio of energy that
’’
s
.
in
,
n
f
s
e fficiency of a machine is the ratio of f to fs If f
s
the e fficiency is equal to unity and the machine is said to be
perfectly efficient In practice some of the work done on the
machine is wasted in it i overco m ing f i ctional forces so that s
is always less than f Th e e fficiency is therefore always less than
unity Th e equation f = f can be used to calculate the
theoretical value of f which would be obtained if the machine
were perfectly efficie t Th e actual value must be less than tha t
so calculated
If we divide the equation f =f s by t the time take to move
through the distances s and s we get
’’
s
Th e
s
.
’’
.
n
s
’’
r
,
.
’ ’
s
.
s
’
,
n
.
.
’ ’
s
n
,
’
,
3
,
s
’
fi f i
v and
H
ere
and
v are the
so
that
v
f
E
?
velocities of the points of application of the forces f and f so we
see that for a perfectly efficient machine the power acti g on the
machine is equal to the power the machine exerts Th machine
therefore serves to transmit power from one point to a other
Th e fficiency of the machine is equal to the power it gives out
divided by the power it receives for
“
s
8
o
’
’
’
i)
,
’
,
n
,
.
e
n
.
e
,
,
f8
'
f
l
!
v
implest type of machine is the lever This consists of a
rigid bar which is held fixed at o e point about which it can turn
S uppose Q P ( Fi g 3 2) is a lever which can turn a b out F
Le t
Th e
S
.
n
.
.
.
Fi g 3 2
.
.
W ORK
]
CH v
.
AN D
E N ERG Y
59
a force f act at P in a dire ction perpendicular to the bar and let
the bar exert a force f at Q al s o perpendicular to the bar N w
let the bar turn through a very small angle 9 P will move
through a distance FP x 9 so that the work done by f will be
F P x 9f
In the same way the work done at Q will be FQ x 9f
Henc e we have
’
o
.
.
,
’
FP
f
S
an
d
ides
x
9f = F Q
x
9f
’
,
or
f
may
both
be
on
the
same
side
of
f
’
’
=
F
FP
fF
Q
.
instead of on opposite
.
Another
imple m a chine is the pulley which consists of a wheel
over which a flexible cord or belt is passed Le t P ( Fi g 3 3 ) be a
S
,
.
.
Fi g 3 3
.
Fi g
.
.
34
.
p lley and suppose a weight W is hung f o m one end of th cord
and that this weight is raised by pulling the other end A If A
’
is m ov e d from A to A the weight will be raised through a
height equal to AA Th e force required at A to balance the
u
e
r
,
.
’
.
I
E
60
M C HAN CS AN D
P ROPE RTI E S
P
A
R
T
I
[
E
OF M ATT R
weight is therefore equal to the weight pro vided there is no
friction in the pulley Th e tension in the cord is therefore the
same on both S ides of the pulley
V arious combinations of pulleys are employed in practice
If the
Co sider for example the combination shown in Fi g 3 4
string at A is pulled dow with a force T there will be a tension
equal to T throughou t the whole length of the stri g so that S ince
the weight W is suppo ted b y five strings the upward fo c e on W
,
.
.
.
n
.
.
n
n
,
r
Fi g 3 5
.
r
Fi g 3 6
.
.
.
will be equal to 5 T Also it is easy to see that if five feet o f s t i n g
is pulled in at A th e weight W will only be raised one fe t so that
the work done at A is equal to that done on the weight Another
arrangement of pulleys is hown in Fi g 3 5 Th e tension in each
tring is marked in the figure ; S O we see that a tension T at A
8 T What is called a differential pulley is
s upports a weight W
shown in Fi g 3 6 Tw pulleys B and A one slightly larger tha
the other are fixed together on the same axle A chain passes
over B round the lower pulley O and then over A Th e pulleys
r
.
e
,
,
.
S
.
.
.
.
,
.
o
n
,
.
.
W O RK
]
CH v
.
AN D
ENE RG Y
61
and B are made with c Ogs which fit i to the links of the chain
them If the chain at P is pulled down
so that it cannot slip
th r ough a distance d the the chain at S is pulled up an equal
distance but the chain at Q is let down by the r tation of A and
i
where denotes the radius of A d
B through a dist nce d
A
n
on
.
n
o
,
g
‘
a
an
rA
,
B
that of B
rB
.
Co n
sequently
C
is raised by an amount 4
,
(
d
d
2)
14
3
force f at P therefore gives a force at W equal to 2f
F
example if r 1 0 inches and
9 5 inches then the force at
W is 40 times that at P but P has to be pulled down 40 times as
far as W moves up
A machine often used in practice is the screw
It consists of
a circular cylinder on the surface of which a spiral groove is cut
Th e
rB
o
or
rA
rA
,
,
.
.
.
Fi g 3 7
.
.
ridge between the adj acent parts of the groove is called the
thread of the screw Th e distance measured parallel to the axis of
th e cylinder from one thread to the next is called the pitch of the
Th e pitch should be constant throughout the length of the
s crew
screw S crews are made by rotat i ng the cyl i nder while a cutting
too l moves along parallel to the axis of the cylinder Th e tool is
m de to move through a distance proportional to the angle
through which the cylinder tu r ns Th screw is used in com
bination with a nut which is a plate with a hole in it which is
g ooved so as to fit the screw
Th e
.
.
.
.
a
.
,
r
.
e
N I CS
E
62
M C HA
AN D
PROP E RTI E S
A
P
R
T
I
[
E
O F M ATT R
shows a screw clamp which is used for p e s s i n g b o di e s
fi rmly together A screw AB turns in a nut at C If the nut is
held fixed and the sc rew turned round once the sc rew advances
through the nut a distan ce equal to its pitch If the screw is
turned by applying a force f at D perpendicu lar to the plane
containing D and the axis of the scre w then if the screw makes o e
revolution the f e f acts through a distance 2 where is
the distance of D from the axis Th e end of the scre wat A then
advances a distance p equal to the pitch Th e force f exerted by
the Screw at A is therefore given by the equation
Fi g 3 7
r
.
.
.
,
.
,
n
,
o rc
7rr
r
,
.
’
.
M
example if
assuming no friction
Fo r
r
,
=2
feet and
p
"
ir
inch
we
get
theo
r
etically
4
,
,
f
’
=
f
x
2w2 x 48 = 604f
"
.
Thus
very large forces can be obtained with screws S cr ews
and nuts are very much used for holding the parts of machinery
firmly together There is a great deal of friction between the
screw and its nut so much that usual ly a force applied to the end
of a screw in the direction of its axis will not make it turn round
in its nut Because of this friction when the nut on a screw has
been tightened it does not easily work loose
.
.
.
.
,
.
,
.
I
E
64
PRO PE RTI E S
M C HAN C S AN D
S uppose
AR
T
P
I
[
E
O F MATT R
now that the body turns rou d through a very small
and ON = p
angle 9 so that Q moves to Q ( Fi g
L e t OQ
Then Q Q
9
Th e work done by the force f is equal to f multiplied by the
resolved part of Q Q in the direc tion parallel to P Q Draw Q M
parallel to Q P and drop Q M perpe dicular to Q M Th work
]Q
do e by f is then equal to f x JV
S i ce 9 is very small we have
N Q O M QQ so that the triangles JlI Q Q and N Q O are similar
Hence
n
’
r
.
.
’
9
.
’
’
.
’
n
e
.
’
n
n
.
’
’
.
’
MQ = Q Q
’
work done by f is therefore equal to f Q Q ; but Q Q
9 so
s
the work is equal to fp 9 p 9 is equal to the distance N would
’
’
Th e
r
,
.
Fi g 3 9
.
.
move parallel to P Q i f ON turned through the very small angle
Thus the work done by the force is proportional to
9
the
p
perpendicular distance from the axis of rotation on to the line of
action of the force If this line of action passes through the axis
0
and the force can do no work on the body and so cannot
p
affect its motion Th e product fp is called the m oment of the
force f about the axis through O
S uppose that a other force acts on the body at R As before we
need only consider the component of this force in a plane pe p e n
di la to the axis for the component parallel to the axis can have
.
.
,
.
.
n
.
r
cu
r
,
CU
.
I
]
VI
I D BOD I ES
65
M EC HAN CS O F R G I
the moti n of the b ody L t S R ( Fi g 40) represent
the c mponent parallel t the plane of the pa per and let it be
equal t f L t p be the length of the pe pendicular between
If th body tu ns through a small angle 9 the
d the axis
SR
work done by f is f p 9 If the total work fp 9 + f p 9 is equal to
zero the e ffect of the force f balances the e ffect of the force f
if no work is done the body its ki etic energy remains
F
unc h a ged so that it mo v es with a constant velocity which shows
that the re ultant action it is
the
zero If therefor e fp
f p:
tw forces f and f balance each
othe so that the b dy if at res t
would be i equi librium We
therefore that the mome t
of a f rce about
axis is the
p r oper measure of the actio of
the f rce i tendi g to turn a
body round the axis In c a l
lating the moment of
force
about an axis we first fi d its
com po ent in a plane p p
di l
to the axis and then
multiply this component by the
length of the perpendicular in
F i g 40
t h is plane from the a is on to
the c mponent in question If any number of forces act on the
body then th c ondi ti f equilibrium is that the sum of all their
moments about the axis hall be zero Th e moments of forces
which turn the body one way are counted positive and the
mome ts of the thers which t r it the Opposite way are counted
egat ve
S uppo e that tw parallel f rces f and f act on the body both
in
a
plane
perpendicular
to
the
axis
about
which
P
ll l F
it is free t rotate
L t P Q ( Fi g 4 1 ) n d RS re p esent the f rces f and f and let
and
be
the
lengths
of
the
perpendiculars
from
on
to
O
P Q and
p
p
RS
If fp + f p = 0 the forces do not tend to turn the body
round the axis at 0 These forces however will tend to move the
w P
5
e fie c
n o
t
o
on
e
.
.
o
o
’
r
e
o
an
r
e
.
’
”
,
’ ’
’
.
’
,
n
on
or
n
,
on
s
’
.
’
o
r
o
,
n
.
n
see
an
o
n
n
n
o
c u
.
an y
n
er
n
en
cu ar
.
.
x
o
.
on
e
,
o
S
o
n
i
n
u
n
.
o
s
a ra
.
o rc e s
e
’
o
,
.
o
e
a
.
’
'
.
.
.
.
.
r
’
o
,
E
66
N I CS
M CH A
AN D
P RO PE RTI E S
E
O F M ATT R
P
I
R
T
A
[
body parallel to their wn directions This motion is stopped by
the axis which we are supposing is fixed Th e force on the axis
o
.
.
F i g 41
.
,
.
which it resists with an equal and opposite reaction is evidently
’
equal to f f This force f + f acting through 0 is the resultant
’
of the two parallel forces f and f ; so that we see that the resultant
,
'
’
CH VI
.
N I CS
]
I I D BO D I E S
M EC HA
67
OF R G
’
of two parallel forces is eq ual to th e i sum and acts along a line in
their plane s u ch that their total moment about any axis through
this line is zero
example let a light e d AB ( Fi g 42) be hung up by a
F
string at its middle point and two weights of say two pounds
Then to keep the o d
d one p und b e h u ng from the rod
balanced the one pound weight must be hung twice as f from
the centre of the d as the two pound weight ; and the tension
in the strin g supporting the rod will be equal to three po u nds
r
.
or
r
,
.
,
an
o
,
r
.
ar
,
re
S uppose
c
ou
p
now that two equal but oppos itely directed parallel
forces
act
on
a
body
which
is
free
to
turn
about
l
an axis at 0 ( Fi g
Le t the plane con
the two forces be perpendicular to the
Le t P Q
es'
.
Fi g 43
.
.
represent one of the forces f and let RS represent the other f
and let the distance between the forces
L t ON = p OM = p
MN = d
Th e moment of f about 0 is fl and that of f is
—
—
d
f
T
h
e
total
moment
is
there
ore
fp f p f ( p p ) f
fp
’
e
,
.
’
,
)
’
’
.
.
5— 2
I
E
68
P RO PE RTI E S
M C HAN CS AN D
P
I
A
R
T
[
E
O F M ATT R
total moment fit is therefore the same for any position of
the axis O ; so that the total moment of two equal and opposite
arallel
forces
is
the
same
about
any
axis
perpendicular
to
their
p
plane and is equal to either force multiplied by the distance
—
between them Their resultant is equal to f f and so is zero
S uch a pair of forces is called a couple Th e pro d tf cl is called the
moment of the couple or sometimes it is called simply the couple
A couple tends to produce rotation but since its resultant is
zero it does not tend to produce any tra slation A couple c
only be balanced by an equal and opposite couple it cannot be
balanced by a single force
Fo example if a light r e d AB ( Fi g 44) is hung up at O by a
string passing over a pulley as shown to which a 4 lb weight O is
attached and if another 4 lb weight D is hung from th e rod then
the rod cannot be in equilibrium in a horizontal position and no
single force can keep it in equilibrium It can be kept in
equilibrium by a force of 2 lbs weight acting do wnwards at B and
a force of 2 lbs weight acting upwards at E if the distance FB is
made twice A O It is then acted on by two couples with moments
4 x A O and 2 x EB which are equal and opposite
Th unit couple is equal to a couple consisting of two uni t
forces unit distance apart
Th e resultant of any number of parallel forces c n be found in
the same way as that of two It is equal to their sum and acts
along a line such that the total moment of all the forces about any
axi s through this line is zero
If all the particles in a body are acted on by parallel forces
proportional
to
their
masses
there
is
a
point
in
the
f M
body through which the resultant force always
passes which is called the centre of mass f the body
Th e weight of a body is made up of a great many almost
exactly parallel forces one f each particle of which the body is
made up Fo each particle is attracted vertically downwards b y
the earth with a force almost exactly proportional to its mass
Th e point through which the resultant of the weights of all the
particles acts is called the centre of gravity of the body Except
for exceed i ngly large bodies the centre of gravity and the centre of
mass practically coincide
Th e
.
.
uc
.
.
,
,
n
.
,
.
r
.
,
.
,
.
,
,
.
.
.
.
.
e
.
a
.
.
,
C e n tr e
a ss
o
.
o
,
or
,
.
.
r
.
.
.
CH
.
VI]
I
E
I I D B OD I ES
M C HAN C S O F R G
69
'
us first consider the position of the centre of mass of two
particl e s A and B of masses m and m at a distance d apart
m
the
forc
s
acting
on
them
be
equal
to
and
e
m
L
e
t
Fi
( g
Th resul tant of these two forces
d act along A A and BB
acts along a li e CC in the same plane as AA and B B and
parallel to them Le t it cut BA at C Draw BMN perpendicular
’
Then we have BM
Bu t
m
to CO and A A
Le t
,
,
a
.
’
an
’
a
,
,
e
.
’
’
’
n
.
.
’
x
.
BC
CA
a
.
e
MN
’
,
BO
.
BM
BC x m
that
CA x a m
Th e resulta nt therefore cuts the line A B at a point
so
,
,
.
C
such that
m,
CA
m,
'
po sition of C is independent of a and of the direction i
wh ich the forces m and a m act O is therefore the centre of
mas s of A and B
Th e
n
a
,
,
.
.
Fi g
Take
.
45
F i g 46
.
.
any pl n s surface EF ( Fi g
perpendicul a rs to it from A B and O
d w res pe ctively
We have
a
,
an
.
,
.
near AB and
Le t their length be
46 )
s
.
BO
m,
OA
m2
w—
'
a,
.
r
a ,
—w
x , , at,
I
E
70
M CHAN CS AN D
He n
P RO PE RTI E S
m , x,
ce
77721 03 1
w
m
i
“
T
A
R
P
I
[
E
M ATT R
OF
,
i 771 n
“
771 2
S uppose
now we have a third particle of mass m at D Th e
resultant of the three forces m m and m c n be got by finding
the resultant of m and th e resultant of m
d a m which acts
Th e resultant of th e three forces
at C and is equal to a ( m
will therefore always pass through a poi t C on CD such that
,
a
a
,,
a
a
,
a
,
a
,
.
,
an
2
,
n
912
CG
m,
This
point C is the centre of mass of m m and m Le t the
perpendicular distance from G on to the plane EF be w and that
of D be m Then as before
2
, ,
,
.
’
,
.
m
( ,
m
( 1
m 2)
5"
m l wl
so that
m,)
m l wl
7722 3 72
W
m,
l
"
‘
77523 72
771 3 3 73
+ m2 + m3
If now we take a fourth particle of mass m we can S how in
the same way that the distance of the centre of m ass of the four
particles f om the plane FF is equal to
4
r
m , as,
m 2x ,
m,
m,
m , x,
m, + m,
,
and so on for any number of particles
If therefore we have particles of masses m m m
m
whose distances from any plane are m m
then the
distance of the centre of mass of all these particles from the
plane is given by the equation
.
n
,
,
,
,
,
4
33
, ,
,
,,
, ,
,
x
centre of mass of two equal masses lies half way between
them ; S O that if a body is symmetric al about a plane its centre of
mass will be in that plane Thus the centre of mass of a uniform
rod is at the middle point of the rod and the centre of mass of a
solid S ph ere of uniform density is at the centre of the sphere
Th e
-
.
.
I
E
72
P ROPE RTI E S
M C HAN C S AN D
This
A
T
P
R
I
[
E
O F M ATT R
result can also be obtained by mea s of the formula
n
2m m
2m
Take
plane passing through
a
and perpendicular to
A
If we suppose the mass of each of the st ips pa allel to B C
be concentrated at their middle points we have at a series
t
of equidistant points along AD a series of masses proportional
their distances from A
t
strips Th distances of the
L t AD = l and let there be
masses from A are the efore l/ 2l/ 3 l/
l/
L t the masses
Th e dista ce of the centre of
m
of th e strips be m 2m 3 m
m ass from A is therefore given by the equation
AD
r
r
.
o
,
o
.
e
n
,
n
r
n
,
,
,
n
,
e
.
n
,
n
n
,
e
.
n
.
l
en is made indefinitely large This agrees with the pre v io
result
When a force acts on a body which is only free t turn about
an
axis
there
is
an
equal
and
parallel
force
R
ti A i
the axis which is resisted by the axis with an equal
and opposite reaction Th e fo ce and this reaction form a couple
T
h w th t wh i m d i d fi it ly l g
wh
us
n
.
o
0 11
ea c
on
f
o
x
s
.
r
.
o
S
o
a
en
n
s
e
a
.
n
e
n
e
ar
e
m+l
n
i
le t th e
C
h
Su
ser e s
an
bt
ge
ra c
m
’
n
be
n o
to
n
n
ti
mn
t
ed
+ 1
b y S ,,
d
an
g
”“
l
(m
1
1)
2
d
an
ri
+
ex pa n
i
d
o
s
n
s
n
e
ua
on
o
e
s
e
n
r e du c e s
on
”
it
,
e
s
ar
ce
Sn
or
ru e
es o
so
to Am “
He n
s
e
m
m
h
w
z
en
ua
’
n
A2 ( 71 +
e tc
.
we g e t
g
ec
n
n
a
-
o
th t th
ffi i
t
Th hi g h t p w
f
d
th A
h
f
t
t
hi g h t
p
n
o
so
,
a
e
.
er
s o
n e
l) n
l
e e
e
e c an
(m
va u e s
a
us
a
a
1)
m
hi q ti i t f ll l
f
f
b th i d f it m t b q l
id i
th t ll th A f hi g h
i d fi it ly l g w
lg t ll p w
T
a
a
( 1 ) fr o m (2)
771.
th t
a ssum e
th t
so
(n
n
m + 1
or
es
er
er s o
n
c oe
e
an
m“
ex c e
e
m
.
i s m a de
i
n
de
fi itly l
n
e
er
o
a rge
.
c en
o
n
on
a re z er o
es
o f ea c
s
.
an
th
p o we r
l ft h
e
e
If
h
n
d th e
-
an
d
i s m a de
e
q ti
ua
on
VI]
CH
.
E
M C HA
NI CS
OF
I I D B OD I E S
73
R G
reaction f the axis does not tend t turn the b dy round
bec use it has no m ment about the axis Th moment of the
force acting on the body ab ut the axis is therefore equal to the
moment of the c uple formed by this force and the reaction f the
axis
A constant angular v elocity is equal to the angle turned through
in uni t time I t is usually denoted by so that 9 wt where 9
de tes the angle turned through in a time t If the angular
v el city is chan ging at a uniform rate then the change i unit
time is called angular acc leration
If denotes the angu lar
acceleratio
Th e
o
o
a
o
o
e
.
o
o
o
.
(0
.
:
,
n o
.
o
n
,
e
a
.
n
a
:
t
where w is the angular velocity at the beginning and
th at at
the end of a time t Th angle 9 described in a time t is given by
t h e equation
o r,
,
e
.
f
2
(o f
-
m
l
m, + m,
l = w, l +
fi
?
at
a
t
In dealing with the ro ta tion of bodies it is usually more
c nvenient to measure angles in circular measure than in
degrees
When a rigid body is ro tating with an angular velocity about
a
fixed
axis
then
the
velocity
of
a
point
P
in
the
R
Ri d
body at a dista ce from the axis is equal to w
If the body has an angular accelerati n a then th e point P has
accelerati n equal t
along the d i rection in which it is moving
that is al ng the tangent at P to the circle which P is describing
about the axis of r tation ( It has also the acceleration v / along
the radiu vector )
A particle f mass m at P must therefore be cted on by a force
havi ng a c mponent equal to m along the ta gent at P Th
m a
moment of this force about the axis is m m x
Le t us now regard the rigid body as di v ided by imaginary
su faces in to an i mmense number of very small parts Le t the
ma sses f these parts be denote d by m m m etc and let their
o
.
or
o
ta t i o n
gi
o
f
,
a
B ° dy
n
r
r
o
o
o
.
an
,
ra
,
o
,
o
s
2
.
.
a
o
o
n
ra
.
r
r
r
fi
.
.
o
r
,
,
, ,
,
,
.
,
e
N I CS
E
74
M C HA
AN D
P RO PE RTI E S
distances from the axis be de oted by
M of the body is the given by
n
r
r
,,
T
P
A
I
R
[
E
O F M ATT R
,
,
r,
,
etc
Th e
.
total mass
n
M
= m,
2m
,
where Em denotes the sum of all the masses m m m etc Th e
th particle requires a force m
to give it its acceleration
Th e total
and the moment of this f rce about the axis is m
moment O required to give the whole body the angular acceleration
i s therefore given by
etc 2m
C m
m
,
,,
n
,, r
n
,
,
.
,
r
a
Q
,, r n
o
a
a
,,
.
a
2
a
,r ,
C
a
Em r
,r ,
2
2
a
r a,
.
2
.
forces acting on the particles of the body are either forces
due to ac tions between one particle and another or extern al forces
acting on the body from outside An y i ternal action is made up
of two equal and opposite forces the moments of which about
the axis are therefore equal and opposite Th e total moment of
the internal forces about the axis is therefore zero ; so that the total
moment of the forces acting on all th e particles of the body is
equal to the moment about the ax s of the external forces acting
on the body
If then C denotes the total moment of the external forces acting
on the body about its axis of rotation we have C Ze
Th e quantity Em
is called the m o m e t f i er ti of the body
h
t
about
the
axis
in
question
T
e
momen
of
inertia
M
t ,
’
of a rigid body about any axis is obtained by
supposing the b ody divided into a great many very small parts
and taking the sum of the products formed by multiplying the
mass of each part by the square of its distance from the axis
It hould be remembered that the moment of inertia of a body
about an axis depe ds o the position of the axis with respect to
the body so that in stati g the moment of inertia of a body the
position of the axis in question must be specified
If we suppose the body divided up into a very large number
of pa rticles all ha v ing equal masses m then the moment of inertia
Th e
n
.
.
1
'
a
,
r
e
I
n e
m
en
ti a
2
n
a
n
o
.
.
o
'
.
S
n
n
n
,
.
n
,
Zm r
where
Le t
Er
?
r
2
,
the average value of
2
r,
2
r
2
e
g
,
4
be equal to
[C
2
so that
Er
2
z
c
n l
2
.
CE
.
,
VI]
N ICS
E
I I D BO DI E S
M C HA
75
OF R G
moment of inertia is therefore equal to m h or M1 if M
denotes the total mass of the body Th moment of inertia is
therefore the same as that of a particle of mass equal to that of the
body at a dista ce t from the axis t is called the radius f gy ti n
of the body about the axis in question h is g iven by the equation
Th e
n
o
o
.
2
0
,
e
.
n
fl
o
ra
o
.
Em r
2
k 2m
2
K
M
where K denotes the mom ent of inertia of any rigid body and M
its mass Th equation O 2 e may be w i tten C K or since
3
.
01
e
a
r
77
a,
:
an example cons ider the case of a S imple pendulum con
sisti ng of a particle P ( Fi g 48 ) of mass m suspended from a fixed
point 0 by a light thread of length l
Le t OC be a vertical lin e and draw P N
perpendi cular to 0 0 If the pendulum
is swingi ng through a small angle in the
plane of the paper the thread remain
straight so we may regard the pendulum
as a rigid body Th e forces acting on P
are the tension in the thread which has
no moment about the axis through 0
about which P rotates and the weight
of P Th e moment of the weight about
an
axis through 0 perpendicular to the
pla e in which P is swinging is m g x P N
Th e moment of inertia of P b o t this axis
is m l Hence we have since C Zm
—
mg PN
m g t sin 9
ml
where 9 is the angle P ON and the
a ngular acceleration of OP round the axis
If 9 is small we may put 9 sin 9 and so get 9 9 l
No w the acceleration a of P along its direction of motion is
As
.
.
.
s
,
.
,
,
.
n
.
a
u
fl
.
a
,
'
c
r
Q
,
2
,
a
.
,
a
.
I
E
76
equal to l and its distance
9l O that
a:
c
,
P RO PE RTI E S
M C HAN CS AN D
from
C
OF
along
P
I
A
R
T
[
E
M ATT R
th e
are
CP
is equal to
S
—
w
—= a
g l
-
.
This
is the same result as that obtained in Chapter I V and from
this it f llows as before that the time T of a complete vibration is
given by
,
o
T
If we put
Q
c,
t
we get
:
277
'
a
c
9
an
d T
:
2w/Vc
.
We see from this that if a body is rotati n g about an axis
a way that its angular acceleration
F i g 49
.
a
such
is given by the equation
in
.
where c is a co stant and 9 its angular displacement from
a fixed position then it scillates about the fixed position and its
time of oscillation is give by
c 9,
a
n
o
,
n
T = Q w/N/C
.
another example consider a wheel W ( Fi g
mounted on
a vertical axle AA having fastened to it one end of a spiral spring
S the other end of the S pring being fixed
If the wheel is tur ned
round th e spring exerts a f rce on it which tends to bring it back to
its original position of equilibrium Th e mome t of this force about
As
.
,
,
.
,
,
o
.
n
CH
.
V I]
N I CS
I I D B OD I E S
77
OF R G
M EC HA
the axis of the wheel is proportio al to the a gle through which
—
the wheel is turned L t this moment then be equal to A 9
where A is a constant
Le t the m ment of inertia f the wheel about its axis f rotati n
Then we have
be K 2
n
n
e
.
,
.
o
o
711 7
o
o
3
.
A9
a
K
,
where a is the angular accelerati n of the wheel If the wheel is
turned round and then let go it will therefore oscillate about its
position of equilibrium d the time of a complete v ibratio wi ll be
o
.
,
,
n
an
balance wheel f a watch is an example of such a wheel
vibrati g under the action f a spiral spring Th e time of vibration
is the sa me for small vibrations as for large ones and so the watch
goes at the same rate whatever the amplitude of vi b ration of its
balance wheel
If a small bo dy of mas m is a ttached to the wheel j ust con
from the axis then the moment of inertia
i de r e d at a distance
Th e time of oscillation will
will be increased from K to K m
therefore be increased to
Th e
o
o
n
.
.
s
s
r
,
,
r
”
.
K
T
l
m
r
2
If we measure T T m a d we can find K and A from the
last two equations We get by di vi di g the two sides of one
equation by the co r responding S ides of the other and squaring
’
,
n
,
r
2
,
n
.
’ 2
T
(T)
E
+
2
771 T
K
(f)
-
f
1
body of mass m may be i the form of a thin hollow
circular cylinder of radius attached t the wheel so that its axis
coincides with the axis f rotation All the matter i the cylinder
from th e axis and so its moment
i s then at the same distance
This gives an accurate way of
of in ertia about the axis is m
finding experimentally th e moment f inertia of a body about an
Th e
n
r
o
,
o
n
.
r
r
,
fi
.
o
E
78
I
P
I
A
T
R
[
E
M C HAN CS AN D P RO P ERTI ES O F M ATT R
axis Instead of mounting the body on an axle and using a spiral
When the body
S pring it may be S imply hung up by a wi r e
is turned round the wi e is twisted d resists the motion with
a couple proportional to the angle of the twist
Th e moments of inertia of bodies having impl e geometrical
As
forms can be calculated by means of the formula K Em
an example take the case of a thin uniform straight rod of length
Le t the axis about whic h the moment of inertia
d mass m
l
is required be perpe dicular to the rod and pass through its
middle point S uppose the rod divided into a very large number
a d mas s m
T
h
moment
e
n of equal parts each of length l/
/
of inertia is then equal to
.
.
r
an
.
S
a
r
an
.
.
n
.
n
n
fi
2
n
(g
2
2
P + 2u
3un
+
u
n
cg
n
.
= 2T
when q is a very large number
N o w consider the moment of inertia of a circular disk of
mass m and radius a about an axis through its centre and per
w
m
a
S
to
its
plane
Its
mass
per
unit
area
is
uppose
e n di c u l
/
p
it is divided up into circular rings of radii a / 2a / 3 a / and
so on where n is a very large number Th e area of the p th ri ng is
.
”
ar
.
.
n
n
,
n
,
.
e qu a l
to
moment of inertia of any ring about the axis in question
is equal to the mass per unit area multiplied by the area of the
ring and multiplied by the square of the radius of the ring Th e
moment of inertia of the disk is therefore equal to
Th e
.
vr a
7ra
n
n
3
3
3
n
n
]
2m a
”
4
2
n
“
ma
2
4
moment of inertia of a thin circular disk about one of its
diameters is equal to m a /4 Moments of inertia are mos t easily
calculated by the mathematical process known as integration
Th e
2
.
.
M C HAN CS AN D
Th e
at
P RO PE RTI ES
O
F
E
P
R
T
A
[
M ATT R
I
distance of the centre of mass of the body from the pla e
n
through
i s
I
E
80
O perpendicular
so that
0
Em a?
Vm
is
27
2
l
to
OP
0
and therefore
Bu t
the centre of ma
Em a
= 0
Hence
.
ss
th e
moment of inertia about the axis through P is equal to K Md
If a rigid body is free to tur only about a horizontal axis and
is acted on by no external forces except its weigh t
o o
dl
and the reaction of the axis then it can only rest
in equilibrium if its centre of g avity is vertically below the axis
Fo otherwise the weight has a moment
about the axis which imparts to the
body an angular acceleration Le t the
horizo tal axis be perpe dicular to the
plane of the paper and pass through O
raw
A
vertically
down
D
O
F
i
( g
wards Le t C be the ce tre of gravity
of the body and let OG = h Th re
t of the weights of all the particles
u lt
which make up the body acts down
wards through G and is equal to Mg
where M is the mass of the body Le t
the angle GOA = 9 Th e moment of
the weight about the axis at O is equal
to Mg h S i 9 Le t the moment of
inertia of the body about the axis at 0
be K Then we have
n
C mp
Pen u
u n
u
d
m
,
r
.
r
.
n
n
.
n
.
e
.
an
s
,
.
.
n
.
.
Mg h S i n 9
Ka
,
Fi g 51
where denotes the angular accelera
tion of the body If the angle 9 is very small we may put
M h9
rs therefore that the body
It
appe
9
9 and so get a
si
p
f
has an angular acceleration proportional to its angular displacement
and in the opposite direction to the displacement Th e body there
fore moves with simple harmonic motion and the time of a
complete vibration is given by
.
a
.
n
.
a
.
a
.
CB
.
VI
I
E
]
I I D BOD I ES
81
M C HAN CS O F R G
rigid body cillating through a small a gle about a h ri
ta l axis under the action of its weight is called a compou d
pe dulum
Le t the mome t of i ertia of the body about an axis through
G parallel to the axis at O be equal to M1 where [ is the adius
Then we have
f gy ation abou t the axis at G
A
o
n
os
n
zo n
n
.
n
n
3
0
r
o
r
c
,
.
Hence
T
27 1
we compare this equation with the equation
If
T
27r
which g ives the time of vibration of a simple pe dulum f length l
we see that the compound pendulum h a s the same time of
n
O
,
k
vibration as a S mple pendulum of le gth l such that l
h
This is called the length of the simple equi v alent pe n dulum
AS an example of a compound pendulum consider a thin
circular ring of radius resting on a kni fe edge at its highest
point I ts moment of inertia about an axis through its centre
and perpendicular to its plane is equal t M where M is the
mass of the ring Its centre of gravity is at its centre Its time
of vibration as a pendulum in its own plane is therefore gi ven by
n
i
.
r
.
o
r
”
,
.
.
T = 27 r
It therefore has the same time of swing as a imple pe dulum
of length equal to the diameter of the ring This
be easily
verifi ed by hanging a ball from the sa m e knife edge so that it
is j ust level with the bottom of the ring Th e two will be found
to swing with equal pe i ods
If we have a body ar ranged so that it can sw i ng on a number
of horizontal parallel axes at di fferent distances from its centre f
gravity then th e equation
S
.
n
ca n
.
r
.
O
,
13
2
4
-
10
2
h
gives the length of the simple equivalent pendulum corresponding
to each value of h ; and i does not cha ge S ince the axes are all
pa allel to the same axis through the centre of gravity In this
wP
6
t:
r
n
.
.
.
E
82
M CHA
N I CS
case we may regard
may be written
l
P RO PE RTI E S
AN D
as a functio of
n
h
l
curve i Fi g
by this equation
Th e
n
.
52
S
A
T
P
R
[
equation giving
Th e
.
l
k
[C
ls
h
E
O F M ATT R
h
'
hows the relation bet ween
and
2
given
.
Fi g 52
.
.
We see that when h /h = 1 then l/l has its smallest po ssible
value which is equal to 2 In this case l = 21
Whe h /h is very large k/h becomes very small so that l = h
approximately and the curve nearly coincides with a straight line
passing through the origin 0
When h /l is very small we h ve approximately
c
3
.
,
n
.
,
,
.
c
a
l
If:
k
h
Is
z
h
'
If h 0 then l = so that the time of vibrati n becomes infinitely
l g which means that the body will then stay at rest in any
position Fo any value of W greater t h an 2 there are two
possible values of h /k Co nsequently two po itions of the axis
00
on
o
,
.
r
e
.
s
CH
VI
.
I
E
]
I I D BO D I E S
83
M C H AN CS O F R G
at di fferent dist nces from the centre of gravity c a be found
for which the times of v ibration are the same Th e equation
n
a
.
h
16
2
2
h
if solved for h gives
,
l
2
If 1 and
1,
h,
l
2
i
k
?
deno te these two values of h we have
,
h,
h,
t
h h
1
and
If then we find experimentally two di ffere t v alues
gi ve equal k n wn times of vibration we can get the
length of the S imple equivalent pendulum l and the
radius of gy ation h
This can be done with a pendulum consisting
of a brass bar of rect ngular cross section having
a series of equidi stant holes bored through it Th e
bar may be about 1 00 cms long 2 m s wide and
0 5 cm thick
Th e holes should be about 06 cm
in diameter n d their centres about 2 e m s apart
end of such a bar is shown in Fi g 5 3 Th e
O
ba
is supported on a horizontal k nife edge put
through one of th e holes as S hown at B Th e time
of vibration of the bar swingi ng in its own plane
th ough a small angle can be found by measuri ng
the time of about 1 00 complete oscillations
Le t the time of vibration be f und with the
knife edge i n each of the holes and also the
di s t nces of the points like A B and C where the
pendulum rests on the knife edge from one end of
Th e position of the centre of gravity of
the b
the bar is also fou d by balan ci g it on the kn i fe
e dge and the dis tance of this from the same end of
the bar is measu e d
Th e dist n c es and correspo n ding times of vibra
tion are then plotted on sq uared paper and a curve
draw through the poin ts S uch a curve is shown
2
,
0
2
.
n
o
,
r
.
a
.
.
e
,
.
,
.
.
.
a
.
n e
.
.
r
.
r
.
o
a
,
,
,
ar
.
n
n
,
r
.
a
n
.
i n Fi g 5 4
.
.
.
which
I
E
84
M C HAN CS AN D
PRO PE RTI E S
E
O F M ATT R
cent r e of gravity is at G A line like A B CD cuts th e
curve in four points for which the times of V ibration are equal
d C are at di fferent distances from the centre of
Th e points A
gravity and on Opposite sides of it ; so that the distance from A
Z In the same w y BD
h = Z By
t C is equal to h + h
h
means of the curve we c n find A C and BD and so get l the le gth
of the simple equivalent pendulum corresp ndi g to the time of
vib ation represented by points on the line A BCD Th e value
of g can then be calculated by the formula
Th e
.
.
an
o
2
,
a
.
a
,
.
n
,
n
o
r
.
T
:
27 7
47 r l/T
?
g
or
'
le gths HD AH h and HB H O
and from them the radius of gyration of
Th e
,
n
:
,
2
17 ,
.
can als b e found
bar about the axis
o
,
Di s ta n
ce
Fi g 54
.
.
through its centre of gravity can be calculated by means of the
Th e mome n t of inertia of a thin uniform rod
equation
about an axis through its middle point and perpendicular to it
is equal to
M
d
2
12
where
is the length of the rod so that the
d
,
radius of gyration of the bar will be found to be nearly equal
to i ts length divided by V1 2 or 2 v3
—
.
CH
.
V I]
I I D BODI ES
E
85
M C HAN I CS O F R G
In the experiment j ust descri b ed the centre of gravity of the
b
i s always below the knife edge so that when th e knife edge
i s in the holes on o e ide of the centre of gravity one end of the
bar is at the top and whe the knife edge is in the holes on
the other side of the centre of gravity the other end is at the top
C are should be taken to measure all the lengths from the same
be inverted like this bar
A pendulum which
e n d of the bar
is called a reversible pendulum Th e most accurate way of findi ng
the value f g is by means of a form of reversible pendulum invented
by Captain Kat er and there fore k own as Kater s pendulum
ar
,
S
n
n
.
c an
.
.
e
’
’
n
Fi g 5 5
.
.
.
a detailed description of the methods of finding g accu
rately with various forms of Ka ter s pendulum P ynting and
M
may
be
con
a
t
t
er
Thoms n s P op er ti e
f
lte d It will suffice here to describe a simple
form of Kater s pe ndulum capable of giv i ng the
value of g correct to W ithin one part in 2000
about
This c onsists of a brass bar A B ( Fi g
1 1 5 m S long having two knife edges K and K
fixed in to it one ear one end and the other
about 99 3 cms from the first These edges
are fixed at right angles to the bar and facing
ea h other as shown in Fi g 5 5 Between A
and K a weight of several pounds is fixed to
the bar
as to bring the centre of gra v ity
much nearer to K than to K Between K and
K there is a small weight which can be slid
along the bar and fixed in any des ired position
with a screw Th e time of vibration of the
F 18 5 6
pe dulum should be very early 2 secs on
either knife edge O e of the knife edges K K ( Fi g 5 6 ) is sup
ported
a firmly fixed U shaped horizontal plane P P and the
pendulum set s wi nging through a s m all angle Th e time of
Fo r
’
,
’
o
su
s
r
o
o
.
’
.
.
’
c
.
,
n
,
.
c
.
.
,
.
so
’
.
’
.
n
n
.
ou
°
.
n
.
-
,
.
~
I
E
86
PROPE RTI ES
M C H AN C S AN D
P
A
I
R
T
[
E
OF M ATT R
v ibration is found by comparing it with the time of vibration
of the pendulum of an accurate clock This is best done by a
method k own as the method of coincidences Th pendulum
is put up in fro t of a clock having a pe dulu m with pe iod
of 2 secs so that its lower end is at the same level as that of
the clock pe dulum and swi gs parallel to it Th time by the
cl ck is noted whe the two pe dulums swing exactly together
this is called the time of a coincide ce If they co ti ue to swing
together for a long time say an hour then their times of vibration
are equal If one gai s
the other then when it has gained
a complete Vibra tio the tw pendulums will again swing exactly
together Th ti m e between two such coi cide ces is found by
the clock let it be t secs In th time t the cl ck pendulum
makes 2 vibrations If the Kater s pendulum loses compared with
2
.
n
n
.
e
.
n
,
r
,
n
n
n
n
o
,
n
,
on
n
,
o
n
e
.
n
n
.
,
.
e
.
n
e
.
,
n
o
’
.
t
the clock then it makes
of vibration
T of
vibrations in the time
1
the Kater s pendulum is therefore
’
i
g
t
.
Th e
time
ven by
‘
Zt
t
t
2
t
i
2
If t is large then a small error in t has very little effect
the
value fou d for T so that this me thod enables T to be found very
accurately Fo example if t 1 000 secs we get
on
n
,
r
.
,
.
2000
2 0040 1
9 98
If we change t to
1 01 0
secs
.
secs we get
.
2020
2 003 9 7
PODS
Thus
secs
.
in this case an error of one per ce t in t only produces
an error in T of one in
Havi g found the time of vibra
tion of the pendulum on one k ife e dge we then in v ert it and fi d
its period o the other By moving the S liding weight these
two periods are made as nearly equal as possible
Th e distance
between the knife edges is then
This distance is
measured accurately and then g can be calculated by the form la
=4 l T
T
o
save time it is a good plan first to measure the
g
/
n
.
n
n
n
n
.
.
u
7r
2
2
.
'
I
E
88
M C H AN CS AN D
P ROPE RTI E S
E
OF M ATT R
A
T
P
I
R
[
In the design of a sensitive balance the beam is designed to
be as light and as small as possible thus keeping M and K
small so that h can be made very small W ithout maki g the period
,
n
,
Fi g 5 7
.
.
inconveniently long Making d large requires M to be increased
so much that no gain in sensitiveness results A se n sitive bala ce
is S hown in Fi g 5 7
.
.
.
.
n
.
I
E
]
C H vi
I I D B OD I E S
89
M C H AN CS O F R G
If the middle knife edge is not exactly half way between the
two end ones then the masses which balance each other are o t
xactly equal Le t the distances from the middle edge to the
’
Then if masses m and m balance each
e d edges b
d and d
m d
If now the mass m is put in the the
d
o ther we have
"
pan and is then balanced by a mass m we ha v e
-
n
,
e
.
'
n
e
.
’ ’
7n
o
.
m
He n
d
a n
ce
m
therefore
In
if
m
th i s
”
'
d
md
d
,
m
‘
d
way the ratio
d
3
and
N/ m
.
d
m
,
r
“
7
be found and also the mass
ca n
m
and m are known masses g standard weights
S uppose
Th e mass m can also be found by another method
masses m and m balance each other If standard weights
so that they also balance m then these
e substituted for
weights are of equal mass to m In this way m can be found
whe ther d and d are equal or not
When a body rotates about a fixed axis it is kept from moving
l i
in any other way by the axis so that there are in
w
ge eral forces on the axis and equal and opposite
forces
the b o dy Th e directions of these forces rotate with
the b dy that they tend to shake the axis F example the
rotating parts f machinery may te d to shake the bearings sup
p rti g them I practice the shafts car y i g rotati g parts cannot
be absolutely fixed so unless the rotati g parts are made in such
way that they do t exert appreciable forces on the shafts
ome shaking results S uch shaking is bad for the bui lding and
m chi ery besides being inconvenient Th e rotating parts of
m chinery are therefore always arranged so that they exert as
little f rce as possible on the shafts carrying them Th e y are
the said to be balanced S uppose a rigid body is rotating with
niform angu lar velocity about an axis 0 ( Fi g 5 8 ) perpendicular
to th e pla e of the paper Consider a particle P of mas s m in
the body at a dis tance fro m the axis Th e force on the axis
to this particle is equal to m / where v is the velocity of the
d
particle B t v = w so this force is equal to m m and acts
m
’
,
e
.
.
.
.
’
.
’
771.
a r
,
.
'
.
B
a n c
a
n
g
B °di e s
in g
“
,
‘
n
on
.
so
o
o
o
n
or
.
.
,
n
n
n
r
n
n
,
a
n o
s
.
a
n
.
a
o
.
n
.
u
(0
n
.
,
.
r
,
.
2
ue
u
.
u
r
,
r
,
)
?
I
E
90
M C H AN CS AN D
P RO PE RTI E S
P
A
T
I
R
[
E
O F MATT R
along OP T k e y pla e A OB contai ing the axis and let
the a gle betwee GP and the plane be 9 Th perp e dicular
i 9
Le t s i 9
m
dista ce from P on to this plane is the
is the resolved part of the force w m
N w m m si 9 = w m
perpendicular to the plane A OB There will be a fo ce like
for each particle in the body Th sum of all their resolved
m
parts perpendicular to the plane A OB may be denoted by m 2m w
If the centre of g v ty of th e body lies in the plane A OB we
have 2m m 0 so that then the resultant force on the axis pe pe
to the plane A OB is zero If the centre of gravity is
di c l
on the axis then in the same way we can show that the resultant
force perpendicular to any plane containing the axis is zero so
that there is then no resulta t force on the axis T balance
a rotating part it must therefore be made so that its centre of
’
n
an
a
.
n
n
n
n
n
a
o
2
n
“
?
e
.
r s n
n
r
.
n
‘
3
x
r co
r
r
.
f
.
2
e
.
2
.
ra
I
r
:
,
u ar
n
.
,
n
Fi g 5 8
.
o
-
.
Fi g
.
.
59
.
gravity lies on the axis about which it rotates There is then
no resultant force on the axis due to its rotation but there may
still be a couple F a couple has a zero resultant
example let 0 0 ( Fi g 5 9 ) be the axis of rotation and let
F
the rotating body consist of two balls A a d B of equal mass
attached to a rod AB whose middle p int C is fixed to the axis
’
Th e centre of gravity Of this b dy is at C so that there will be no
resultant force on the axis when the body rotates Bu t the force
due to B d the force due to A will form a couple tending to
turn the axis round from its position
If A OB were t right
.
.
or
.
'
or
,
.
n
o
,
,
'
.
o
.
an
a
I
E
92
M C HAN CS AND
P ROP ERTI E S
E
[
O F M ATT R
PART
I
the force acting on the wheel multiplied by the distance through
which it acts If K be the moment of inertia of the wheel we have
where is the angular acceleration Th e acceleration
T =K
of the weight is T so that
.
r
a
a
,
.
a
M
( g
Mr a ) r
Mg r
K
(
=
Ka
Mr ) a
,
?
.
the wheel start from rest and move through an a gle
let its a gular velocity then be
Then we have
9
a
Le t
9
n
n
on
.
s
a
wh ch
g
i
,
M gr
9
?
K Mr
1 ve S
9Mg r
and
I
2
a)
r
M
§
w
K
4
5
2
2
wh
is the work done by the weight so that we see that
this is equal to the t tal kinetic energy of the wh eel and the
mass as it should be according to the principle of the conservation
O f e ergy
If a constant couple of moment 0 acts
a body which can
rotate
about
an
axis
perpendicular
to
the
plane
w
l
of the couple then the work done by the couple
whe the body turns through
angle 9 is 09 If the body
r tates with angular velocity w the couple does work equal t
Th e power of the couple is therefore equal
Cw per u it time
to Cw Power is ofte transmitted by means of rotating S hafts
If a shaft makes turns in unit time its angular velocity is
C where C is the
2 m so that the power it transmits is a
couple driving it
S u ppose a shaft is driven by a belt passing over a pulley
Le t the tension in the belt on one side
on the shaft ( Fi g
Then the
o f the pulley be T and that on the other side T
moment of the tensions ab ut the axis of the shaft is ( T
is the radius of the wheel Th e power transmitted is
where
therefore 2 mm ( T
example if = 2 feet T 1 05
F
pounds weight and T = 5 pounds weight and is 25 0 turns
Bu t
Mg r 9
o
—
n
.
on
po
er of
e m p es
'
,
an
n
o
.
o
,
n
.
n
.
.
n
7
,
.
.
'
.
o
r
r
.
or
7
r
,
’
,
,
n
:
C H v1
.
I I D B O D I ES
E
]
93
M C H AN I C S O F R G
per minute the p wer is
per minute This is equal to
27 r
o
,
x
25 0
‘
1 00
x
x
foot pounds
2
-
.
3 1 41 5 9
horse power
3 3 000
.
ower is Often tra mitted from
shaft to another by means
f two pulleys and a belt
If one pulley has a radius
d makes
revolutions in unit time and the other has a radius and
makes revolutio s in unit time then if the belt does not slip
we have
because the sa me length of belt passes over
P
on e
n s
o
r
.
an
'
r
n
'
n
n
,
Fi g 6 1
.
.
"
each pulley If C is the c ouple in the first shaft and C that
in the second then if there were no loss of power we should have
.
,
On
C
’
n
'
,
and therefore
Th e couple required to transmit a given amount of power
along a shaft is inverse ly proportional to the rate of revolution
of the shaft
Th e equations obtained for rotating bodies may be co mpared
h
for
particles
moving
along
traight
W
i
t
t
h
se
i f
C
t
E
li n es
We have
.
.
o
m pa
q
ua
i
r son
on s
o
o
.
.
S
I
E
94
P RO PE RTI E S
M C HAN CS AN D
9a
k(
1
2
60 2
Ka
0
A
R
T
P
I
[
E
M ATT R
OF
,
K w,
s
t
Kinetic
energy
W = fs
f
s
P
(
I) ,
2
energy
W
09
C9
11
5
,
m
%
1
K inetic
m
v
,
§
2
0
10
5
2
,
,
0
5
2
2
7
2
P
We see that moment of inertia K takes the place of mass
M
m
and
couple
0
that
of
force
me
tum
f
l
A
M
m
is replaced by K w This quantity K w there
fore may be called the g la m o m e t m of the body It is
also sometimes called the moment of momentum S i ce G E
we see that if 0 0 or if the moment of the forces acti g on the
body about its axis of rotation is zero then its angular acceleration
0 then it follows that w and therefore K is
is also zero If
constant Th e angular momentum of a rigid body about any axis
therefore remains unchanged so long as no forces act n it having
any moment about that axis
If a heavy wheel is mounted on a S haft which is free to turn in
any direction about the centre of gravity of the
wheel then if the wheel is made to rotate rapidly
it is found that the direction of its axis remains fixed An
apparatus of this kind is shown in Fi g 6 2 It is called a g y o
Th e wheel Wi s mounted on a shaft A B which is supported
sc pe
’
This ring is supported by two bearings at C and D
by a ring
in f a m e which can turn about a vertical axis in a bearing
at F Th e wheel can turn about the vertical axis through F
and lSo about the two perpendicular horizontal axes CD and AB
so that it is free to t urn in a n y direction If the wheel is set
spi ning rapidly about its axis AB then the direction of A B will
remain parallel to itself when the apparatus is moved about in any
manner
If n e w a weight is hung from the ring CADB at A so that it
tends to tur the rotating wheel about CD it is found that the
n
gu
a r
tu m
Oa
?)
n
o
.
,
.
an
u
n
r
u
.
.
u
n
n
,
a
.
co
.
o
.
Gy ro s c
o
pe
.
,
.
.
o
r
.
.
a
r
'
.
a
,
.
n
,
.
n
,
‘
96
E
I
M C HAN CS AN D
P ROPE RTI E S
Fi g
Fi g
.
.
63
64
.
.
OF
E
M ATT R
A
P
[ RT
CH
.
VI]
I
E
I I D BO D I ES
97
M C HAN C S O F R G
momentum about the axis 0 0 ; P essing it down at A gives
it a gular momentum about the axis EF which is perpendicular
to 0 0 We can sh w that the e ffect of thi is to tur 0 0 fr m
i ts original direction towa d F
Its direction
An g ular velocity has magnitude and direction
It may therefore be
is the direction of the axis of rotation
represented by a straight line drawn parallel to the axis and
Th e line is drawn
f a len g th proportional to its magnitude
fr om the body that
lo king b ck along it towards the body
the body ap pears to be ota ting in the opposite direction to the
ha ds of a watch
Th e angular m mentum Km is p Op ti n l to the angular
velocity so it also can be represe ted in magnitude d direction
by a straight line A gular momentum is there
fore vector d t h e resultant of the two angular
momenta can be found in the same way as the
resultant of any two vectors for example two
forces or two displacemen ts Le t 0 P ( Fi g 6 5 )
represe t the angular momentum of the t p
bef e it is pressed down at A OP is then
pa allel to the axis of rotation of the top When
it is pressed down at A it is given angular
mome tum about the perpendicular axis EF
Le t this additi nal angular momentum be e p e
sented by OQ Th resultant a gular momentum
is then represe ted by the diago al OR of the
Fi g 6 5
parallel gram OP RQ Thus the axis of ro tation
mo v es from OP t the di rection OH We have OQ Ct If the
angle P OR is mall it is equal to P R/OP
Le t K w be the origi al angular mome tum represented by
d let P O
OP
The we have
9
P
r
n
s
o
.
r
s
n
o
.
.
.
o
.
on
SO
o
a
r
n
.
o
I
or
o
a
n
,
an
n
.
a
'
an
,
.
.
n
o
or
.
r
.
n
.
o
r
n
e
.
r
n
n
.
o
.
o
.
.
s
.
n
an
.
n
n
.
Gt
Km
If the force exerted at
so that
.
P
.
9
0
t
Kw
,
A
t
W
.
is f
Kw
.
an
.
d AO =
r
we hav e f
r
:
0
NICS
E
98
M C HA
P ROPE RTI E S
AN D
T
P
AR
I
[
M ATTER
OF
is the an gular velocity with which the t p d of the axis
of rotatio tur s towards F about AB
axis whe A is pre ssed
down with the f rce f This angular velocity may be made very
mall by making very large
d has a radius of o
S uppose the top weighs 1 0 pounds
be y 8 inches so that its
foot
L t its radius of gyrati
in pou ds d feet S uppose
f inertia is 1 0 ( T? )
m oment
it makes 3 0 revoluti s per seco d
that w = 2 x 3 0 1 8 8
e pou d weight
L t f be equal t
3 2 pou dals and be
applied 6 inches from the ce tre of the top Then we have
9/t
en
o
n
as
n
n
o
s
.
a)
.
an
.
on
e
8
o
,
n
t
32
44 4
7r
or
p
57
.
n
1
approximately
52
1 88
.
Thus
er
:
.
xg
x
.
SO
n
9
an
n
n
on
o
,
,
2
on
e
sa
,
n e
the axis will move towards F at the rate of gg degrees
second since the u it a gle in circular measure is about
degrees
-
n
,
n
.
F
N
RE ERE CES
Ma tter a n d Mo ti o n
Mec h a n i c s , C o x
,
J
.
C
l
erk
-
M a x we
ll
.
.
i n g To p s , Pe r r y
Ele m e n ta ry Ri g i d Dy n
Spin
n
P r o p er ti es
.
a
m i c s,
a tter , P o n
M
y
f
o
i
t
n
E
.
g
J
an
.
t
Ro u h
h
d T
.
om son
.
N I CS
E
1 00
M CH A
PRO PE RTI E S
AND
A
P
T
I
R
[
E
O F M ATT R
earth the distance between the earth s centre and the moon s
and m the earth s mass we have
’
,
’
r2
,
’
,
where denotes the acceleration with which a body falls to the
earth at its surface and a the acceleration of the moon towards
the earth Bu t a g so that
a1
2
1
.
an
i
9
We have
He n
98 0
g
8
;
2
2
2
6 37
r1
x
10
8
em s
,
and
r2
38 4
x 1 0 cms
10
.
ce
the m o on moves round the earth nearly in a circle of
radius
3 8 4 x 1 0 cms and its period of revolution T 3 9 3 43
minutes Hence its acceleration towards the earth S g1 ve by
No w
10
r2
.
I
.
2
v
T2
47 r
2
r2
T
47
2
X
38 4
( 3 9 3 43
2
x
n
10
10
X
This
agrees almost exactly with the value calculated by assuming
the acceleration to be inversely proportional to the square of the
distance from the centre of the earth Thus we see that the
attraction between the moon and the earth is of the same kind as
the attraction between the earth and bodies at its surface
.
.
the
values
of
and
are
known
g
%
di h
so that if we knew G we could get m the
m
E
mass of the earth To find G it is necessary to
measure the force with which tw bodies of k own masses attract
each other when they are at a know distance apart This was
first done by Cavendish about the year 1 7 9 7 Th following is
a description of an apparatus S imilar in principle to that used by
Cavendish which e ables a r ugh determinatio of the constant G
and so of the mass of the earth to be done in a few minutes as
a lecture experiment
In the equation
Ca ve n
x
Pe r i m
s
e
= G
g
r1
,
’
s
'
.
n
o
n
.
.
,
,
n
o
.
n
e
CH
.
V II ]
V I TATI ON
G RA
wire AB ( Fi g 6 6 ) abo ut 5 cms long has at each end
a small S ilver sphere weigh i ng about one gram This is supported
by a Wire W which carries a mirror M and the whole is hung
up by a fine fibre E made of fused quartz which is about
This apparatus is enclosed in a case
Th e part of
6 0 cms long
the case surrounding A B is a glass box with double walls and is
only wide enough to allow the spheres to move horizontally through
a few millimetres Th e wire AB can oscillate about a vertical axis
A
.
.
.
,
,
.
.
.
.
Fi g 66
.
.
under the action of the couple exerted on it by the quartz fi b re
If m is the mass of each sphere and 2d the distance between the
centres of the spheres then the moment of inertia about the ax s
of rotation is approxi m ately 2 d so that the time T of a complete
oscillation is given by
.
1
,
7n
2
or
T = 27 r
C
8fl
-
3
7
d
z
1
where C i s the couple exerted by the fibre when it is twisted
th r ough u n it angle i circular measure S uppose d 2 5 m
m = 1 gram and T
6 00 sec s then we get
n
:
.
e
s
,
,
14
S uppose
x
A
dynes
cts
on
the
sphere
at
in
a
horizontal
a
f
di rection perpendi cular to AB and an equal and opposite force
'
a
fo r c e
MEC HANI CS
PROPE RTI E S
AND
P
I
T
R
A
[
E
O F M ATT R
acts on the sphere at B Th e couple due to these forces is
If this couple turns A B round through an angle 9 we have
.
e
,
5f = C9
14
x
or one degree then f is given by
14 x 10
x
dyne
}
f
x 5
,
1
73
5
.
we see that the arrangement described provides an extra
ordi arily sensitive means of measuring very small forces Th
angle turned through by AB is found by reflecting a beam Of light
from the mirror M on to a g raduated scale
If the spot o f light on
the scale m oves through a distance then AB has turned through
an angle 9 given by
Thus
n
e
.
.
8,
s
20
’
denotes the distance from the mirror to the scale
supposed to be perpendicu lar to the beam of
D = 200 m s and
cm Then we have
.
e
.
.
1
4000
so
’
that
f
14
5
x
x
10
4000
%
dyne
10
x
.
Thus
with this apparatus we can detect a force on the spheres
equal to one ten millionth part of a dyne Th e arrangement
described is called a torsion balance T use this apparatus t
measure the attraction betw e en two bodies we put up on eithe r
side of the box containing the suspended spheres A and B two
large lead spheres each about 8 e m s in diamete r Th e positions
A B is the wire and
f these large spheres are S hown in Fi g 6 7
mall spheres in the box OD Th e quartz fibre is perpe ndicular to
the plane of the paper at 0 Th e large S pheres E and F a e
carried by parallel horizontal r ds GH and MN on which they can
lide freely Th e sphere E is first p t e tly opposite the small
sphere A and th e sphere F is put pposite B Th e attractions
between E and A and between F and B cause AB to turn through
a small angle and it oscillates about its equilibrium po sition The
-
.
o
o
.
,
,
.
.
o
.
s
.
.
r
.
o
‘
S
u
.
xac
[
O
.
.
I
E
1 04
PRO PE RTI E S
M CHAN CS AN D
A
R
T
P
I
[
E
O F M ATT R
There
is also a couple in the opposite direction due to the attrae
tio s between E d B and A and F This couple is equal to
an
n
.
G
Th e
total couple
0
0
_
is
G
Mm
Mm
2
E
‘
sd
z
5 6 6d
'
'
therefore given by
hi m { 1
l
1
G
—
0 3 23
x
,
,
and twice this is equal to
8 71
-2
s
8 7 r m ol
2
so that
T
3
20
2
2
G
9
-
fil m
0 3 23
x
d
s
Hence
d
T
20
2
in
2
d
3
T DM
2
If M 3 000 grams d 25 m s T = 6 00 sec s D 200 m s t
will be found that is about 1 5 cms This makes G about
x 10
Th e value of G has been caref lly determined by a
number of observers using a variety f me th ds Th most
probable v alue of it is
G 6 66 x 1 0
A full acc unt of the di ffere t methods of finding G will be
found in Poynting and Thomson s P p er ti f M tte
e
:
,
,
,
s
‘
e
:
I
.
s
u
.
o
o
e
.
— 8
.
n
o
’
es o
r o
P u ttl n g
= 98 0
g
and
0
see
G
2
6 37
r1
,
6 66
x
10
8
r
a
em s
.
,
x
=G
in the equation
g
m
we get m 6 x
g ams which is therefore the mass of the
earth Th e average densi ty of the earth is got by di viding its mass
by its volume and is equal to
5
grams
per
c
c
m
5
4
S ince it is found that the weight f body at any particular place
depends only on its mass and is independent of its condition it
follows that the gravitational attraction between any two bodies
depends only on their masses and the distance between them
7
r
:
,
.
so
-
71 7 3
3
.
.
,
O
a
'
,
.
F
N
RE ERE CE
P r op e r ti es
f
o
i
fil a tter , Po y n t
n
g
an
h
d T
o m so n
.
C HAP T ER
.
V III
Y
EL AS TI C I T
ELAS TI C I TY
is that property of matter in virtue of which force
is required to hange the s hape or the volume of a
fi it
E
m
piece of matter and to maintain the change Fo
example if a weight is hung from one e d of a piece of india
rubber cord the co d becomes longer and thinner This change
of shape p rsists so long as the weight is not removed but when
the weight is removed the i di a u b b e goes back to its original
shape If a straight woode bar 8 supported horizontally at its
ends and a weight hung from the middle it becomes bent When
the weight I S removed it springs back to its origi nal shape S olid
bodies require force to change their shape but fluid bodies do not
resist a change of shape with a permanent force While th e
shape of fluid bodies is being changed there are forces which
resist the change b u t these disappear when the change is com
l
et
and
are
smaller
the
smaller
the
rate
at
which
the
change
is
p
made If the change of shape is made extremely slowly the
with fluids the forces required to produce it become extremely
small whereas with solids the force requi ed is nearly as great for
a S low cha ge of shape as for a rapid change Fluids resist a
change of volume with a permanent f rce and spri g back to the i r
original volume when the force is removed S olid bodies may be
said to possess ela ticity of shape and of volume while fluids
possess no elasticity of shape but only elasticity of volume
When the shape or size f a body is altered it is said to
be strained and the change Of shape or size is called
t i
a strain
When a body is fi t strained by the applicati n of forces to it
and then the fo ces are removed if the body then springs back to
De
la
io n
n
n y
s
o
c
f
‘
r
.
n
,
r
,
.
e
,
n
.
r
n
.
r
1
.
,
.
,
.
,
e
.
n
,
r
,
n
.
n
o
.
s
,
.
o
S
ra
n
.
.
o
rs
r
,
I
E
1 06
M C HAN CS AN D
P ROPE RTI E S
P
AR
I
T
[
E
O F M ATT R
its ori gi nal ze and shape so that the strain completely dis
appears the body is said to be perfectly elastic for the strain
in question Fluids are f u d to be perfectly elastic for changes
of volume
If the volume of a body is V and it is cha ged to V without
Si
,
o
.
n
.
’
n
an
y
V
change of shape then the strain is taken equal to
V
’
,
V
that is to the change of vol u me per u it volume Fo example if
a sphere of radius is changed unto a S phere of radius by m
pressing i t equally in all directions the strain is
n
r
.
r
,
r
T
,
'
co
,
3
4
“
5
If V is less than V the strain is negative A change of volume
without change of shape is called a uniform dilatation
If a rectangular block AB CDEF GH ( Fi g 6 9) is strained
by movi g the top parallel to the base fro m AB CD to
while the base EFGH remains fixed then it is changed into
'
.
-
.
.
n
,
Fi g 6 9
.
.
a parallelepiped
the volume of which is equal
to the original volume of th block This sort of strain alters the
A
Shape without altering the volume and it is called a s h e
shear is measured by the angle AEA which is called th e angle of
shear
It can be shown that any strain is m ade up of a dilatation
and a shear
e
.
ar
’
.
.
.
I
E
1 08
PRO PE RTI E S
M C HAN CS A ND
These
E
OF M ATT R
A
R
T
P
I
[
are the values of
and l when the stresses are
expressed in dynes per sq cm
A an example suppose a cubical block of steel with sides
1 00 m s lo g is acted on by a uniform pressure all over its
surface equal to 1 000 kilograms weight per sq m This is a
x 1 0 dynes per sq cm Th e change
pressure of 1 0 x 9 8 0 or
of volume of the block is given by
o
n
.
s
.
,
e
n
.
.
6
c
.
8
.
.
98
10 =
8
x
16
—
x
V
10
'
V
—
10
6
which gi v es V V 6 1 0 cubic centimetres
A pressure of 1 000 kilograms weight per sq cm is equal to
pounds weight per sq inch We see then that only ve ry
small changes can be pr oduced i the volumes of most solids and
liquids even by enormous pres ures
Fluids as we h ave seen are distinguished from solids by the
fact that they have no elasticity of shape Fluids are of two
kinds liquids and gases Liquids have a definite volume ; so that
when some liquid is contai ed in a vessel of volume greater than
that of the liquid the liquid occupies only a portion of the
volume of the vessel A gas al ways completely fills up the vessel
in which i t is contained ; so that i t volume is equal to that of the
vessel containing it Th gas exerts a pressure on the walls of the
vessel containing it which depends on the volume of the vessel
and the amount of gas in it If the volume is changed from
’
V to V and the pressure changes fro m p to p then if V
V is
very small it is f und that
’
.
.
.
.
.
n
s
.
«
,
,
.
.
,
n
,
.
s
.
[
e
.
.
’
’
,
o
P
—
=
p
—
k
V
where 1 is a constant I is the bulk modulus of elasticity of the
gas Th e properties of a gas depe d greatly on whether it is hot
or cold that is
its temperature so that it is best to deal with
them in the chapters Heat
0
t
.
n
.
,
on
,
on
.
F
N
RE ERE CES
f Ma tter
P r o p er ti es
o
Ex p er i m e n
ta l
,
i
Po y n t
Ela s ti c i ty , G
.
n
g
F
.
an
hm
l
d T
C Scar
.
o
e.
so n
.
C HAP T ER
THE P ROPERTI ES
IX
OF L I Q U I DS
IQ U I DS ha v e no elastici ty f shape b t occupy a definite
v olume Consider a small area on the surface
of a solid body immersed in a liquid which is at
rest relative to
Th force exerted by the liquid on the area
must be equal and opposite to the force exerted by the area on
the liquid This force must be perpendicular to the area for if
not it would have a comp ent parallel to the surface which
would tend to produce a sheari g strain in the liquid and si ce a
liquid h
no elasticity of shape it can ot resist a force tending to
hear it without bei g set in motion Th force exerted by a liquid
when at rest any surface in co tact with it is therefore normal
to the surface ; the pressure on the surface is measured by the
force per uni t area If we c n ider y small area in the liquid
the the liquid on one side of it exerts a force on the liquid on the
other side This force als when the liquid is at re t must be
normal to the area Th force per unit area is called the pressure
in the liquid at the place where the small area is situated We
how that this pressure i the same i all directions at any
B
oint
in
the
liquid
nsider
a
ve
y
small
triangle
A
O
Fi
7
0
C
(
)
p
g
in a liquid at rest Draw AA BB d CC perpendicular to the
plane of the triangle d f equal le gths J i A B B G and
CA
Th liquid i side the prism
is in equilibrium
under the action of the f rce exe ted on it by the surr unding
liquid and its weight If we take the prism small enough its
weight can be neglected f its weight is pr portional to its
v olume while the f ces
i t due to the surroundi g liquid are
proporti nal t the areas of i t surfaces Th forces
the
surfaces of the prism must theref re be i equilibrium and they
are normal to the urfaces on which they act and prop rtional t
L
O
u
,
a
.
a
e
.
a
a
a,
.
on
n
n
,
n
as
S
n
e
.
n
on
an
s
o
.
,
n
o
.
s
e
.
.
ca n
S
n
s
r
o
.
.
’
’
an
.
an
’
n
o
’
o n
.
’
’
’
,
’
e
.
n
o
o
r
s
.
,
,
or
o
o
on
o
or
n
,
s
.
o
s
,
on
e
n
o
o
I
E
110
M C HAN C S AN D
P ROPE RTI E S
A
R
P
I
T
[
E
O F M ATT R
d the opposite force on A B C
the areas Th e force
A BC
must bala ce each ther because the other th ee forces on ABB A
’
d
A A C C and B B C C are perpe dicula t the forces on A B C
Th e three fo ces
A BB A A A C C and BB C C must
theref re be i equilib ium B t they are perpendicular to the
th ee sides f the t ia gle AB C
that if this tria gle we e
turned through a right a gle
its own pla e its sides would
be parallel to the th ee f ce B t when three forces in
an
on
.
’
’
’
’
n
on
r
n
o
’
u
.
n
SO
,
n
In
n
or
r
’
’
,
n
r
O
r
’
’
’
r
an
o
r
’
’
r
o
n
’
s
r
,
u
'
.
Fi g 7 0
.
.
equilibrium are parallel to the ides of a triangle then the
sides of the tria gle are proportional to the forces B t th
’
sides f AB C are proportio al to the areas of ABB A A A C C
d BB C C so that the forces
the e areas must be proportional
to the e
Th force per u it area or the pressure is therefore
the same o each of the three areas A BB A AA C C and
be supposed placed i any
BB C C
S i ce the small prism
position it follows that the p essure at a poi t in a liquid is
the same i all directi ns
be illust ated i the foll wing way Th top of
This fact
a small cyli drical metal b AB ( Fi g 7 1 ) is cl sed by a thin
rubber sheet To the middle f this sheet a wire K is fastened
which turns a balanced lever EF G about a bearing at F Th
end f the le v er moves over a graduated cale CD which is
supported by an m attached t the box If this apparatus
is immersed in water th pressure of the water on the rubber
S
,
n
’
n
O
’
an
’
as
on
,
,
n
’
'
,
,
can
n
.
’
’
'
’
,
s
n
e
.
’
’
,
,
ar
n
n
r
,
n
o
.
r
c an
e
u
.
n
o
ox
n
.
.
e
o
.
O
.
.
o
s
o
ar
,
e
.
e
I
E
112
M C HAN C S AN D
PRO PE RTI ES
P
I
A
R
T
[
E
O F M ATT R
to all parts of the liquid This is known as Pascal s principle
after its disc verer This property of liquids is made use of
in a machine called the hydraulic press by mea s of which
very large forces can be btai ed
Fi g 7 2 shows a hydraulic press
P is a cyli drical piston
which can slide up and down in a cylinder CC Oi l or some other
liquid is forc ed into the cyli der through a pipe S If the pressure
i the oil is p and the area of cross ection of the piston P is
then the upward force on the pisto is p
This force dri v es the
piston up and any body A placed on the top of the piston is
’
.
o
.
n
,
o
n
.
.
n
.
.
n
.
n
s
a,
n
a
.
,
d a plate F supported by stro g
squeezed between the pisto
bars
i l is pumped at high pressure into the cylinder C C
Th e
by means of a pump with a small pisto Q which can be pressed
d w by a lever LMN V and V are tw valves consisting
of metal co es which fit
to conical surfaces and are held down
by spri gs as shown Whe Q is moved dow the v alve V rises
t
d allows the i l to pass i to the cyli der CC but it does
allow y i l to flow back Whe Q is raised the valve V rises
and allo ws oil to flow i from a tank T but it does not allow y
to flow back i to T when Q i pressed d wn Thus when the
pisto Q is worked up and down the oil is f rced fr m the tank
n
an
n
.
O
n
’
o
n
on
n
n
o
an
n
n
.
an
n
o
n o
,
’
n
.
n
,
n
n
n
o
.
an
,
o
s
,
.
o
o
CH I x
-
.
]
P ROPERTI ES
THE
OE
L IQ U I D S
113
into the cylinder CC a n d the piston P rises If a cock B is
opened the oil flows ba ck into the tank through a pipe BD
Th e oil is prevented from
d the piston P moves down
escaping round the S ides of P by a leather ring RR of U shaped
cro ss section Th e pressure of the Oi l forces this leather ring
against the sides of P so that the oil cannot escape Le t the
downward force on the lever at L be denoted by F Le t N L d
and N M d Th e for ce on the top of the piston Q is therefore
Ed/d
Le t the area of cross section of Q be a then the pressure
is
given
by
in
the
oil
due
to
the
piston
Q
p
.
an
.
-
.
.
.
'
.
’
’
.
,
Ed
P
a
i
!
d
upward fo r ce exerted by the piston P is therefore equal
to
can easily be made much larger than
and
d than d so that this force may be enormously greater than
example suppose P is two feet i diameter and Q
F
Fo
one inch in diameter Then
Th e
a
a
’
’
,
r
.
n
,
.
a
a
2
4
( )
’
Also
2
— o 76
.
suppo se d is ten times d i Then the force exerted by
the large piston is 5 7 60 times the force F applied at L In
this case a fo rc e equal to 1 00 po u nds weight at L gives a force
of
pounds weight on the body A or about 26 0 tons
weight Various types of hydraulic presses are much used in
modern engineering prac tice and presses capable of exerting a
forc e equal to
tons weight are in use Large masses of
steel are pressed while red hot into the desired shape by means
of the hyd aulic press This method is found much superior
to the old plan of using a steam hammer
Th e free surface of a liquid at rest in a vessel lies in a
horizontal plane T prove this consider a p rticle
W
of the liquid at the surface Th e weight of the
pa ticle is directed downwards so that if the surface were not
horizontal the weight would have a compo ent parallel t the
surface which would set the liquid in motion f a liquid cannot
resist a ta ge tial force Th e surface therefore must be hori
o ta l when the liquid is at rest
Th e air or atmosphere exerts a
'
.
,
.
,
.
r
.
.
pm
s
di fi e n
m
n
c
a
t
t d
.
‘h
S
o
a
'
.
r
,
n
,
n
z
n
n
.
.
W
.
P
.
o
or
N I CS
E
114
M CHA
P RO PE RTI E S
AN D
P
I
A
T
R
[
E
OF M ATT R
pressure on all bodies in contact with it which is equal to about
1 5 pounds weight on the square inch or one million dyne s
per sq cm Th e press re at the free surface of a liquid when
exposed to the air is equal to the
A
B
atmospheric pressure Le t us now
consider the pressure i a liquid with
a free surface e xposed to the i at
di fferent depths below this surface
Le t 6 denote the p ressure of the air
In Fi g 7 3 let EF be a vessel o
taini g any liquid Imagine c yli de
A B CD with its S ides vertical and one
end AB at the free surface of the liquid
and the other at a depth h be low the
free surface Th e liquid in this cylinder
is supported by the forces exerted on it
Fi g 7 3
by the surrounding liquid Th e forces
on the sides of the cylinder are horizontal and so do not help
to support the weight of the cylinder Le t the horizontal cross
Then there is a downward force
section of the cylinder be
on the top of it due to th e pressure of the air equal to b a
the bottom of it equal to p a
d there is an upward force o
where p is the pressure in the liquid at the depth h
If w is the weigh t of the cylinder we have therefore
w pa
b
,
.
u
.
.
n
a r
,
.
.
“
.
c
,
n
a
.
n
n
'
r
.
.
.
.
‘
.
a
.
,
an
n
,
.
,
a
w
(p
b)
a
.
the mass of unit volume of the liquid be p so that the weight
of unit volume in dynamical units of force is pg Then since the
volume of the cylinder is h we have i t) h pg ;
hence
h pg
b
(p
)
h
or
b
p
pg
substance
is
called
its
density
Th e mass of unit volume of
y
It appears therefore that the di fference between the pressure t
a point in a liquid at rest and the pressure at its free surface is
proportional to the depth of the point and to the density of the
liquid Th e pressure is therefore the same at all points in a n y
Le t
,
.
a,
a
a,
a
.
an
.
a
.
E
116
N I CS
M CHA
AND
P ROPE RTI E S
A
R
T
P
I
[
E
O F M ATT R
thing is tried with the vessels B and C it is found that the depths
at which the lever tips are the same with all three vessels although
th e weight of water in A is much greater than that in C and the
weight of water in B is much less than that i n C Th e forces
exerted on the water in A by the sloping sides of the vessel have
upward components which help to support the weight of the water ;
so that the pressure on the piston is the same as when the vessel
,
,
.
Fi g 7 5
.
.
is used In the same way th e forces exerted by the sides of B
have downward components which make the force on the piston
greater than the weight of the water in the ve ssel
When a solid body is immersed in a liquid the forces exerted
on it by the liquid have a resultant which is
o
b d
i
d
directed upwards so tha t they tend to move it
upwards If this upward force is greater than the weight of the
body the body moves up and floats but if the weight is greater
then the body sinks
C
.
.
,
F
rc e
m m
o n
e rs e
o
y
,
'
,
.
,
,
.
CH I x
.
]
P ROPE RTI ES
THE
L IQUI D S
OF
117
be a solid body of any S hape immersed in
’
a liquid with its free surface at RS Take a small area AA equal
to on the free surface and de s cribe
a cylinder AB CG B A with vertical
sides and cross section a Le t this
cy linder cut the surface of the body
at BB and CC Le t the area of BB
"
Le t the
be and that of CC b e
angle between a normal to the area
BB and the vertical length of the
’
cylinder be 9 and that between the
normal to CC and the vertical be
Th e pres
h and A C h
Le t A B
sure a t B i s h pg + b where p is the
density of the liquid and b the atmo
spheric pressu re Th e force on the area
b
and
the
h
BB is there fore
)
( pg
compo nent of this vertically down
wards is ( h pg b ) cos
In the
same way the component of the force
"
’
on CC vertically upwards is ( h pg b ) cos
Bu t a c o s 9
a
"
a so that the upward force on the body due to the
cos
a d a
forces on the two areas BB and CC i s
Le t P Q
F
i
7
6
( g
)
.
.
a
,
’
’
.
’
’
'
.
a
’
’
a
.
’
’
’
”
.
'
,
.
’
’
a
a
’
’
’
’
a
’
”
n
’
a
(
"
P9
b)
’
b)
a
a
h
pe (
"
h
’
)
volume of the cylinder A CC A inside the body between
B and C is equal to a ( h — h ) so that a pg ( h — h ) is equal to
the weight in dynami cal u nits of force of a volume of the liquid
equal to the volume of the cylinder inside the body Th e whole
of the free surface of the liquid above the body may be supposed
di vided into small are as like and correspo nding to each small
area there is an upward force on the body equal to the weight
of a volume of the liquid equal to the volume of the body
vertica lly below the small area Thus we see that the total
u pwar d force on the body i s equal to the weight of a volume
of the liquid equal to the whole volume of the bo dy If the
volume of the body is V then the upward force on it due to the
liquid is equal to Vpg
’
’
Th e
”
’
’
,
,
.
a
,
.
.
,
.
N I CS
E
118
M CHA
If
P ROPE RTI E S
AN D
P
I
A
R
T
[
E
O F M ATT R
'
mass of the body is m then the r e sultant force on it
is equal to Vpg — m g upwards If the density of the body i s
e sultant upward force F on it is
then
so
that
the
r
m
V
p
p
Thus if p p the body
given by F Vpg Vp g = Vg ( p
will be in equilibrium and will not move either up or down If
is
greater
than
it
will
sink
to
the
bottom
and
if
is
greater
p
p
p
than p it will rise to the surface and float
S O far we have only considered the vertical force on the body
It can easily be S hown that there is no resultant horizontal force
on it for if a horizontal cyl i nder is drawn through it like the
’
vertical cylinder A CC A since the pressure is the same in the
horizontal cylinder on both sid e s of the body we see that the hori
o ta l compon ents Of the forces on the two sides are equal and
opposite
If we imagine the body P Q replaced by an equ al volu m e of
the liquid then we see at once that the forces on it are j ust those
required to support this equal volume of the liquid so that the
resultant force on the body due to the liquid is evidently equal
and opposite to the weight of an equal volume of the liquid Th e
line of acti on of the resultant force on the body therefore passes
through th e c en t e of mass of a body of uniform density occupyi ng
the same space as the body I f the body is not of uniform density
its centre Of mass may not be on the line of action of the resulta nt
force on it due to the liquid In this case it will tend to turn
round u ntil its centre of mass does lie on the line of action of
the resultant;
Th e upward force on a body immersed in a l iquid may be
measured by suspending it from one of the pans of a balance
by a fine wire and allowing it to hang in a vessel of the liquid If
its weight is w and the weight required to balance it when com
l
e te l immersed is w then w
w
is
the
upward
force
on
it
due
p
y
to the liquid We have therefore
w LU Vpg
where p is the density of the liquid and V the volume of the
body Also w Vp g where p is the density of the body Hence
th e
,
.
’
’
:
,
’
’
.
.
'
,
’
.
.
,
’
,
,
z
n
.
.
.
,
,
.
r
.
’
.
.
’
’
,
.
”
,
:
’
’
.
,
.
—
w W
p
w
p
h
’
is
N I CS
E
1 20
M C HA
’
AA
At
.
therefore
therefore
a point
h + (h
dpg
,
AND
P
P ROPE RTI E S
at a distance
’
Th e
where
p
PB
.
a
from
B th e
force on unit area
is the density of
Fi g 7 7
T
P
A
R
I
[
E
O F MATT R
th e
liquid
d epth
at P
is
.
.
Divide
the area AB into a large number of n arrow strips
of equal width by parallel straight lines drawn across it p e pe
Le t these strips be numbered
di c la to the plane of the paper
starting at B Th e area of each strip is
1 2 3 4 etc up to
n
r
r
u
,
,
,
lb
.
Th e
.
,
Th e
77,
n
n
,
force on the
.
,
l 0th
strip is
total force on all the strips is therefore
lb pg
lb pg
since
n
is very large
h +
,
h +h
’
2
Thus
the force is the same as if the depth all over the area was
equal to the average depth Th e force of course is perpendicular
to the area To find the point at which the resultant acts we
.
‘
.
CH i x
.
]
THE
P RO PERTI E S
L IQUI DS
OF
1 21
serve in the first place that it i nust lie on a l i ne parallel to AB
down the middle of the area Also since all the forces are
parallel the moment of the resultant about any l i ne must be the
same as the total moment of all the forces
Divide the area into strips as before
Th e moment of the
force on the l 0th strip about the tOp side of the area is
Ob
.
.
.
10
h + (R — h )
I
7
bl
7.
pg
Z
X
;
10
total moment about this line is therefore
Th e
1
—
l
'
n
2
l bpg
2
h
-
l
h
g
-
j f,
here x denotes the distance Of the point of action of the resultant
from the top side of the area Hence we get
w
.
,
This
reduces to
a:
i
h
2h
3
h
surfac e of the liquid then
,
h
A+
(
’
h — h
2
3
i
’
’
J
h =0
If the top of the area I s at the
and
IrS =
g
l
.
Also
if the area
g
is horizontal so that h h we get
so that then the resultant
acts at the middle point Of the area
S ince the pressure due to a column of liquid of height h is
equal
to
h pg where p is the density of the liquid
M
t
a column of liquid in a glass tube is often used
as a means of measuring pressures S uch an arrangement is
called a manometer A simple form of manometer is shown in
Fi g 7 8
This consists of a glass tube bent as show and about
half full of liquid Mercury is Often used in manometers because
it gives Off very little vapour i t does not wet the glass and the
position Of its surface c be easily seen A graduated scale
is fixed alongside the manome ter tube which should be vertical
I f the press re of the gas in A is greater than the atmospheric
’
a
.
a n o
m
,
,
e
er
.
.
.
.
n
.
.
,
,
an
.
,
u
.
I
E
1 22
P ROPERTI ES
M CHAN CS AN D
P
I
A
T
R
[
E
M ATT R
OF
pressure in the open end B then the liquid sta nds higher in B
tha n in A If the di fference of level is h then the difference
between the p essure in A and the pressure i B is equal to
h pg
If the end of the tube at B is closed and there is no gas
in th e t u be B C above th e liquid
then the pressu re there is zero so
that the pressure in A is then
equal to h pg
In practical work a pressure is
often recorded by stating the height
of a column of liquid which pro
duces an equal press ure Fo ex
ample we may record a pressure
as equal to 7 6 0 mms of mercury
P ressures stated in this way can
be red u ced to dynes per sq cm
by multiplying by pg p bei n g the
density of the liquid in grams
per a c and g the acceleration of
gra v ity at the place where the pres
F
i
g
sure was measu ed This method
of rec rding pressures is often convenient but it has the dis
advantage that the pressure due to a column of liquid of given
height depends on the temperature of the liquid and on the value
of g at the plac e Unless the temperature and the value of g are
given as well as the height of the column of liquid it s mpossible
to find out the exac t value of the pre ssure
Th e pressure of the atmosphere is usually meas ured by means
of
a
special
type
of
mercury
manometer
called
a
t
barometer A simple form of barometer is shown
in Fi g 7 9 It consists of a glass tube A CBED having wide parts
A C and ED about 2 e m s in diameter j oined by a narrow tube
Th e par ts A C and DE are in the same straight line
OBE
Th e
tube is closed at A and open at D Th e tube is filled with pure
dry mercury which is boiled in it to expel all air When it is
pl a ced in a vertical position the mercury in A C falls le a v mg
a vacuum above it Th e di fference of level between the surface
of the mercury in A C and the surface in DE is read o ff on a
,
.
,
r
n
.
.
,
,
.
r
.
'
,
.
.
.
.
,
.
r
.
.
o
,
.
I
,
I
.
Ba
ro
m
e er
.
.
.
.
,
,
.
,
.
.
.
.
,
,
.
.
'
E
N I CS
M CHA
Su
AN D
b
PROPE RTI E S
A
R
T
P
I
[
E
O E M ATT R
t
s an c e
viscosity of most liquids gets less as their temperature rises
F
example the viscosity of water at 1 5 C is 00 1 1 5 at 3 0 C
it is 00 08 0 and at 45 C it is 00 060
If a liquid is flowing at a constant rate along a pipe of uniform
diameter there is a resistance to the flow due to the V s c o s i ty
of the liquid In consequence of this the pressure in the liquid
falls as we pass along the pipe in the direction of the flow Th e
difference between the pressures at two points in the pipe multi
plied by the area o f cross section of the pipe is the force driving
the liquid between the two points and is balanced by the viscous
resistance
When the velocity with which a stream of liquid moves is
changing
so
that
the
liquid
has
an
acceleration
i
St
then force is required to produce the acceleration
in the same way as for any other body F example suppose
a steady stream of liquid is flowing through a pipe AB ( Fi g 8 0)
Th e
.
°
°
or
.
,
.
°
.
.
I
,
.
.
.
p
res s u r e
rea
m
n
a
,
,
'
or
.
,
.
Fi 0
o' .
80
.
having a greater diameter between C and D as shown In going
from A to P the velocity falls since the tube gets wider so there
must be a force acting on the liquid in the direction from P
to A Th e pressur e in the liquid at P must therefore be greater
than the pressure at A In the same way the velocity at B
is greater tha n at P ; so that the pressure at P must be greater
than at B Thus it appears that in a stream of liquid the pressure
is greater at places where the velocity is smaller
,
.
,
.
.
.
.
CH I x
.
]
THE
P RO PE RTI ES
OF
L IQ U I DS
1 25
If a liquid issues from an apertu r e in the side of a vessel
c ontaini ng it then if the pressure inside the vessel
near the aperture is grea te r than the atmospheric
press ure by an amount p the work done on the liquid by this
excess of pressu re is p V where V is the volume of liqui d which
comes out If the velocity of the issuing j et of liquid is r then
its kine tic energy is my where m is its mass Th e work done
on it is nearly equal to i ts kinetic energy so that we have
,
,
,
.
,
”
.
,
,
V
p
Also
m
Vp
,
where
z
.
is
the
density
of
the
liquid
so
that
p
,
P
If the pressure
S O that
m
u
fi
‘
l
’
g
v
p
i
is due to a depth of liquid
p
h
,
then
h pg
p
,
h pg
V2g h
U
or
I f a body falls f eely from a height h its velocity at the bottom
i s equal to V2g h ; so that it appears that whe a liquid issues from
ape rture in the side of a vessel at a depth h below the free
an
surface in the vessel the velocity of the issuing j et of liquid is
near ly equal to the velocity acquired by a body in falli ng freely
a height h S ome of the work done on the liquid as it comes out
is used up in overcoming the viscous resistance to the motion
of the liquid so that the velocity is always less than Mfgh If
a very Visco us liquid is used the velocity may be very small but
with water issuing from a fairly large hole it is ne rly equal to
ME Le t B C ( Fi g 8 1 ) be a vessel of water with a hole at A
Then the water comes out at A with a horizontal
Le t F A = h
velocity MEI Le t AD = d and DE = l L t t be the time a
particle of the water takes to fall from A to E Then since the
initial velocity at A is horizontal the time t is equal to the time
a body would take to fall vertically from A to D starting from
.
r
n
.
.
,
a
.
.
.
.
—
i
e
.
.
.
,
Also l = V2g h t S ince while the water
est Hence t
is falling it moves horizontally with velocity v2g h He c e
i
r
.
,
.
n
I
E
1 26
M C HAN CS AN D
P ROPE RTI ES
P
I
AR
T
[
E
OF M ATT R
If A is half way between E and D S O that d h we get l = 2h
It is easy to measure the distances h d and l so that the formula
2\/clh can be verifi ed experimentally
l
If the area of the aperture from which the liquid issues is a then
i t is found that the volume comi g out in unit time s less than
ME is the velocity of the j et This is due to the
where
c
cross section of the j et being less than that of the aperture Close
to the aperture the liquid moves towards the axis of the aperture
so that the diameter of the j et dimini shes and becomes less than
that of the aperture
If a j et Of water strikes against a body it exerts a force on the
body If the body moves owing to the action of this force the
-
,
,
.
,
_
.
,
n
I
a,
.
.
.
.
,
,
water does work on it and so imparts to the body some of its
kinetic energy Th e greatest possible amount of work is done
on the body by the water when the water is red ced to rest Th e
force exert e d by a j et of water is made use of in a form fo f water
wheel called Pelton s wheel shown i Fi g 8 2 Th water issuin g
at A impinges on a series of buckets forming the wheel which are
designed so as to reduce the water early to rest when the wheel
is rotating at the { proper speed If the pressure of the water
supply to the wheel is due to a depth h feet and pounds of
water enter the wheel per minute then the theoretically possible
.
u
’
n
,
.
.
.
e
n
.
n
,
horse power of the wheel is equal to
-
n
h
i ce one horse power
S n
-
N I CS
E
1 28
M CHA
AND
P ROPE RTI ES
P
AR
T
I
[
E
O F M ATT R
These
facts suggest that the surface of a liquid behaves as
if it were covered by an elastic skin which tends to contract
S uch phenomena are said to be due to surface tension
Th e
surface of any liquid is in a state of tension and tends to contract
so as to make the surface as small as possible S ome liquids
like soap solution can exist in thin film s which tend to contract
S oap bubbles are a beautiful example of this
Th e force per unit
length which the surface of a liquid can exert is taken as a
measure of its surface tension A water surface in air tends to
contract with a force of about 7 5 dynes pe r centimetre Owing
to the tension in a thin film of soap solution there i s an excess of
pressure in a soap bubble over that outside
8 4 shows
.
.
,
.
,
,
.
.
.
.
.
Fi g 8 4
.
.
apparatus with which this excess of pressure can be measured
It consists of a T shaped glass tube bent as shown so that the
part DEF forms a small water manometer A piece of india
rubber tubing is fastened to the branch C and its end is closed by
a glass rod H Th e narrow neck at B which should be not more
than 0 2 cm wide is dipped into a little soap solution and then
by slightly pushing in the rod H a small soap bubble a few mms
in diameter is blown at B Th e di ffere ce between the water levels
in the manometer and also the diameter of the bubble are
measured It is found that the pressure in the bubble is inversely
proportional to its diameter
Le t A B C ( Fi g 8 5 ) be a soap bubble and let A B be an
.
-
,
.
.
.
,
,
,
.
n
.
.
.
.
'
CH
.
]
Ix
THE
P RO PE RTI E S
LI Q UI DS
OF
1 29
imagin ry plane passing through its cent e L t its radius be
and let the di fference betwee the p essure inside the bubble
and that outside be denoted by p If the surface te n sio f the
soap solution is T then each cm of the thin film which has two
surfaces exerts a force 2T Th e two hal ves of the bubble on either
ide of the pla e A B are theref re pulled together by a force
the
This force is balanced by the pressure p acti g
2T x 2
area e that
4
or
T
v
p
p
r
a
r
e
r
n
,
.
n
.
o
,
.
.
,
o
n
S
7 rr
n
.
on
SO
7
‘
7r r
rr
If we get p i dynes per sq i and in cms this formula gives T
in dy es per cm F soap solution T is about 25 dynes per cm
n
.
n
This
0 25
r
.
r
.
or
.
Thus I f
c n
.
cm we get p
.
4
25
5
0 25
400
dynes per sq cm
.
.
would give a di fference of level in the water manometer of
0 408
cm
.
Fi g 8 5
.
Fi g
.
Another
.
86
.
way of getting T for a soap film is with the apparatus
show in Fi g 8 6 Th i s consists of tw pieces o f thin aluminium
wire AB and CD bent as shown and connected by two fin e silk
threads A EC and BED of equal le gth If this arrangement
w P
9
n
.
o
.
n
.
.
.
I
E
1 30
PRO PE RTI E S
M C HAN CS AN D
P
T
I
R
A
[
E
O F M ATT R
is held by the ring G dipped into soap solution and then with
drawn
soap film is formed over the area A B CD This film
pulls in the threads so that they form arcs of circles as show
Also let A C = BD = h and let EF = d
Le t A B = CD b
Th wire CD is
L t the mass of the wire CD be m grams
supported by the surface tensio of the film and by the tensions
in the two threads A EC d B ED Th e weight of the soap film
can be neglected so that consideri g the forces across a horizontal
line at EF we see that
,
a
,
.
n
:
.
.
.
e
e
.
n
an
.
n
,
2Td,
22
5
mg
where t is the te sion in each thread If we consider a complete
circle of thread surrou ded by a soap film it is easy to see that
the tension in the thread is given by the equation
n
.
n
,
t = 2r T,
where is the radius of the circle It can a lso easily be show
from the geometry Of the circle that the radius of the arcs AEC
and B ED is given by the equation
r
h
SO
n
.
(b
2
d)
4 (b
d)
h + b
— al
2
that we have
2
m g = 21
,
2(
b
2
means Of this formula T can be found h b d and m are
measured
Th e height h to which a liquid rises in a arrow tube which
it wets can easily be calculated At A
the
pressure
is
less
than
8
7
F
i
)
( g
atmospheric by an amount h pg where
h is the height of A abo v e the f ee
surface of the liquid at B Also i ce
the surface of the liquid in the tube is
spherical the pressure due to its surface
tension is equal to 2T/ where is the
Fi g 8 7
radius Of the tube Hence h pg = 2T/
or h = 2T/pg Th e pressure due to the surface tension is only
half that in a soap bubble of radius r because the soap film has
two surfaces Th value of h can also be obtained by equating
By
,
,
.
n
.
.
,
r
S n
.
,
r
r
.
.
r
r
.
,
.
e
.
I
E
132
M CHAN CS AN D
PROPE RTI E S
P
R
T
I
A
[
E
O F M ATT R
the solution Th e potassium perma ganate has an intense purp le
colour and it is found that the colour gradually rises through th
water until after a long time the whole of the water i the j ar
becomes e qually coloured
Th molecules of a liquid are supposed to be in motio rela
particular
molecule
moves
about
ti ve ly to each other so that
y
in the liquid and so in the course of a very long time will have
been as long in one part of the vessel co tai ing the liquid s
in any other part Th e molecules of substa ces like sugar or
potassium perma ganate also mo v e about in the liquid i the
same way so that they gradually distribute the m selves equally
th oughout the volume of the liquid This process by which one
substance penetrates into another is called diffusion In liquids
it is a very slow process
By stirri g up the liquid the u ifor m distribution f a sub
stance dissolved in part of it can be brought about in a few
seconds
n
.
e
,
n
.
e
n
an
,
n
n
.
n
a
n
,
n
,
,
r
.
.
.
n
.
n
O
P AR T I I
HEAT
C HAP T ER I
TEMPERATURE
words hot warm cold cool are used to indicate different
states
of
bodies
with
which
everyone
is
more
T
t
less familiar If a cold body is put near a fire
it gradually becomes warm and then hot and if put in the fire
it may become red hot or even white hot Thus a body can pass
from a state i which it is cold to a state in which it is white hot
by a conti uous process so that there is an indefinitely large
number of possible states intermediate between a cold state and
a hot state Th e temperature Of a body is a quantity which
indicates how hot or how cold the body is I order to deal
with temperatures scientifically it is necessary to adopt some
measurable property of a body as a measure of its temperature
n d to define a unit of temperature
It is found that bodies expand or get larger when they are
made hotter and that liquids usually expand more
than solids Fo example if some mercury is put
in a bulb blown o n the end of a glass tube of
small bore so that the mercury completely fills the bulb and a part
of the tube then it is found that as this a rangement gets b e tte
the mercury increases in volume so that it occupies a greater part
of the tube S uch a bulb and tube containing mercury is called
a mercury i glass thermometer Th e position of the end of the
mercury column i the tube is taken as
indication of the tem
t
u e of the thermometer
We
may
suppose
a
scale
of
equal
e
p
THE
,
,
,
or
em
pe ra
u re
.
.
,
.
n
n
,
.
.
n
,
a
.
r
.
,
r
,
.
-
n
-
.
n
ra
r
an
.
r
A
T
P
R
II
[
E
1 34
H AT
lengths marked on the tube so that the position of the mercury
can be read ff on the scale If the bulb is put in contact with
hot body then the mercury rises to a certain point and if it is then
put in contact with a cold body it falls If the hot body d th e
cold body are put in contact it is found that the hot body get s
colder d the cold body gets b tte and after a time the mercury
in the thermometer is found to sta d at the same level when th
bulb is in contact with either body If a number of bodies some
hot and some cold are put in a wooden box lined with copper d
left there for a time then the thermometer will give the same
indication when in contact with any of the bodies in the box
We co clude then that bodies in co tact with each other when
protected from any disturbi g cause which migh t te d to make
some of them hotter than the others all take up equal tem
i
s
When
a
thermometer
bulb
is
immersed
in
a
liquid
t
es
t
p
temperature soon becomes the same as that of the liquid d
when the b u lb is put into a hole bored in a solid body it takes up
the temperature of the body A thermometer therefore serves to
indicate not only its own temperature but also the temperature of
any body with which it is in sufficiently intimate contact
If a thermometer is put into a mixture of crushed ice and
pure water it is found that the mercury stands at a certain
le v el which remains constant so long as there is plenty of i c e
mixed with the water Th e indication of the thermometer doe s
not change if the mixture is taken from a cold p lace to a warm
place although this causes the ice to begin to melt If the same
thermometer is put into mixtures of ice and water m d e up in
di fferent places and at di fferent times it is found always to
i dicate the same temperature We conclude therefore t h at the
temperature of a mixture of ice and water is a co stant definit
temperature This temperature is called the melti g point of
ice In the same way if a thermometer is put i n to the steam
risi g from boiling water at a pressure of 7 6 cms of mercury then
it always indicates the same temperature This temperature i s
called the boili g point of water at 7 6 cms of mercury pressure
In order to establish a scale of temperature it is necessary t
devise some way of graduati g thermometers so that they will
all agree in their i dications Th e plan adopted is to make u s e
o
a
.
,
,
an
.
r,
e
an
n
e
.
,
an
,
,
.
n
n
,
n
n
,
er a
ur
.
,
an
.
.
,
.
.
,
a
,
n
.
n
e
n
.
.
n
.
,
.
n
.
.
o
n
n
.
136
E
H AT
A
P
[ RT II
I]
OH
.
E PE RATU RE
137
T M
undertake such comp risons A therm ometer c a be sent to the
Bureau of S tanda ds and it is there compar ed with a standard
thermometer d a table giving the di ffere ces between its indi
c ations and those Of the standard is made and sent back with it
In this way the small di ffere ces between di fferent thermometers
be eliminated We shall see later that it is possible to define
ca
a scale of tempe rature which does not depend on the properties of
particular substan ces like glass and mercury This scale is called
the absolute scale of temperature It is found that the scale of
temperature given by a goo d mercury i n glass thermometer agrees
very e rly with the absolute scale between 0 C and 1 00 C
Mercury freezes at
so that a mercur y in glas s ther
m o m e te cannot be used below this temperature
Thermometers
c ontaining alcohol instead of mercury can be used do wn to about
1 1 2 C and thermometers containing pentane down to — 200 C
M ercury i n glass thermometers cannot be used much abo v e
3 00 C but mercury in fused quartz thermometers can be used up
to 45 0 C Th e tubes of ther m ometers for use at high tempera
tures are filled with n i trogen gas the pressure of which prevents
the mercury from boiling Temperatures above 3 00 C are usually
not measured with mercury thermometers but by other methods
some of which will be described in later chapters
Th
following are some important temperatures on the
c entigrade scale
L w t p
ib l t m p t
B ili g p i t f li q i d h y d g
B ili g p i t f li q i d y g
M lti g p i t f m
y
M lti g p i t f i
B ili g p i t f w t
B ili g p i t f l p h
M lt i g p i t f g l d
M lti g p i t f p l ti m
a
r
n
.
,
n
an
.
n
n
.
.
.
-
-
a
n
°
°
.
.
-
r
-
.
°
°
.
,
.
-
-
°
.
,
°
.
,
°
.
.
,
.
e
:
o
es
o ss
o
n
o n
o
u
o
n
o n
o
u
e
e
n
o n
o
e
n
o n
o
o
n
o n
o
o
n
o n
o
e
er a
o n
o
e
n
o n
o
ox
en
en
ce
er
su
n
ro
erc u r
a
e
ure
ur
o
a
n u
F
N
RE ERE CES
Th eo ry
H
ea t
f
o
Hea t, Po y n
i
t
,
n
g
J Cl k M
d J J Th
an
.
er
.
-
.
l
a x we l
o m so n
.
.
C HAP T ER I I
THE EX PA
N S I ON
B ODI ES
OF S O L I D
WI TH
RI S E OF
TEM PERATU RE
is found that solid bodies get larger whe they are mad e
hotter Fo example an iron rod which is 1 00 e m long whe in
a mixture of ice and water becomes about 1 00 1 2 m s long whe
surrounded by steam from boiling water It is found that the
increase of length per unit le gth is approxi m ately proportional
to the rise in temperature Th s if l denotes the le gth of
body at t C and Z the length at t C then
IT
n
r
.
s
,
e
,
n
.
n
.
.
n
u
.
1
a
°
°
1
n
2
,
.
l
a l1
1
(t
.
,
t, )
2
or
l = l ll
(t
where a is a constant a is equal to the increase in length of
unit length for one degree rise in temperature ; it is called the
coe fficient Of linear expa sio
Th e expansion of rods can be
measured with the apparatus shown i Fi g 2 A d A B about
(1
l
2
2
.
n
n
.
n
.
.
re
Fi g 2
.
.
cms long and 05 e m in diameter is contained in a brass tub e
CD about 2 e m s in diameter and a few millimetres shorter tha
the rod Th e ends of the tube are closed by short corks through
which the ends of the rod proj ect Th e temperature of the rod is
50
.
.
n
.
.
,
.
1 40
II
P
R
A
T
[
H EAT
coincide with the tra sverse lines o the bar as seen through the
microscopes Th other bar in its ta k is then brough t beneath
the microscopes and its positio adj usted until one of the transverse
li es on it appears to coincide with one of the cross hairs If the
other line also appears to coincide with the other o s hair then
the distance betwee the lines on the second bar is eq al to that
on the first bar If it does t appear to coi cide then the c
hair is moved until it does appear to coi cide and the dista ce it
has to b e moved through is measured with the micrometer screw
If a standard scale divided into millimetres is bserved with the
microscopes then it is easy to find how far their cross hairs have to
be moved to make them appear to go fr m one scale division to the
next o e Th e difference be tween the lengths of the two bars
corresponding to the dista ce the cross hair had to be moved can
then be calculated In this way the change in the lengt h of the
bar in the tank of water when the temperature of the water is
changed can be accurately determined Th e change in the length
can be found in thi way to less than 00 01 millimetre Th e length
of the bar in the ice and water can be found b y comparing it with
a standard metre also kept in a tank of ice and water or a standard
metre may be used instead of this bar Th e apparatus described
is called a comparator and is the most accurate instrument known
for finding the difference between two nearly equal lengths It is
always used for comparing the lengths of standards of length
Th e f llowing table gives the values of the coefficients of linear
expansion for bars made of several substa ces
0 0 000 1 9
B
n
n
e
.
n
n
n
.
cr
s
n
,
u
n o
.
n
r o ss
,
n
n
.
O
,
o
n
.
,
n
,
.
.
s
.
.
.
.
o
n
ra ss
I ro n
0 0 0001 2
l
a ss
0 0 00009
Co ppe r
0 0 0001 7
G
P
l ti
a
n u
0 0 0 0009
m
00 00000 7
F dq t
t Ni
0 0 000009
Ni k l t l 3 6 p
A brass bar 1 00 m long at 0 C increases in le gth by 1 9 m m
when heated to 1 OO C while a bar of fused quartz of the same
length increases by only 00 7 mm It is found that di fferent
samples of the same solid subs tance d not all expand exactly
equally so that when it is necessary to know the coe fficient of
uar
u se
c
z
er
s ee
e
e
s
c en
.
°
n
.
.
°
.
.
o
,
s
.
c a n
]
ExP ANS I O N O F S O
L I D BO D I ES WI TH RI S E O F TE MPE RATU RE 1 41
exp ansion of a particular sample ex ac tly it h a s t be meas ur ed
Th e ex p nsion of so lids with rise of te mperature is o t exact ly
unifo m so that the formula
o
.
n
a
r
,
[s
Ai l +
—
f
”
( tz
a
~
—
ti )
l
is not exa tly true When t t is not more than say 1 00 C
this formula is su fficiently nearly true f most purpos e s Th e
v lue of is not qui te i dependent of the tempe ature
A more ex t formula is
°
c
g
.
,
,
,
.
or
.
n
a
a
,
r
.
ac
l, ( 1
0 15+
i
where l de o te s th e le gth t t C l that at 0
are const n t Fo platinum between 0 C a n d
°
n
n
,
a
.
.
,
0
°
s
a
a
.
r
.
0 0 00008 8 7
an
d B
C
°
.
,
an
d
1 000 C
a
an
d
B
°
.
00 0000000 1 3 24
.
Although
the expansion of solids with ri se of tempe ature is
small it is nevertheless of considerable practi c al importance in
engin eeri g and other b anches of appli ed science Large steel
st uctures lik e bridges have to be designed so that they
ex pand and contrac t free ly without dam ge and the rails of
lin es and railways have to be laid with short ga ps betwe en their
ends to allow room for exp nsion
Th e te mperature out of doo rs may v ary f om say — 3 0 C to
Th e change of length of a s teel strue
40 C a range of 7 0 C
ture 1 000 feet long when its temperature change s 7 0 C is ab out
1 0 inches
It may therefore be n e cessar y to allow fo possible
c h ange s in the lengths of ste el or iron structure s of about one inch
1
feet
00
e
p
Th e con tr ac tion of bodi e s on cooling is sometimes m ad e use
of fo fi in g things firmly to gether Fo example the iron tires
of wheels are made a little smaller than the wheel and then s lipped
In the manufac ture of glas s a ticles it is necess ry
on while hot
to cool them V ery slowly becaus e otherwi s e some parts cool quicker
than others so that the ar ticl es a re liable to be broken by the
nequal contrac ti on Ar ticles made of fus ed quar tz c a n be heated
and cooled in any manner without brea k i ng because the expans ion
of fu s ed quartz is so small
Th e changes in the length of the pendulum of a clock with
changes of te mpera ture ca u se its ti me of Vib ation to alte r so that
th e rate of th e clock varies wi th the tempe ature
Acc u r ate clocks
r
,
r
n
.
r
a
a
ea r
,
.
°
r
.
°
°
.
,
°
.
r
.
r
.
r
x
.
r
,
a
r
.
,
,
u
.
.
,
r
,
r
.
T
P
A
R
II
[
E
1 42
H AT
are therefore provided with pe dulums co structed so that their
le gths remai constant i spite of temperature changes S uch
pe dulums are called compensated pe dulums O e form of com
e s te d pendulum has a bob consisting of a glass vessel containing
p
mercu y When the temperature rises the pendulum gets longer
but at the same time the me cu y expands and so its level in the
vessel rises S i ce mercury expa ds more tha solids like brass
and iron it is possible to design the pe dulum so that the rise in
the level of the mercury compensates for the increase in the length
of the pendulum rod If the pendulum rod is made of fused
quartz or 3 6 / nickel steel its expansion is so small that it can
be neglected except in the case of clocks intended for very exact
measurements Wood has a rather small coefficient of expa si n
so it is often used for making the rods of clock pendulums If the
rod is made of a substa ce like wood which has a small coeffi cient
of expa sio then the pendulum can be compe sated by maki g
the bob of some material like brass which has a larger coefficient
Th e rod is passed through a hole in the h o b and the bob is sup
ported by nut screwed on the end of the rod below the bob
When
the
temperature
rises
the
rod
gets
longer
but
at
Fi
( g
the same time the bob expands so that its centre of gravity rises to
a greater height above the nut the rod By making the bob the
proper height the two e ffects
be made to compensate each other
Th time of vibration of the balance wheel of a watch is als
a ffected by changes of temperature A the temperature rises the
wheel gets larger and also the restoring couple due to the spiral
hair spring gets smaller Both these e ffects make the time Of
vibration longer Th e e ffect due to the change in the elasticity
of the hair spring is much the greater of the two These e ffects
are compensated by maki g the rim of the balance wheel i two
separate halves each supported by a poke at one end o ly ( Fi g
Each half of the rim is made of a strip of brass which is on the
outside with a strip of steel attached to it on the inside When
the temperature rises the brass expands more than the steel so that
the curvature of the rim is increased and the free ends of each half
This diminishes the time of vibration
m ove in towards the axle
and by properly designing the wheel this e ffect can be made to
compensate the increase in the time due to the other two effects
n
n
n
n
n
.
n
n
n
n
.
a
r
.
,
r
r
n
.
n
n
n
.
0
O
n
.
o
,
.
n
n
n
n
,
n
.
,
a
.
,
on
.
c an
.
e
o
s
.
.
.
.
n
n
n
S
,
.
,
.
,
,
.
,
.
1 44
A
P
I
I
R
T
[
H EAT
where i s the coefficient of li ear expansio of the block
volume at t C is therefore equal to
a
n
n
.
I ts
°
.
a o 60 00
S ince
Le t
a
1
(
is a small quantity this is early equal to
so that
and a b c
c
a tb t t
n
ao
bo c o ( l
S a t)
.
f
ut
o
f
ut
o
o
= v0 ( 1
3 1 t)
where 8 3
8 is called the coefficient of cubical expansion and is equal to thre e
times the coefficient of li ear expansion
If a brass sphere is made so that it can j ust pass through a
brass ring W hen both are cold then on heating the sphere it i s
found to be too large to go through the ring If the ri g is the
heated it becomes big enough to let through the hot sphere
f
ut
v0
1
(
Bt)
,
a
,
.
,
n
.
,
.
n
n
.
.
C HAP T ER III
WI TH
THE EX PANS I ON OF L I Q U I DS
RI S E OF TEMP ERATU RE
is found that liquid s expand when they get hotter Th e
fact that mercury expands more than glass is evident from the rise
of the mercury in a thermometer tube when the temperature of
the thermometer gets higher
Le t th e volume of a thermometer bulb and tube at t ( 1 up to
the freezing po i t mark be denoted by b and let b b ( 1
where b is the volume at 0 C and 8 the coe fficient of cubical
expansion of the glass Le t the volume of mercury at t C be
denoted by V Then since at 0 C the mercury j ust fills the bulb
and tube up to the freezing po i nt mark we have V 6 Le t then
V = b (l
f
where
is
the
coef
icient
of
cubical
expansion
of
t
)
y
7
the mercury Th volume of the mercury in the tube beyond the
freezi g point mark when the thermometer is at t C is then
given by
IT
.
.
°
,
n
t
t,
,
0
°
0
.
,
°
.
.
°
t
.
.
0
t
:
0.
f
0
,
e
.
°
n
.
,
Vt
—
l
—
B)
,
.
If 8 were equal to 7 then V b would be zero so that the
mercury would t rise in the tube at all Th e quantity y B is
called the coefficient of apparent expansion of mercury in glass
because the amount of mercury which passes the freezing point
mark is the same as it would be if the glass did not expand at all
and the mercury had a coe fficient of cubical expansion equal
to 7 8
Th e expansion of liquids can be easily measured by means of
an apparatus S imilar to a thermometer bulb and tube called a
di latometer O e form of dilatometer is shown in Fi g 5 It
consists imply of a glass bulb of about 5 c ca pacity blown on a
tube about 1 5 m S long and about 1 mm inte r nal diameter Th e
tube is graduated into millimetres Th e bulb and tube can be
10
wP
t
,
—
t
,
n o
,
,
.
n
.
.
S
.
.
.
.
.
c
c
.
r
.
.
E
T
A
R
P
I
I
[
H AT
1 46
easily filled with any liquid by means of a glass funnel with a long
eck only 0 5 mm in diameter made by drawi g out a glass tube
in a blowpipe flame S uch a funnel is S hown in Fi g 6 To empty
the dilatometer the funnel is connected to a vac um pump and the
liquid drawn out by means of it Th e volume of the dilatometer
n
n
.
.
.
.
u
.
Fi g 6
.
.
at 0 C up to any division on the tube can be found by filling it
with mercury at 0 C and getting the weight of the mercury
of mercury at 0 C weighs
grams If a liquid is
On e
put in the dilatometer its apparent volume at any temperature is
equal to the known volume at 0 C of the portion of the dilato
meter which it occupies Th e true volume can be calculated from
V (1
t h e volume at 0 C by means of the equation V
°
.
°
.
.
°
.
.
°
.
.
°
.
t
:
O
E
1 48
P
1
1
A
R
T
[
H AT
of the mercury in AB 6 that of the mercury in
h be the height of the column of
mercu y in AB and h that of the
column of mercury in CD Then since
the two columns of mercury balance
each other the pressures due to them
must be equal so that
where p is the density of the mercury
in AB and p that of the mercury in
CD
We have therefore
where v and v denote the volumes of
one gram of mercury at the tempera
tures t and 15 Also
1
and
v (1
t
v
v
y )
(
where y is the coe fficient of cubical
expansion of the mercury Hence
'
2
,
OD
,
and let
1
r
2
.
,
1
2
.
,
2
2.
1
f
0
l
0
2
.
11 2
1
f
This
t
y2
equation enables y to be c lc u
lated from k h 15 and t
With an apparatus similar in prin
c i le to that j ust described Regnault
p
found the density of mercury at tem
e
t
u e s from 0 C
to
3 60 C
Hi s
p
results are represented by the formula
g,
ra
a
«
]
2
,
V0
.
°
r
Vt
,
°
.
.
P0
1
Pt
2
3
’
where the o s have the values g iven in the table above Th e
following table gives the volume of 1 00 grams of mercury at
several temperatures
T mp
t
.
e
er a
ure
0 C
°
.
50
1 00
1 50
200
25 0
3 00
CH
EXPANS I ON
II I ]
.
L IQ U I D S WI TH RI S E
OF
E PE RATU RE
1 49
OF T M
third column gives the di fferences between the numbers i
the second column We see that the increase in v olume for a rise
of 5 0 C is not constant but gradually gets greater as the tem
is
nearly
equal
to
ra tu e rises
T
h
increase
from
to
5
e
e
0
0
C
C
p
that fr m 5 0 C to 1 OO C so that over the range f temperature
0 C to 1 00 C the simple equation V
represents
the
V (1
t
B)
volume of mercu y very well 8 is equal to about 00 001 8 22
Th e density of a liquid at di fferent temperatures can be found
by suspendin g a body in it by a fine w i re from one pan Of a
balance and getting its ap parent weight at di ffere t temperatures
If V denotes the volume Of the body and p the density of the
liquid then the apparent weight is less than the actual weight by
f
grams
If
the
coe
ficient
f
c
bical
expansion
the
body
is
O
f
t
V (1
t
so
that
if
W denotes the apparent loss of
B then V
fi)
weight we ha ve
Th e
n
.
°
.
°
°
r
°
°
o
.
°
.
.
.
.
O
,
°
.
,
.
r
.
O
.
,
n
t
.
t
,
i
O
.
t
,
0
u
t
,
W
t
Pt
V0 ( 1
fit)
or
W/V ( 1 Bt)
p
If the body is made Of fused quartz 3 is equal to only 00 7 and
so can be neglected except in ve y exact work We have
=
W
V
O
that
p
/
t
t
0
.
6
,
r
o
o
o,
,
.
S
P
t
P
0
W
W (1
0
t
Bi )
This
method has been used to find the density of water at di fferent
temperatures Th e following table gives some values of the density
of water i n grams per cubic centimetre and also Of the vol u me Of
one gram
.
Vl
o um e o f
1 gra m
1 0 00 1 3 2
1 0 0003 2
1 0 00000
1 0 0003 2
1 0 00 1 24
1 0 00 27 3
1 0 01 7 7 3
1 0043 46
‘
1 0 1 20 7
1 0 227 0
1 043 43
‘
E
1 50
R
P
A
T
1
1
[
H AT
It will be seen that from 0 C to 1 0 C the density is nearly
constant and equal to one but it is slightly less at 0 C than at
at 4 C Thus the density of
4 C and slightly less at 8 C tha
water has a maximum v alue at 4 C Above 8 C the density
diminishes as the temperature rises but not at a uniform rate
Th e rate f diminution increases as the temperature rises
If a tall j ar of water is surrou nded by crushed ice contained
in a larger j then the water will slowly cool down Le t the
temperature of the water at several depths below its free surface
be Observed by means of a thermometer supported in it If the
water is not disturbed it will be found that at first the water at
the bottom is colder than that at the top Of the j ar When the
water at the bottom gets down to about 4 C however it stops
getting colder and the water at the top gets colder than that at
the bottom Th e water at the top eventually gets down to 0 C
and this temperature gradually spreads from the top down to the
bottom of the j ar These facts can be easily explained When
all the water is above 4 C the coldest water is also th e heaviest
and so S inks to the bottom but when some of the water gets down
to 4 C then it is as dense as possible and water below 4 C floats
above it Th e coldest water therefore stays at the top when the
temperature gets belo w 4 C
Lakes and rivers in cold weather cool down like the j ar of
water j ust considered When all the water has got down to 4 C
water colder than this stays at the top so that ice forms first
at the top Th e water below the surface after it has got down to
4 C is not further cooled by streams Of colder water sinki g down
so that lakes and rivers do not usually cool below 4 C except near
the surface even in very severe winters
Th e contraction when water at 0 C is heated can be S hown by
means of a fused quartz dilatometer If one is filled with water
and slowly cooled the level of the water falls very slowly when
the temperature approaches 4 C and then it stops falling and
eventually rises as the temperature falls below 4 C down to 0 C
On warming the dilatometer the water first falls and then rises
agai n
°
°
.
.
°
.
,
°
°
.
n
.
,
°
.
°
°
.
.
,
.
O
.
ar,
.
.
.
°
.
,
,
,
°
.
.
.
.
°
.
,
°
°
.
,
.
.
°
.
°
.
.
,
.
°
n
.
,
°
.
,
.
,
°
.
.
°
.
,
°
°
.
.
.
P
II
A
R
T
[
E
1 52
H AT
at a constant temperature A millimetre scale is marked on
the tube AB or else one is fixed up alongside it If the tube AB
is Of unifor m bore the length of the part f it which is full Of gas
is proportional to the volume of the gas Th e pressure of the gas
in AB is equal to the atmospheric pr essure outside plus the
pressure due to the di fference of level between the mercury in CD
and that in AB By raising or loweri g the vessel E the level of
the mercury in 01) can be altered and so the pressure on the gas
in AB varied Th e following table gi v es a series of results Obtained
with such an apparatus at a temperature of 1 5 C
AB
.
.
O
.
n
.
.
°
.
( 2)
i ht f m
CD b v th t i
He g
a
O
o
e
20
c
P
in
AB
er c u r y
a
ms
n
r o du c
l
t
of n um
c o um n s
1
( )
an
b
ers
in
d (3 )
1 5 40
.
27 8
1 5 42
79 0
1 5 40
height of the barometer was 7 5 0 cms of mercury Th e
pressures n column ( 3 ) are Obtained by adding the numbers in
column ( 2) to the barometer height Th e third column contains
the products formed by multiplying these pressures by the corre
B
T
A
lengths
of
the
gas
in
hese
products
are
nearly
s o di
g
p
constant which shows that the volume f the gas is inversely
as its pressure Th e density Of a gas is inversely proportional
to its volume so that another way of stating Boyle s law is to
say that the density Of a gas is nearly proportional to its pressure
If a definite mass of a gas say one gram is admitted into an
empty vessel it fills the vessel d exerts a cert in pressure on
the walls If now another gram of the same gas is admi tted
the density of the gas in the vessel is doubled and the pressure
is also doubled Thus the second g ram produces the same increase
of pressure that it would have produced if the first gram had not
been there If a third gram of the gas is let in the pressure
becomes three times that due to one gram so that the third gra m
produces the same increase of pressure as it would have produced
if the vessel had been empty Thus it appears that the pressure
due to a gas is equal to the sum Of the pressures each part Of the
gas would exert if present by itself in the same vessel
It is supposed that gases consist Of molecules all in rapid
Th e
.
.
I
.
n
n
.
O
,
.
’
,
.
,
,
an
a
.
.
.
,
,
.
.
.
]
PROPE RTI ES
THE
CH I v
O F G AS
ES
1 53
motion i n di fferent di rections and that the total volume of all
the molecules in a gas at ordi na y pressures is very small com
pared with the volume Of the vessel contain i ng the gas Thus
the greater part of the volume Of the vessel is really empty even
when the vessel is filled with a gas If more gas is let in the
molecules composi g it move about in the empty spaces between
th e other molecules and so produce the same pressure on the walls
that they would have produced if the vessel had been empty
Although the volume actually occupied by the molecules is small
they are S O numerous that very many thousands are contained
i
every cubic mil limetre of the volume of the vessel
It is f und that Boyle s law is not exactly true Th e follo wing
ta ble gives some values of the product of the pressure p and
v olume 1) for di fferent gases If Boyle s law were true these
products would be independent of the pressure
r
.
.
n
.
,
.
n
.
’
o
.
’
.
.
p in
a
t m ph
os
er e s
1
1 00
200
Hy dr og en
1) i n
a
tm
o sp
h
a
°
t 0
eres
1
200
400
pressure p in the table above is expressed in atmospheres
O e atmosphere is equal to about 1 0 dynes per sq cm or 1 5
pounds weight per sq inch We see that when gases like air
and hydrogen are exp sed to enormous pressures the product
does not remain cons tant but varies considerably At 8 00 atmo
spheres about six tons weight per square inch the product p v
air at 0 C is 7 5 greater than at one atmosphere At small
f
press ures of not more than say 200 cms f mercury Boyle s law
is very nearly true for air oxygen nitrogen hydrogen helium and
Th e
.
6
n
.
.
.
.
o
.
Or
°
or
.
.
’
.
,
,
O
,
,
E
1 54
P
1
1
A
R
T
[
H AT
some other gases Other gases including carbon dioxide sulphur
dioxide and ethylene show considerable deviations from Boyle s
law even at moderate pressures
Boyle s law is approximately true at any constant temperature
provided the pressure is not made too great If steam is passed
through the tube B B in the apparatus shown in Fi g 8 the gas in
the tube AB is heated to about 1 00 C and Boyle s law can be
shown to be true at this temperature Th e following table gives
some results Obtained with the same amount of air as was used in
the experiment previously described Temperature 1 00 C Height
of barometer 7 5 cms
L g th f AB
ti i g i
P
P d ct
.
,
,
’
,
,
.
’
,
,
.
.
’
0
.
.
°
.
.
.
en
O
c on
a n
25
n
c
m
s
ro
r e ssu r e
a r
c
.
ms
u
1 99 5
.
20
99 8
1 99 6
15
1330
1 99 5
It appears that the product of the pressure p and the volume 1)
at 1 00 C is constant as it was at 1 5 C but it is equal to
1 9 9 5 at 1 00 C and 1 5 40 at 1 5 C
It is found that the product
denotes
the
v
increases
uniformly
with
the
temperature
If
p
p
pressure and v the volume Of a definite mass Of gas at t C and
then
and
the
pressure
and
volume
f
the
same
mass
at
0
C
t
p
°
°
.
.
,
,
°
°
.
.
t
.
°
t
.
°
,
O
o
v
t
p t
.
)
1
m
,
(
p
at
,
,
where a is a constant If we substitute the values
and 1 540 at 1 5 O we get
.
1 99 5
at
1 00 C
°
.
°
.
,
1 99 5
1 5 40
which gives
or
1 000:
1
1 5a
1
equal to nearly
It is found that for gases
like air oxygen nitrogen hyd ogen helium argon carbon mon
oxide etc for which Boyle s law is nearly true at moderate
,
,
r
,
,
,
,
o
’
.
,
pressures the constant has n e
a
,
h
e
t
y
a rl
same value
1
00 03 6 6
2
7 3
.
t) shows that if the volume is
equation p m p v ( 1
kept constant so that v t then p p ( 1 a t) and if the
a t)
t (l
pressure is kept constant so that p p then
t
1
a
Boyle s law is not exactly obeyed and the equations p
)
p (
a t are also not quite exact
and
v (1
)
Th e
:
,
a
o o
,
t
o,
,
,
t
,
o
o,
’
t
,
.
o
.
P
II
A
R
T
[
E
1 56
H AT
from boiling water at 7 6 cms of mercury pressure the value
constant a in the equation
,
.
p m0
1
p (
o
Of
the
1 0001 )
can be found from these two pressures without using the ther
T
m
o m e te r
a
following table gives some values of a found by Regnault
is called the coefficient of pressure increase at constant volume
t t)
G
(v l m
.
Th e
.
.
as
a
e c on s an
o u
Hy dr o ge n
00 03 66 7
Ai r
i
N tr o g e n
Ca r b o n m o n o x de
Ca r b o n d o x de
S u p u r d o x de
Cy a n o ge n
i i
i i
l h
i
0 0 03 8 29
It will be observed that the gases which Obey Boyle s law closely
’
have val u es
Of a
nearly equal to
00 03 6 6
or
$
3
.
Th e
others
have larger values Of
Th e variation Of the volume Of a gas when its pressure is kept
constant can be observed with the apparatus shown in Fi g 1 0
A glass or fused quartz bulb V is connected by a narrow tube
to a tube AB on which is graduated a scale of equal volumes
This tube is j oined to an open tube DE
At the bottom a side
tube 0 is connected by means Of a rubber tube to a vessel F
containing mercury A three way stopcock K enables gas to
be removed from V or admitted into it Th capacity of the
bulb V is found by weighing it empty and then full Of water
It is filled with dry air or other gas by means of an air pump
and the stopcock K Th e bulb V is immersed in a mixture of
ice and water and the amount Of gas in it is adj usted S O that
the mercury then stands at the zero of the scale on AB and is
at the same level in ED Th e cock K is then closed If now
the bulb V is immersed in steam from boiling water the gas i it
expands d f c e s down the mercury in AB By loweri g F the
mercury can b e k e pt at the same level in DE and AB In this
way the pressure Of the gas is kept constant at atmospheric pressure
and the volume which comes out of V into AB c be read Off on
the scale
a.
.
.
.
.
-
.
e
.
.
.
.
.
n
an
or
n
.
.
an
.
]
CH I v
.
P RO PE RTI E S
THE
O F G AS
ES
1 57
the volume Of the bulb be V
If it is made Of fused
quartz its expans ion can be neglected Le t the volume of the gas
which comes out into AB be 1) and let the temperature Of AB
'
Le t
.
Fi g
be
10
.
emperature can be found by means Of a ther
m o m e te put in c ntact with AB
Le t a denote the coefficient
of expansion of the gas at constant pressure so that
°
t
Th i s
.
C
.
t
’
r
o
.
V0 ( 1
Vt
Th e
volume
7)
Of
gas at
a volume
V at
15
O
.
1 + ta
0 C
°
.
5
0 at
0
ta
I
1)
1
(
at
1 00 C
o
.
becomes
l
1
.
’
’
10
Hence
t)
would become
l 00a
1
v
Thus
°
a
l
.
V(1
ta
1 00a
1 00 C
°
’
)
’
1 00a V ( 1
E
A
R
T
P
I
I
[
H AT
1 58
means Of this equation can be calculated from the values
found for v V and t Th e following table gives some values
’
of a found by Regnault
t
t
G
)
p
(
By
a
.
,
,
'
.
as
a
’
re ss u r e c o n s a n
Hy dr o g e n
0 0 03 6 6 1
Ai r
00 0 3 6 7 0
Ca r b o n m o n
id
ox
0 0 03 6 69
e
i
Ca r b o n di o x de
l h
i
Su p ur d
Cy a n o g e n
Th e
00 03 7 1 0
o x i de
00 0 3 903
0 0 03 8 7 7
gases which Obey Boyle s law closely have values of a nearly
equal to
’
’
1
while the others have larger values
27 3
.
gases which obey Boyle s law closely the pressure c o
efficient at is very nearly equal to the coe fficient of expansion
F o these gases
’
Fo r
a
'
.
r
pm
=
ov0
p
1
(
)
at
,
when
7
)
is
constant
)
p
a t when
is
constant
v (1
u
)
p
Th e density of a gas of course is inversely as its volume S O
that
in
stating
the
density
Of a gas it is necessary
D it G
to state its pressure and temperature Th e den
7
of
gases
is
usually
stated
at
a
pressure
of
cms
si t
O f mercury
6
y
and a temperature Of 0 C Th e de sity of a gas can be found by
weighing a large glass bulb first when exhausted by means of
an air pump and then when filled with the gas at a known pres
sure and temperature Th e di fference between the two weights
gives the mass of the gas and this divided by the volume Of the
bulb is equal to the density Le t p denote the density at a
pressure p cms Of mercury and temperat u re 15 C Th e density
f
and
cms
mercury
is
then
given
by
the
equation
0
7
at
C
6
p
t
f
1
p (
at
,
t
0
.
,
en s
y
of
a ses
.
.
.
°
n
.
.
,
.
°
.
.
°
0
.
.
O
P
p
P
1
( +
)
at
o
where
1
a
27 3
,
provided the gas is one Of those which obeys Boyle s law closely
at pressures between 7 6 and p Th e following table gives the
values of the ratios Of the densities Of several gases to the density
’
.
P
II
T
AR
[
E
1 60
H AT
Hence
Pt
p,
pm
i
—
p
OO
t
1
p
t = 1 00
7
1 1 00 1
"
"
“
30
means of this equation the value of the temperature
corresponding to any pressure can be calculat e d Th e apparatus
described above for measuring the pressure of a gas at constant
volume can therefore be u sed as a thermometer to determine
temperatures It is then called a constant volume gas thermometer
Th e scale Of temperature given by such a thermometer is not exactly
the same as the scale of temperature given by a mercury thermo
meter and di fferent gases gi ve slightly di fferent scales Of tempera
ture Th e scale also depends but very slightly on the initial
pressure of the gas at 0 C Fo accurate work the scale f
temperature adopted as the standard is the scale given by a
constant volume hydrogen gas thermometer in which the pressure
of the hydrogen at 0 C is equal to 1 00 cms of mercury Th e
standard hydrogen thermometer of the Bureau International at
S evres France gives the scale of temperature adopted as the
standard scale in all civilised countries Mercury thermometers
are compared with this standard thermometer and a table of
corrections made for each mercury thermometer giving the
difference between the temperatures indicated by the mercury
thermometer and the temperatures indicated by the standard
hydrogen thermometer In this way it is possi b le to measure
temperatures anywhere on the scale Of the standard hydrogen
thermometer at S evres
Th e di fferences bet ween the scale Of an ordinary mercury ther
and the standard scale of temperatures are very small
m o m e te
between 0 C a d 1 00 C so that except in very exact work
they can be neglected
Th e following table sho ws the corrections that must be added
to the readings of a mercury thermometer of hard glass made by
To n e lo t to Obtain the temperatures on the standard hydrogen
scale
By
.
-
.
.
.
,
,
°
.
O
r
°
.
.
,
,
.
.
.
r
°
.
°
n
.
.
n
,
.
CH I v
.
]
THE
Me rc u r y
th
PROPE RTI E S
E
1 61
O F G AS S
t
er m o m e e r
0 C
°
.
20
50
70
1 00
if the merc ry thermometer in question indicates 50 C the
C
temperat re on the s tandard scale is
Th e apparatus described above for measuring the i crease O f
volume of a gas at co stant pressure can also be used to measure
temperatures If
d v are found then the equation
1
gives
t
)
(
71
t
t = 1 00
Thus
°
u
.
u
.
n
n
an
.
,
a
to
I) ,
0
0
v1 00
"
00
scale gi v en by a cons tant pressure gas thermometer is not
exactly the same as the scale Of temperature given by a constant
v olume thermometer containing the same gas
In experiments with gases it is ofte necessary to remove
as much as possible of the gas from a closed
A
P
vessel S O as to leave it nearly empty If all
the gas in a vessel could be removed so as to leave it completely
empty th e empty space would be what is called a perfect
v acuum It is possible by means of modern air pumps to reduce
the amount of gas in a vessel to a one thousa dth millionth part
Of the amount presen t when the vessel is filled with
the gas
at atmospheric pressure Even then the vessel co tains many
milli ons Of molecules per cubic centimetre A perfect vacuum
cannot be Obtai ed by y k own process
A very con v enient air pump for many purposes is S hown
diagrammatically i Fi g 1 1 It was inve n ted by G aede
and is k ow as G e de s rotary box pump A B is a circular
cylindrical box of brass Inside this a steel cylinder CD is
mou ted on
axle at 0 about which it can be made to rotate
by means of a small electric motor not S ho wn in the figure Th e
cyli der CD has a slot i it in which two plates E d F c a n
slide These plates fit closely th e inside of the box A B d are
pressed outwards by spri gs in the lot Tw tubes G and H
lead into the box as hown and the aperture at H is closed by
w P
11
Th e
.
n
ir
u
m ps
.
.
,
.
n
-
-
n
.
.
n
n
n
n
n
an
.
.
.
’
a
.
.
n
an
.
,
n
n
an
,
an
.
n
S
.
.
S
,
.
o
A
P
T
II
R
[
E
1 62
H AT
a valve V When the cylinder is rotating in the di rection of the
arro w the plates E and F push the gas round so that it is forced out
through the valve V A closed vessel connected to G is very quickly
exhausted S that the gas pressure in it is reduced to 00 1 mm
If air or other gas is let in at G it is blown out
Of mercury
at H This pump therefore can be used to pump gas into a
vessel by connecting the vessel to H While the pump is working
.
.
O
,
.
.
,
,
.
,
.
Fi g 1 1
.
,
Fi g 1 2
.
.
.
is continually supplied to the cylinder at its rotating shaft and
this i l is driven out through the valve along with the air Th e
Oi l supply is essential to the proper working Of the pump
To Obtain very low pressures or very high vacua another form
O f pump also due to G aede may be used
In Fi g 1 2 AB is
a cylindrical box i side which a circular cylinder CD is mounted
on an axle at 0 Two tubes lead into the box at E and F and
there is a proj ection at G on the inside Of AB which nearly
touches CD If the cylinder CD is made to rotate at a very
high speed say 1 00 revolutions per second the gas in the box
is dragged round by the cylinder so that the gas pressure becomes
higher at F than at E It is found that the difference between
th e pressure at F and the pressure at E is nearly independent
of the pressure at F Thus if the pressure at F is 7 6 e m s of
mercury that at E may be 7 4 cms when it is connected to a
closed vessel If the pressure at F is reduced to 20 e m s then
Oi l
,
O
.
.
.
.
,
n
.
.
,
,
.
.
.
.
.
.
C HAP T ER
Q UAN TI TY OF HEAT
V
S PECI
.
FI C
HEAT
a hot body is placed in conta ct with a cold body the cold
b ody gets hotter and the hot body colder
We
this is
due to heat passi g from the hot to the cold body S uppose for
example that a piece of copper heated to 1 00 C i the steam
fro m boiling water is placed in a vessel of water at 0 C and that
the temperature Of the wa ter then rises to 4 C Th e temperature
f the copper at the same time falls from 1 00 C to 4 C
A
certain amount f heat has passed fro m the copper into the water
In order to deal with quantities of heat scientifically it is necessary
to adopt a unit quantity Of heat and to devise methods of
measuring quantities of heat in terms Of the unit adopted It i s
found that heat is a measura b le quantity as we hall see presently
Th e unit quantity of heat generally used in scientific work is the
amount t eat required to raise the temperature Of one gram Of
water from 1 5 O to 1 6 C This amount of heat is called a
c lo i e
Th e a mo unt of heat required to raise the temperature Of
1
1
5
6
number
of
grams
of
water
from
to
is
equal
to
C
O
y
calories
It is found that the amount of heat required to raise the
temperature of one gram of water one degree at y temperature
for example from 25 C to 26 C or from 6 0 C to 6 1 C is nearly
equal to the amount required to raise it from 1 5 C to 1 6 C If
two equal masses of wate one at a temperature 1 6 C and the
other at a temperature 1 4 C are mixed together the mixture is
found to have a temperature nearly equal to 1 5 O We suppose
as it is nat ral to do that when a hot body d a colder body are
put in contact the heat lost by the hot body to the cold body is
eq al in amount to the heat gained by the cold body from the
IF
,
.
n
.
°
n
.
°
.
,
°
.
°
°
O
.
.
O
.
,
.
S
.
°
°
.
.
a
r
.
.
°
°
an
n
n
.
.
.
an
°
°
°
.
,
.
°
.
.
,
°
°
.
.
°
r,
.
°
.
,
,
°
.
u
,
,
u
an
,
CH
.
Q U AN TITY
v]
E
H AT
OF
S
.
PEC I FI C
1 65
H EAT
‘
hot body and that the amount of heat gained by a b ody when its
temperature rises from t C to t C is equal to the amount of
heat lost by the body when its temperature falls from t C to
Accordi g to this the heat lost by the water at 1 6 O
t C
is equal to the heat gained by the water at 1 4 O S i nce the
te mperatures of the two equal m a ss es Of water are both changed
by 1 O it follows that the heat requi ed to raise a gram of water
from 1 4 O to 1 5 O is nearly equal to that required to raise it
from 1 5 C to 1 6 O If eq ual masses Of water at any tempera
tures are mixed together it is found that the temperature O f the
mixture is nearly equal td the m e o f the temperatures before
m i xing Thus if 1 000 grams of water at 50 C are mixed with
This
1 000 grams at 1 0 C the mixture will be at nearly 3 0 C
shows that the amount of heat requ i red to rais e the temperature
of one gram of water one degree is nearly the same whatever the
initial temperature of the water Exact experiments how that
this is not exactly true but it is su fficiently nearly true for most
purposes
If a mass of m grams of water has its temperature changed
from 1 C to t C the amount of heat required for the change is
therefore approximately equal to m ( 15 t ) calories Q uantit ies of
heat are very Often measured by finding how much they change
the temperature of a known mas s of water
S uppose for example that a piece of lead weighi g 1 000 grams
at a temperature of 1 OO C is put into 1 000 grams of water at
15 C
If the water is stirred up the lead and water soon arrive
at a temperature of about 1 7 5 O It appears that the heat given
out by 1 000 g ams of lead when it cools from 1 00 to
or
through a range of 8 25 degrees is equal to the heat required
to warm 1 000 grams Of water from 1 5 to 1 7 5 or o ly 25 degrees
It is clear that a mass of lead requires much less hea t to warm it
th ough any range of temperature than an equal mass of water
Th e a m e un tp f h e t required to raise the temperature of unit mass
O f any substance one degree is
called the specific heat of the
substance Th e product of the mass of a body and its specific
hea t is called its capacity for heat ; it is a lso sometimes called
the water equivalent Of the body If— 8 de n otes the spec i fic heat
of any substance then the amoun t O f heat re quired to aise the
°
°
1
2
.
.
°
2
.
°
°
,
n
.
.
°
.
°
r
.
°
°
.
.
°
°
.
.
an
°
.
.
°
°
.
.
S
.
.
°
3
°
2
.
.
2
i
.
.
n
°
.
°
‘
.
,
°
.
°
r
,
°
°
n
.
r
.
a
.
.
,
r
A
R
T
P
II
[
E
1 66
H AT
te m perature Of a mass m Of it from 16 to t is equal to 3 m ( t
calories In the experiment j ust described we have therefore
1
tl )
2
2
.
1 000 ( 1 7 5
where
1 5)
s
1 000 ( 1 00
is the specific heat of the lead This equation gives
Th e following table gives the values Of
00 3 approximately
3
the specific heats Of some substances at ordinary temperatures
S b t
S p ifi H t
s
.
:
.
ec
s an ce
u
Co p p e r
00 9 5
I ron
01 1 4
L ea d
A um
00 3 1
l
i
S od
i i
n
um
02 1 9
um
c
h
l id
or
ea
c
03 1 3
e
Ice
0 5 02
Q
0 1 91
u a r tz
Wa te r
1 0 00
S pecific
heats are Often found experimentally by what is called
the method Of mixtures This consists in mixing a mass of the
substance at a known temperature with a quantity of water and
finding the change in the temperature f the water TO Obtain
exact results a number Of details must be attended to Th e tem
e
e Of the vessel containing the water also changes so that
t
p
the heat required for this must be allowed for D uring the
experimen t there may be some exchange Of heat between the
vessel Of water and surrounding bodies and the substance may
lose heat while it is being put into the water Th e vessel used
to contain the water in such experiments is called a calorimeter
Fi g 1 3 S hows a vertical section of a simple form Of calorimeter
A AA A is a double walled cylindrical vessel open at the top
This is made Of brass nickel plated and polished inside and out
Th e space between the walls is filled with water the temperature
’
Of which is measured with a thermo m eter T passed through a
cork as S hown Th e vessel AAAA is called the water j acket
Th e opening at the top of it can be closed with a round wooden
cover P P Th e calorimeter proper is a thin walled cylindrical
vessel 0 0 made O f brass or copper nickel plated and polished It
rests inside the water j acket on three wooden pegs or corks It is
about three fourths filled with water the temperature Of which is
measured with a thermometer T passed through a cork in a hole in
.
O
.
.
ra
ur
,
.
,
.
.
.
.
-
.
,
.
,
.
.
-
.
.
,
.
-
,
P
II
A
R
T
[
E
1 68
H AT
surfaces lose heat much l ess readily than dull surfaces so that the
outside of th e calorimeter and the inside Of the water j acket are
brightly polished Th top is covered with a wooden lid to
prevent air currents blowing over the calorimeter which would
tend to cool it down Th e water in th e calorimeter is always well
stirred before Obser v ing its temperature ; if this is not done parts
of the water may be m uch hotter than other parts
T find the specific heat of a substance a known mass of it is
heated to a definite temperature and then put into the calori
meter I f the substance is a solid body which is not acted on by
water it may be used in the form f a hollow cylinder with a
f holes bored through its sides so as to expose a large
n umber
surface to the water and to allow free circulation Of the water
through it Th cyli der may be heated in a double walled vessel
or h eater between the walls Of which steam is passed Th e top
and bottom of the vessel are closed with corks and a thermometer
passed through the top cork gives the temperature f the cylinder
Th e bulb of this thermometer should be placed in the middle f
the hollow cylinder Th e cylinder is suspended by a thread in
the middle of the heater and when its temperature has remained
constant for some time the heater is brought above the calori
meter and the cylinder quickly let down into the water Th e
heater is then removed
Th e temperature of the water in the calorimeter i s Observed
every minute or half minute for some time before the hot body is
put into it Th e time at which the body is put in is noted d
the temperature of the water is Observed every half minute for
some time afterwards Th e water in the calorimeter is kept well
stirred
S uppose that before the hot body was put in the temperature
O f the water fell at the rate of
Le t the tem
C per minute
e a tu e of the water j ust before be t C
uppose
that
after
the
S
p
hot body was put in the temperature of the water rose in a time
T minutes to 15 C and then began to fall at the rate of 8 C per
minute S uppose also that during the whole experiment the water
in the water j acket was at 15 C Th e rate at which the temperature
Of the water in the calorimeter falls is nearly proportional to th e
di fference between it and the temperature of the water j acket
‘
,
e
.
,
.
,
.
O
.
O
O
n
e
.
-
,
.
,
,
O
.
O
.
,
.
.
an
.
.
.
,
a
r
°
.
.
°
r
1
.
,
°
2
°
.
,
.
.
°
3
.
.
Q U ANTI TY
v]
CH
.
During
OF H
EAT
S
.
PE C I F I C HE AT
1 69
the rise from 5 to t the average temperature difference is
the mean O f t i and t t At t the rate of fall is a and at t the
rate is 8 During the rise from 15 to 15 the average rate of fall is
so that the total fall during the
therefore roughly equal to 5 (
time T is nearly equal to } ( 8 ) T If there had been no loss of
heat from the calorimeter its temperature would therefore have
risen to
T Le t t = t + §
t is ca lled
the corrected final temperature Of the water in the calorimeter
Let t be the temperature f the hot body before it was put into
the calorimeter Th e heat given out by th e hot body is equal
—
to 3 m ( t t ) where s is i tS specific heat and
its mass Th e
heat received by the water and calorimeter etc is equal to
—
t ) where w is the mass
f the water in the calori
( 20 + w ) ( t
meter and w denotes the total heat capacity of the calorimeter
stirrer and thermometer We have therefore
“
11
s.
2
s
1
2
]
1
.
2
,
2
a
a
1
.
,
’
’
,
2
.
.
\
O
\
.
777.
g
.
,
'
.
'
2
O
,
’
,
.
3
(
m
w
(
t2)
15
’
w ) ( t;
—
t1 )
.
means Of this equation 3 can be calculated Th e heat capacity
of the cal rimeter stirrer and thermometer can be calculated
by adding up the products f their masses and specific heats or it
can be found experime tally In fin din g i t experimentally the
c alorimeter is about
one fifth filled with hot water and its
t
temperature bser v ed Le t it
water
at
is
ld
u
o
c
h
En g g
then poured in until the calorimeter is about three quarters full
L t the mixture take up a temperature t
Then we ha v e
By
.
o
,
o
n
,
.
fi
/
-
O
2
.
-
.
e
3
(
w ) ( tl
,
M
t3 )
m
( tg
.
tz)
.
Here w de n
otes the mass Of the water in the calorimeter at first
and m the mas s Of water poured
best to start with hot
water at early 1 00 C
r and to pour in cold
water because the total capacity for heat w Of the calorimeter
stirrer etc is usually very small
Th e specific heat f a liquid can be found in the same way as
that of a solid Th e liquid is enclosed in a metal bottle or a glass
bulb Th heat g iven out by the bottle or bulb must of course
be allowed for Another way is to use the liquid the specific heat
of which is required instead of water in the calorimeter add
to it a hot body of known heat ca pacity and observe the rise of
temperature
,
°
n
.
,
’
,
,
.
,
.
O
.
.
e
.
,
,
.
,
P
II
A
T
R
[
E
1 70
H AT
and P etit discovered that for many elements the
product atomic weight S pecific heat is nearly constant This
rule is called Dulong a d P etit s law Th e following table gives
some values of the product in question It is not exactly constant
At m i W i g ht
El m t
P d t
S p ifi H t
Dulong
.
’
n
.
.
.
e
en
S o di u m
Su p u r
I ron
ro
c
ec
ro
ea
n e
er
L ea d
uc
6 75
23
l h
B mi
S ilv
e
c
o
32
01 78
56
0 1 1 40
80
00 8 43
6 74
1 08
00 570
6 16
207
00 3 1 4
6 50
50 8
It is found that the specific heats of most substances are smaller
at low temperatures than at high temperatures At very low
temperatures near 27 3 C which is the lowest possibl e tempera
ture the specific heats become very small AS the temperature
rises the specific heats rise at first more or less rapidly and then
more slowly and at higher temperatures become early constant
Dulong and P etit s law applies to the nearly constant values
attained at the higher temperatures Th e specific heats of
metals have nearly reached their constant values at ordinary tem
a tu e s but those Of carbon silicon and some other bodies do
e
p
not become nearly constant until very high temperatures are
reached Fo example the specific heat of graphite a form of
carbon has the following values
T mp
t
S p ifi H t
.
°
.
,
,
.
n
,
.
’
.
r
r
,
.
,
r
,
,
ec
c
er a u r e
e
ea
0 C
°
.
0 443
6 00
0 45 3
8 00
0 46 7
1 000
specific heat of platinum between
by the equation
Th e
00 3 1 7
3
w
and
0 C
°
.
0 00001 2t
‘
,
here t denotes the temperature
.
F
N
RE ERE CES
Hea t, Po y n
Adva n
c ed
i
t
n
g
an
d
J J
.
.
Th o m s o n
P r a c ti c a l P lay s zc s ,
.
Wa ts o n
.
°
1 200
is given
P
A
T
II
R
[
E
1 72
H AT
the mixture so m e Of the ice melts a d if heat is taken away some
of the water freezes If heat enters or leaves the mixture the
amounts Of ice and water in it remain constant We say that ice
and water are in equilibrium with each other a t the temperature
0 C
Th e melting point of a substance may be defined as the
temperature at which the solid and liquid states can exist together
in equilibrium TO raise the temperature of a mixture of the
solid and liquid states f a substance above the melti g point it is
necessary first to melt the solid ; and to cool the mixture below the
melting point it is necessary first to freeze all the liquid Th e
solid d liquid cannot exist together in equilibrium except at
the melting point
TO find the melting point Of any substance the bes t way is to
prepare an intimate m ixture Of the solid and liquid states and
find its temperature with a thermometer Fo example to find
the melting point of tin heat a quantity in a nickel crucible over
a Bunsen burner until it has all melted Then tur down the
flame so that the liquid cools slowly and stir it up with a
thermometer After a time the liquid will begin to solidify ; and
by stirring an intimate mixture of the solid and liquid can be
Obtained
While such a mixture exists the thermo m eter will
indicate a constant temperature which is the melti g point of
the tin
Th e latent heat of ice can be found experimentally by putting
a k own mass Of dry ice at 0 C into a calorimeter containing
warm water A good way is to wrap up a piece of clear ice in
blotting paper and weigh the ice and paper together Then slip
the ice out of the paper into the calorimeter and weigh the paper
again Th e paper absorbs any water formed from the ice so that
the differe ce between the two weights gives the weight of the
ice put into the calorimeter Th water in the calorimeter is
stirred up until the ice h s all melted L e t t be the temperature
of the water before the ice was put I n and 15 the temperature after
the ice had melted Then we have
n
,
n o
.
.
°
.
.
n
O
.
an
.
r
.
,
n
.
,
,
.
,
.
n
,
.
°
n
.
.
.
.
,
n
e
.
a
1
.
2
.
mF
w
(
m i2
’
w ) ( t1
tz) ,
where m is the mass Of th e ice and F its latent heat w the mass
O f water in
the calorimeter and w the heat capacity Of the
,
’
,
CH
‘
.
VI ]
C HAN G
E
O F S TAT
E
1 73
cal orimeter stirrer and thermometer Th heat lost b y th e
—
water and calorimeter e tc is
t ) ; the heat required
to melt the ice into water at 0 C is m F and the heat required
to warm the water formed from the ice from 0 C to the fi al
temperature is m t
Th latent heat Of other substances can be fo u d i n a S imilar
way by putting a piece Of the solid into so me of the liquid and
bserving the resulting change of temperature when the solid has
all melted If 3 denotes the specific heat Of the liquid and 3 that
f the solid then we have
'
,
e
.
,
2
.
,
°
.
°
'
n
.
z.
‘
n
e
O
’
'
.
O
,
mF
Here
3
m
t
(2
to)
s
'
m ( to
t, )
(
sw
w ) ( tl
tg )
.
is the mass f the solid F its latent heat t the final
temperature f the liquid t the melt i ng point of the solid t the
i ni tial temperature o f the solid w the mass of the liquid w the
heat c pacity of the calorimeter etc and t the in itial tempera
ture O f the li quid This method is suitable only for substances
which melt t temperat u res below about 40 C I all such
experiments it i s necessary to correct the Observed temperatures S O
as to allow for the heat ga i ned or lost by the calorimeter from the
water j acket This is done in the way described in the previous
chapter
Th e following table gives the latent heats and melting points
f some solids
L t t h t ff i
S b t
C l i
p
g m
m
O
,
O
,
2
,
o
,
,
’
,
,
a
.
,
1
,
.
'
°
a
n
.
.
.
O
a en
s an c e
u
h ph
S lph
B mi
S di m thi
os
u
ilv
ra
80
5
or us
n e
u
osu
l ph t
a
162
3 70
e
Ti n
S
er
us on
ur
ro
o
O
a o r es
Ice
P
ea
14
er
cha ge from the liquid state into the solid state is usually
accompanied by a change of volume F example unit volume
f water becomes about 1 0 9 u it volumes of ice
Thus the de sity
of the solid at the melting point is usually di fferent from the density
Th e
n
.
O
n
or
,
.
n
P
II
AR
T
[
E
1 74
H AT
of the liquid at the same temperature Th e following table gives
some values Of the densities in gra m s per c c at the melting points
M pt
D ity f li d D ity f liq i d
S b t
0 C
0 9 1 6 03
Wt
0 9 99 8 6 8
.
.
.
s an ce
u
en s
.
.
O
so
en s
o
u
°
a er
.
L ea d
3 25
Sod um
i
97 0
Me r c u r y
389
09 52
0 9 29
In most cases the density f the liquid is less than that of the
solid so that the solid s i nks in the liq u id Water is an exception
to this rule so that ice floats on water It is found that the
melting points of substances such as I c e the volume Of which
changes when they melt depend on the pressure at which the
melting takes place Usually the pressure is the ordinary atmo
spheric pressure about 7 6 cms of mercury At this pressure ice
melts at 0 C At a pressure of 1 00 atmospheres or about
1 5 00 pou ds weight per square inch ice melts at
0 7 4 C instead
Th e melting points of substances which expand when
Of at 0 C
they melt are made higher by great pressure while those which
contract like ice have their melting points lo wered
Th e change O f volume when ice melts can be measured with
the apparatus shown in Fi g 1 4 This consists Of a glass bulb of
about 1 00 c c capaci ty blown on a tube about 05 cm in diameter
"
About 3 0 c c of water are put in the bulb and it is then filled up
to a poi t in the tube with paraffin Oi l Th e bulb is put in a
mixt re Of crushed ice and hydrochloric acid which a cts as
freezing mixture and gives a temperature of about 20 C Th e
water then freezes and its expansion causes the Oi l to rise con
If the rise and the diameter of the tube
s i de
b ly in the tube
are measured the cha ge of volume can be easily calculated
Water pipes are Often burst in cold weather when the water in
them freezes on account of the greater volume Of the ice formed
If a thick walled cast iron bottle closed by an iron plug screwed
in is filled with water and put in a freezing mixture the bottle
will burst when the water freezes If the pressure required to
burst the bottle is say 200 atmospheres then the bottle and the
water i it have to be cooled down to 1 4 8 C before the freezing
can go on sufficiently to produce this pressure
Many substa ces such as glass wrought iron and sealing
O
.
,
.
,
,
,
,
.
.
,
.
°
.
°
n
.
°
.
,
.
.
.
,
,
.
.
.
.
,
n
,
.
,
u
a
°
.
ra
.
n
.
,
.
,
,
,
.
,
,
,
°
n
.
.
n
,
~
‘
“
R
T
A
P
II
[
E
1 76
H AT
li quid s can be cooled down to te mperatures considerably below
the me lting point without solidifyi ng provi ded no solid is pre sent
If a glas s tube a fe w
This p henomenon is called super cooling
millimetres i diameter and say 1 5 cms long closed at one end is
filled about two thirds full of liquid carbolic acid phosphor s
sulphur salol or other substa ce it can be cooled below the melting
poi t without the liquid solidifying Th melting point f salol is
be
41 C but liquid salol in a tube like that j ust described
cooled down to 1 O C without solidifyi g If a small piece of
solid salol is held in the super cooled liquid close to its surface
then solidification takes place at the surface f the solid so that
it grows down the tube with a definite velocity which is called the
velocity Of solidification This velocity depends on th e tempera
ture of the surface of the solid As the super cooling is increased
the velocity increases at first rapidly but the more slowly and
the rema ns c Os t t over a considerable range of temperature
Th e following t a ble gives the velocities of solidification of salol at
several tempera tures
.
,
-
.
n
.
,
,
u
-
,
n
,
,
n
O
e
.
°
.
,
c an
,
°
n
.
.
-
,
O
.
-
.
n
,
i
n
n
,
an
.
Other
substances behave in a similar way Fo example benzoic
a hydride which melts at 42 C gives a maximum velocity f
solidification of 3 5 mms per minute Water can be super cooled
considerably its velocity of solidification is large but has not yet
been determined accurately
n
.
r
,
°
.
,
o
,
-
.
.
,
.
C HAP T ER
C HA
N GE
VI I
OF S TATE
Va p o u r
Li qu id
a dish of water is left standing in the Open air the water
radually
disappears
It
is
converted
i
to
water
vapour
which
g
mixes with the air and is carried away by the wind T study
this process more closely the appa
ratus shown in Fi g 1 5 may be used
B is a large three necked bottle At
A a tube with a cock i it leads i to
the bottle This can be connected
t an air pump and the bottle ex
hansted At 0 is a tap funnel con
taini g some liquid such as water
alcohol or ether At D a closed
mercury manometer MN is con
e te d to the bottle and indicates
the press e f the gas in the bottle
S uppose that the bottle is al m ost
completely exhausted so that the
me cury stands at the same level i
the tubes M and N Th e pressure
in N is supposed to be zero If now
one drop of the liquid ether say is
let i to the bo ttle by pening the
Fi g 1 5
t p of the tap f nnel for a moment
i t is found that the pressure rises by a small amount and the dr p O f
ether all evaporates and so becomes i visible Th e bottle is then
full of ether vapour at a small pressure which is indicated by the
manometer If now another drop is let in it also e v aporates and
wP
12
IF
n
.
o
.
.
.
-
.
n
n
.
o
.
n
,
.
n
c
O
ur
.
n
r
.
.
,
n
,
o
a
u
°
~
,
o
n
.
.
.
.
,
P
R
II
A
T
[
E
1 78
H AT
the pressure is about doubled If a third drop is let in it e v p
rates and the pressure becomes about three times that due to the
first drop If more drops of ether are let in the pressure goes on
increasing and the drops evaporate until a certain defi ite pressure
is reached ; and then on letting in more ether no further increase
in the pressure takes place and the ether does not evaporate but
remains in the liquid state on the bottom Of the bottle Th e final
value of the pressure depends on the temperature
If the temperature of the bottle is 20 C the final pressure is
about 44 cms of mercury when the liquid used is ether If alcohol
is used the final value is 44 cms Of mercury and if water is us ed
it is only
cms of mercury at 20 C It appears that a liq uid in
a closed vessel evaporates until the vapour attains a cer tain pressure
d vapour remain together in equilibrium
d then the liquid
and no further evaporation tak e s place Th e pressure of the
vapo r at which the liquid and vapour can exist together in
equilibrium is called the vapour pressure Of the liquid Th e
vapo r pressure of a liquid depends on the temperature Of the
liquid It increases apidly as the temperature rises Instead
of starting with the bottle B co m pletely exhausted we may start
with some gas such as air or hydrogen in it S uppose at the
start the bottle contains air at 20 cms pressure If now ether is
let in it is fou d that the pressure rises j ust as it did before but
the final pressure attained is nearly 6 4 cms of mercury instead Of
Th e increase in the pressure due to the ether is nearly the
44 cms
same as before It appears therefore that in the presence Of a gas
a liquid evaporates until the pressure due to its vapour is nearly
the same as i t would have been if the gas had not been there
Exact experiments show that the presence f the gas very slightly
dimi ishes the vapour pressure but the e ffect is negligible for
most purposes
S uppose we start with the bottle B completely exhausted and
let in ether until the maximum pressure is reached and no more
ether evaporates Then pour mercury into the tap funnel and
allo w it to
into the bottle Mercury gives ff practically
vapour In this way the volume Of the S pace occupied by the
vapour in the bottle c be diminished It is found that as the
mercury is run in the pressure indicated by the manometer remains
.
a
,
.
o
,
n
,
.
.
°
.
,
.
.
.
,
°
.
.
,
an
an
.
u
.
u
r
.
.
.
.
.
n
,
.
.
.
.
O
n
,
.
.
ru n
n o
O
.
.
an
.
AR
P
II
T
[
E
H AT
1 80
mercury by mea s of a small pipette like those used for filling
fou tain pens Th e liquid floats to the top f the me cury column
and the vacuum is filled with the vapour Enough liquid must be
introduced for some of it to remain in the liquid state Th e
vapour at the top of A B is then saturated and its pressure force
do wn the mercury a distance 1 such that the vapour pressure is
equal to the pressure due to a column Of m ercury Of height h
By passing water at known temperatures through CD the d
pression Of the mercury can be found at di fferent temperatures
Th e following table gives the vapour pressure Of some liquids at
di fferent temperatures
n
n
,
O
.
r
.
.
s
1
.
.
Va p o u r P r ess u r e i n
mms
M
er c u r
f
y
o
.
Wa t er
E th
.
Mer c u r y
er
29 2
442
1 27 6
229 4
48 5 9
7 6 00 0
1 3 28 1
3 58 1 0
1 1 688 0
6 7 6 20 0
an open vessel containing water is heated over a flame the
temperature f the water rises steadily until it reaches about
Th e water then boils and its temperature remains con
1 00 C
stant at 1 00 C Bubbles Of vapour are formed beneath the surface
f the water and these grow larger and rise to the surface where
they burst Th e more rapidly heat is supplied to the water the
more vigorously it boils When boiling the water is being con
verted into steam or water vapour In order that a liquid may boil
it must be heated till its vapour pressure is equal to the pressure
At 7 6 cms f mercury pressure water boils
f the gas above it
at 1 00 C If the atmospheric pressure is less tha 7 6 cms water
boils below 1 OO C and if the pressure is greater than 7 6 cms it
boils above 1 00 C A very good way to fi d the vapour pressure
f a liquid at di fferent temperatures is to find its boiling point at
di fferent pressures This can be done with the apparatus sh w
in Fi g 1 7 Th e liquid is contained in a metal boiler B heated by
If
O
°
.
°
.
O
.
.
,
.
O
.
.
O
°
n
.
.
°
.
°
.
.
n
o
.
.
.
o
n
CH
.
V II ]
C H ANG
E
E
1 81
OF S TAT
a bur er P Th e temperature of the va pour above the boiling
liquid is m easured with the thermometer T Th e boiler is con
u s eted to a tube GH which slopes upwards This tube passes
th ough a wider tube 0 0 through which cold water is passed by
means of ide tubes at E and F Th e vapour from the boiler con
denses in GH and the liquid formed runs back into the boiler
At H the tube GH leads into a large vessel V This is connecte d
to a mercury manometer M d to an air pump throu g h the tube
A
By mea s of the pump the a i pressure in the apparatus can
be brought to any desired value which can be measured with the
n
.
.
.
r
S
.
.
.
an
r
n
.
,
Fi g
.
17
.
manometer Th e liquid in the boiler is kept boiling steadily d
the temperat u re of its vapour is read Off o the thermometer F
example if the liquid used is water and the pressure is adj usted to
3 5 8 1 m s of mercury it is found that the water boils at 1 5 0 C
Th e vapour pressure of water at 1 5 0 C is therefore
e m s of
me r cury
It is found that if the water is not pure but h as salts o
other s u bstances dissolved i it the temperature Of the boiling
solution is slightly higher than that of pure water boiling at the
same pressure Th temperature Of the v apour rising from the
bo ili ng solution however is found to be the same as that of
an
.
n
or
.
,
°
e
.
.
°
.
.
.
r
n
.
e
P
II
AR
T
[
E
1 82
H AT
the vapour from pure water It is therefore best to measure
the temperature Of the vapour above the boiling liquid when
fi ding the boiling point of a liquid at any pressure
Th e following table gives the boiling points Of water at several
pressures near to 7 6 0 m m s Of mercury
.
n
.
.
P
mms
r es su r e ,
of
.
m
er c u r y
7 40
7 45
7 50
755
7 60
76 5
770
775
780
When it is necessary to test a thermometer to see if the
upper fixed point is correctly marked on it the thermometer is
immersed in the steam fro m boiling water in a suitable vessel
from which the steam can escape freely Th e temperature i
di te d by the thermometer is then Observed and also the heigh t
f the barometer
If the height of the barometer is 7 6 cms then
the thermometer should indicate 1 00 C but if the barometer is
not at 7 6 cms then the thermometer should show the boiling point
of water at the pressure indicated by the barometer Fo example
if the barometer sta ds at 7 45 mms the thermometer should
indicate 9 9 4 43 if it is correctly graduated
When an open vessel f water or other liquid is heated over
a
flame
so
that
is
kept
boili
g
its
tempera
i
t
H
f
E
fi
ture does not rise although heat is continually
entering it This heat is used up in converting the liquid into
v apour To condense the vapour back into liquid form it is
necessary to take out of it an amount Of heat equal to that which
was used in converting the liquid into vapour Th e amount of
heat required to c onvert a unit mass Of a liquid at any temperature
i to vapour at the same temperature is called the latent heat of
evaporation of the liquid at that temperature TO convert one
gram of water at 1 00 C into steam at 1 00 C requires 53 6 calories
Th e latent heat of evaporation Of a liquid such as water or
n
.
ca
O
.
.
°
.
,
.
r
.
n
,
.
.
O
L a te n t
va
ea
p °r a
t o
n
.
,
°“
.
.
.
n
.
°
°
.
.
.
P
II
A
R
T
[
E
1 84
H AT
water in the calorimeter be t the final temperature be 15 and the
Th e heat received by the calori
corrected final temperature
meter co denser stirrer and thermometer is equal to that given
out by the vapour in condensing We have therefore
1
n
,
,
2
,
,
.
mL
Here
ms
t
(
’
w ) ( t;
w
(
t2)
is the mass of the vapour condensed L th e latent heat of
the v apour the specific heat Of the liquid formed t the tempera
ture of the vapour w the mass of water in the calorimeter and w
the heat capacity of the calorimeter condenser stirrer etc m is
found by weighing the S piral co de ser before and after the
experiment Th e vapour first condenses into liquid at a tem
t
e 15 giving out m L heat units
d
then
the
liquid
is
cooled
e
p
down from t to 15 giving out m s ( t t ) heat units
Th e following table gives the values of the latent heats of
evaporation Of some liquids in calories per gram at the tem
e
t
u e s stated
p
L iq i d
W t
777.
,
s
,
,
’
,
,
,
n
,
.
n
.
ra
ur
an
,
o
2
.
,
ra
r
:
u
a
er
Wa te r
l
l
A c oh o
A c oh o
h
Et
l
l
er
M er c u r y
l h
Th absorption of heat when a liquid evaporates can be illustrated
by pouring some methyl chloride into a small metal crucible stand
ing in a little water Th methyl chloride evaporates rapidly and
absorbs heat from the crucible and water that the water soo
freezes
Th following is another illustration
A little water is put
into a platinum basin which is supported above a dish containi g
strong sulphuric acid Th basi and dish are covered with a bell
j ar which is then exhausted by means of an air pump to a pres ure
less than one millimetre f mercury Th water then boils and
its vapour is absorbed by the acid Th rapid evaporation cools
the water so that it soon freezes although it is boiling
Th
volume f vapour formed when a liquid evaporates is
usually much greater than that of the liquid Th density of the
Su p
ur
e
e
.
n
SO
.
e
.
n
.
e
n
s
O
e
.
.
e
.
e
O
.
e
C H VI I
.
E
]
C HAN G E O F S TAT
1 85
vapour is therefore much smaller than that of the liquid F
example one gram f water gives 1 6 6 3 cubic centimetres f
saturated v apour at 1 00 C
Th e relation betwee the pressure and volume f a quantity of
any substance can be represented on a diagram by tak i ng distances
measured v ertically upwards from a horizontal line to represent
the pressures and the distances measured horizontally from a
ve tical line to represent the corresponding volumes In Fi g 1 9
the point P represents the state of a substance when its volume
or
.
O
O
°
.
n
O
'
r
.
.
Fi g
.
19
.
represented by MP or ON and its pressure by N P or
S uppose we represent in this way the relation between the
pressure in a vessel containing a liquid and its vapour and
the v olume of the vessel If the volume is diminished some f
the v apour condenses and if the temperature is kept constant
the pressure remains constant S O long as both liquid and vapour
are present the relation betwee pressure and volume at constant
If
temperature is therefore a straight horizontal line like A B
the v olume is diminished u til all the vapour has condensed and
nothing but the liquid remains in the vessel then increasing the
is
O
.
,
.
n
.
n
,
P
II
R
A
T
[
E
1 86
H AT
pressure will produce only a small diminution Of volume because
liquids are very nearly incompressible If A represents the state
when all the vapour has condensed then the line showing the
relation between pressure and volume at constant te m perature
for volumes less than that corresponding to A will be nearly
vertical like AL If the volume is increased until all the liquid
has ev porated at the constant temperature then on further
increasing the volume the pressure will fall because the vapour
will then not be saturated S uppose that B represents the state
when all the liquid has j ust evaporated and there is nothing
but saturated vapour present in the vessel Fo volumes greater
than that corresponding to B a curve like B V will represe t the
relation between p d v at constant temperat u re Th e complete
curve LAB V is called an isothermal curve Th e part LA e
presents the state Of the substance when it is all liqu i d the part
A B when it is partly liquid and partly vapour and the part B V
when it is all vapour
It is found that the relation between the pressure and the
v olume is re presented by a curve like LAB V at all temperatures
below a certain temperature which is di fferent for di ffere n t sub
stances As the temperature is raised the pressure of the saturated
vapour rises and its density i creases and the volume of the liquid
increases Consequently at higher temperatures the horizontal
part Of the curve is higher up and shorter like
Th e
part lik e AB gets shorter as the temperature rises until it dis
appears and then there is no longer a stage in which the substance
separates into the liquid and gaseous states Th e highest tem
e
t
u e at which this separation can take place is called the
p
Above this temperature
e Of the substance
c i ti c a l temp e a t
the substance cannot be liquefied that is it cannot be made to
separate into two di fferent states with a surface of separation
betwee them
Th e existence of the critical temperature can be demonstrated
with the apparatus shown in Fi g 20 This consists Of a strong
glass tube AB closed at A and j oined to a bulb at B Th e bulb
and tube are fi lled with pure carbon dioxide gas and the b u lb is
immersed in mercury contained in a steel tube CD At the lower
end Of CD there is a screw by means of which the mercury can
.
,
.
,
a
,
.
.
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.
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an
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.
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.
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.
.
n
.
,
.
ra
r
r
r
ur
.
,
,
n
.
.
.
.
.
P
II
A
R
T
[
E
1 88
H AT
known volumes the critical density and p ressure can also be
determined Th e following table gives the values Of the critical
quantities for several substances
i
P
D ity
T mp
At m p h
S b t
t
m
G
p
(
)
1 95 C
36
Eth
03
31
di i d
C b
77
3 60
200
W t
05
,
.
r essu r e
u
s an c e
e
er a u r e
a
en s
er e s
ra
s
er c c
.
.
°
er
ar
os
n
.
e
ox
on
er
Ox y g e n
1 19
56
Hy dr o g e n
23 8
15
In order to liquefy a gas it must be cooled below its critical
temperature Thus oxygen cannot be liquefied above 1 1 9 C
and carbon dioxide above 3 1 C All gases including hydrogen and
helium have been liquefied Th gases like air oxygen hydro
gen and helium which are diffi cult to liquefy because their critical
temperatures are low can be liquefied by an ingenious process
invented independently by Hampson and Linde This process
depends on the f c t that when a gas expands without doing work it
is cooled slightly All gases show this e ffect at low temperatures
This small cooling e ffect is believed to be due to the molecules of
the gas attracting each other so that when the volume f the gas
increases and the molecules separate to greater distances apart
their velocity is diminished In H m ps
apparatus for pro
d i g liquid air the air is com pressed by a pump to about 1 8 0
atmospheres pressure Compressing air makes it h t because work
is done on it and so it has to be cooled by passing it thro ugh spirals
of copper pipe immersed in cold water Th e air at 1 8 0 atmospheres
pressure is freed from carbon dioxide and water vapour by passi g
it through a vessel containing sticks of solid caustic potash ; it
then enters a long coil Of narrow copper tubing contained i a
box surrounded by layers of felt Th e air is allowed to escape
from the copper tubing through a valve and its pressure falls to
slightly more than one atmosphere Th e valve is near the bottom
of the box with the coil of tubing above it Th e air after escaping
through the valve passes f rom the bottom to the top of the box in
between the coils of the copper tubing and then goes from the top
Th e
f the box back to the pump where it is again compressed
expansion from 1 8 0 atmospheres pressure down to about one
°
.
.
°
.
.
,
e
,
,
.
a
.
.
O
uc n
on
a
.
’
s
,
o
.
.
n
n
.
.
.
,
,
,
O
.
CH
.
V II ]
C HANG
E
OP
E
S TAT
atmosphere cools the
and the cooled air as it passes up
between the coils of tubi g cools these d wn and so cools down
the air moving through the tubing towards the v alve I this
way the temperature at the valve is made to fall steadily until it
reaches about 1 90 C when about fi v e per cent of the escaping air
Liquid
li q e fie and the liquid collects at the bottom Of the box
Its boiling po m t at 7 6 cms
i is a colourless liquid like water
f mercury pressure depends on the percentage of oxygen in it and
v aries between about 1 8 0 and 1 90 C Hydrogen and helium
be liquefied by a similar pr cess but in the case of these gases
ca
it is necessa y to cool the compressed gas to a very low tem
t
e before letting it into the liquefying apparatus
L
iquid
p
hydrogen is a colourless liquid which boils at about 25 3 C at
By allowing liquid hydrogen to
7 6 cms Of mercury pressure
evaporate at a pressure of only a few mms of mercury it can be
frozen into a transparent colourless solid which melts at about
Liquid helium has a lower boiling poi t than liquid
25 8 C
hydrogen
a i r,
o
n
n
.
°
.
.
u
s
.
a r
.
.
o
°
°
.
o
n
,
r
er a
ur
.
°
.
.
.
\
.
°
n
.
.
F
N
RE ERE CE
l
e
Ex p er i m en
ta
l S tu dy
f
o
Ga s es , Tr a
v
ers
.
C HAP T ER
CO
N VECTI ON
V I II
AN D CO
N DU CTI O N
this chapter we S hall di cuss the two ways in which heat
can
move
from
one
point
to
another
If
a
hot
body
C
v ti
is moved from one place to another its h e a tg o e s with
it This way of moving heat is called co vectio
If a v essel Of
water is heated from below the hot water at the bottom is less
dense than the colder water higher up Th e hot water therefore
floats up towards the top d the cold water sinks A circulation
of the water is thus set up so that all parts f the water in turn
come near to the bottom and get heated there This circulation
causes the water in the vessel to get hot much more quickly than
it would if the water remained at rest If a glass beaker full f
water is heated from below with a small flame the circulatio Of the
water can be easily seen if a few crystals of potassium perma ganate
are put at the bottom of the beaker Th permanga ate colours
the water as it passes by the crystals and the stream of coloured
water rising up the middle of the beaker d falli n g down its sides
ca
be clearly seen A similar circulatio is produced when a closed
vessel full of any gas is heated from below
B uildings are often warmed in cold weather by mea s Of hot
water which is passed through coils of pipe in each room I this
way heat is transferred from the furnace where the water is heated
to all parts of the buildi g Ho t air is sometimes used instead of
hot water Th e cold air from outside passes through pipes heated
by a furnace in the basement which lead from the furnace to each
room Th e air enters the room n d then escapes by the doors and
windows or through special openings to the outside
Thus there
is a continuous circulation of the air This circulation is kept up
j ust like the circulation i n a closed vessel heated below Th e hot
IN
s
.
on
ec
on
.
n
n
.
.
,
.
an
.
O
,
.
O
.
n
,
n
n
e
.
an
n
n
.
.
n
n
.
n
.
.
,
,
.
a
.
.
.
P
T
II
AR
[
E
1 92
H AT
inserted into small holes bored into the bar Th e whole
apparatus is thickly covered with felt which is a very bad
conductor of heat and therefore practically stops any heat from
escaping If the mass Of cold water flowing per second through
the box at B is m grams then the heat it received per second is
equal to m ( t
where t is the temperature of the water before
and t that after going through the box If the temperature of
the stream of water flowing through the box at A is varied then
it is found that the amount of heat flowing along the bar is
T2
.
,
.
,
,
3
4
4
.
,
,
,
,
,
Fi g
.
21
.
proportional to the difference between the temperature 15 at O and
the temperature t at D Thus it appears that the flow of heat is
proportional to the change of temperature in any distance along
the direction Of the flow
S uppose we have a thick slab of any substance bounded by
two parallel planes Le t the temperature f one side of it be
maintained constant at t and the other at t Le t the thick ess
f the slab be d
Th e amount Of heat which flows through an
area A Of the slab in a time T is given by the equation
'
2
1
.
.
O
.
2
o
,
n
.
.
H
:
k TA ( t,
where k is a co stant depending only on the nat re
n
u
Of
the material
CH
.
VIII ]
CO N
VECTI ON
AN D CO N
DU CTI ON
1 93
of which the slab is made I i s called the conducti v ity for heat
—
1 sq cm t
I fA
t =1 C d
1 cm
f the material f the slab
and T 1 second we get H ; k
Thus the conductivity for heat of a substance may be defined
as the am unt of heat which flows in unit time through unit area
of a slab of uni t thickness when the di ff ere ce between the
temperatures of the two sides f the slab is one de g ree
Th apparatus shown in Fi g 21 above can be used to find the
heat conductivity Of the copper bar We have
t
.
°
O
O
.
.
:
.
2
,
1
.
,
.
,
.
,
o
n
O
e
.
.
.
m
(
154
t)
?
Of
[5 1 1
0
2
/
ti ) d,
where A is now the area
cross section of the bar and d the
di stance from C to D Th e follo wing table gives some values of
the co nductivities for heat of di fferent substances
S b t
C d ti ity
.
u
s an c es
on
uc
v
Co ppe r
I ro n
00 8
0 0 02
0 0 006
0 0 05
00 0004
00 01 4
0 0 003
0 0 00048
0 0 003 2
W
.
P
.
C HAP T ER
HEAT A
F ORM
IX
OF E
N ERGY
a block of lead is hammered on an anvil with a heavy
hammer it gets hot although the anvil and hammer were both cold
If a brass tube is moun t ed on a vertical shaft so that it can be
rapidly rotated by means Of a pulley and belt driven by a motor
and is squeezed between two wood blocks while it is rotating it
rapidly gets hot If some water is put in the tube it soon begi s
to boil Whenever two bodies are made to slide over each other
so that work is done against the frictional resistance to the motion
heat is produced and the bodies get hot Fo example if the
bearings of the axles on a railway c a are not properly Oiled they
may become red hot when the car is running In such cases work
is done and heat appears Thus the quantity of heat in a system
is increased although no heat has come in from outside whe n
some of the energy in the system is used up in doing work against
frictional forces F example if a heavy body is allowed to fall
on to a block of lead the lead gets hotter Th e system consisting
of the earth the body and the block of lead has lost some potential
energy but has gained some heat It is found that when energy
is used up in this way so as to produce heat the amount Of heat
produced is proportional to the mount Of energy used up or to
the work done If an amount Of work W is done and nothing but
heat produced then
IF
.
,
,
n
.
.
,
r
.
,
r
.
.
,
,
or
.
,
.
,
.
a
.
where H is the amount of heat produced and J is a constant
J is called the mechanical equivalent of heat it is equal to the
amount of work or energy which must be expended to produce
one unit Of heat On e calorie can be produced by the expenditure
of 42 x 1 0 ergs Hence We g s 42 x 1 0 H calories
.
,
.
7
.
r
7
.
T
P
AR
II
[
E
1 96
H AT
block is mou ted on a vertical shaft which c be m ade
to rotate rapidly by means of a pulley and belt Th e shaft
carries a worm wheel which drives a other wheel havi g 1 00
teeth This serves to i dicate the number f revolutions made
by the shaft Th copper calorimeter consists of two conical
v essels one of which fits into th e other Th e inside one carries
a wooden wheel to which a thread is fastened Th e thread i s
wrapped round the wheel about once and then passes over a pulley
Th e
an
n
.
.
n
n
O
n
.
e
.
.
,
.
F i g 22
.
.
and a weight is hung from it Th e inside par t of the calorimeter
contains water the temperature Of which is measured wi th a
thermometer A little i l should be put between the two conical
vessels so that the inner one c a turn round freely in the other
Th e outside vessel is made to rotate at such a speed that
the couple exerted by it on the inner vessel is j ust e ough to
support the weigh t
If r is the radius of the wheel in cms the couple is then equal
to m g dyne em s where m is the mass of the weight in grams
Th work done per revolution is therefore equal to 2
If
mg
the outer vessel makes revolutions the total work done i s
.
,
O
.
n
.
'
n
.
.
r
-
.
e
7r r
n
.
CH I x
.
E
]
H AT A
F ORM
OF
E N E RG Y
1 97
If the temperature of the water in the calorimeter
r ise s from 13 to t we have
27 r 71 r m g
.
1
2
277
°
n r
mg
— t
t
( 2 i ),
J (w
where w is the mass of the water in grams and w the heat capacity
of the calorimeter and thermometer In this way J can be found
within about 2 per cent Th e temperature 13 should be corrected
for loss f heat during the expe riment in the usual way
Rowland mad e a series of very exact measurements Of J at
B ltimore
in the years 1 8 7 7 — 7 8 by a method similar in
principle to that j ust d e s cribed Hi s calorimeter held about
9 000 grams of water and its temperature could be raised 25 C in
40 minutes
A steam engine was used to dri v e the apparatus
Rowlan d found that the energy required to raise one gram of water
n e degree centigrade was not exactly the
same for di fferent
degrees of temperature Th e following table gives some of
Rowland s results
E gy q i d t
i g m fw t 1 C
'
.
2
.
O
a
.
,
.
°
.
.
.
o
.
’
n er
ra s e a
ra
4 203
re
o
x
u re
o
°
a er
.
1 07
4 1 96
4 181
4 1 74
S ubsequent
researches have shown Rowland s results to be
nearly exact Th e umber of ergs required to raise one gram Of
water from 1 5 C to 1 6 C is very nearly
x 1 0
Th e number
o f foot po u nds of work
requ i red to raise the temperature of
a pound of water one deg r ee Fahr enheit is about 7 7 8
’
n
.
°
°
.
7
.
.
-
.
C HAP T ER
THE CO
N VERS I O N
X
OF HEAT I
N TO
WORK
a gas is compressed it gets hotter This may be shown by
means Of what is called a fire syringe A fire
ifi h t f
S
syri
ge
is
merely
a
small
brass
cyli
der
with
n
G
a piston and pis ton rod If a piece of tinder i s
put in the cylinder and the piston quickly forced down so as
suddenly to compress the air in the cylinder to a small volume
the air gets so hot that the tinder catches fire When a gas is
compressed work is done o it which is converted into heat If
a gas is allowed to expand so that it does work on the walls of the
vessel containing it the gas gets colder and heat must be imparte d
to the gas to bring it back to its original temperature Th e
amount Of heat required to raise the temperature of a gas therefor e
depends on whether the volume is kept constant or is allowed to
change when the gas is heated If the volume is kept constant
the gas does no work so that the heat required is used up in
raising the temperature Of the gas Th e amount of heat required
to raise the temperature Of a unit mass of gas one degree when its
volume is kept constant is called its specific heat at constant
vol me and will be denoted by G Th e specific heat at constan t
volume can be found by first measuring the heat capacity f
a vessel filled with the gas and then the heat capacity of the
same vessel when empty Joly measured the specific heat at
constant volume of several gases in this way Th e heat capacity
was found by immersing the vessel in steam and finding the mass
of water which conden sed on it while its temperature rose to the
temperature of the steam If 777 grams of water condense then
IF
.
.
c
pe c
a s es
ea
o
n
.
.
,
.
n
.
,
.
.
,
.
u
”.
O
.
.
.
.
m
L
'
(w
G0”)
,
(
—
t1
t2
)
,
A
P
R
T
II
[
E
200
H AT
It will be seen that the values of 0 got in this way di ffer
slightly from those found by J oly
Le t A B ( Fi g 23 ) be a circular hollow cylinder closed at B and
let CD be a piston which can slide freely along
M
l l
the
cylinder
uppose
that
the
space
below
the
S
fJ
piston is filled with a gas such as air or hydr ogen
Le t the pressure of the gas be denoted by p and its temperature
by t S uppose that the cylinder is heated
and that as the temperature rises the piston
moves up so that the pressure Of the gas is
kept constant If A denotes the area of
cross section Of the cylinder the force on
the piston is Ap and if the piston moves
up a distance d the work done by the force
is Ap d Bu t Ad is the increase in the
volume Of the gas so that if the volume
increases from v to v the work done by the
gas on the piston is 3) (v
Le t the temperature rise from t to t
when the volume of the gas increases from
at constant pressure Then we have
v to i
,
.
,
.
a
ti o
y
’
er s
Ca
c u
a
.
n
o
.
.
,
.
.
,
.
,
,
,
2
1
,
t
2
.
1
(
01
v,
o,
= v0 1
(
a
F i g 23
.
tg ) ,
.
where v is the volume of the gas at 0 C and or its coe fficient of
expansion at constant pressure Hence v 0 0 ( t t ) so that
W the work done is given by W= p v ( t — t )
When the gas is heated at constant volume it does no work
but when it is heated at constant pressure it does the work j ust
calculated In 1 8 42 before J oule s researches Mayer suggested
that the greater amount Of heat required to heat a ga s at constant
pressure is necessary because some of the heat is used up in doing
the work S uppose the mass of the gas is m then the di fference
between the heat required at constant pressure and that at constant
—
m
(7
volume is equal to (
( t t ) If J denotes the amount
of work equivalent to one unit of heat then Mayer supposed that
°
,
.
2
.
oa
0a
1
1
2
l
,
.
,
’
.
,
,
.
,
'
2
p
1
.
,
v
a
o
p
Thus Mayer
(
—
t1
tg
=
7
772
U
)
( p
(
—
t
t
2
1)
(
.
supposed that when a gas is heated at constant
CH x
.
]
T HE C O N
VE RS I O N
OF H
E AT I NTO W ORK
20 1
pressure the heat actually re m aining in the gas is the same as at
onstant volume and that the additional heat required at constant
pressure does not stay in the gas but is used up in doing the work
Th e equation j ust Obtained by
t h e gas does when it expands
m aking this i m portant assumption reduces to
c
“
.
a
v
o
p
O
( p
m
where
is
the
density
of
the
gas
at
?
7
p
0
air we have
Also i f p
7 6 0 mm
p
Fo r
.
p
C
0
p ( p
.
,
02 3 8 9
p
00 01 29 3 ;
0
76
Hence
and
01 ,
1 3 5 96
x
G,
0 C
°
and
x
00 0 1 29 3
.
and at the pressure
0 1 7 00
tr
calories per gram
00 03 6 7
:
.
.
00 03 6 7
9 8 00
x
a
00 6 8 9
ergs per calorie
This result agrees very well with the value of J found by
direct experiment which S hows that Mayer s assu m ptio
was
nearly correct According to Mayer s assumption we should
e xpect that if the gas were allowed to expand without doing any
work then the heat required to raise its temperature from t to 15
would be the same as at constant volume We may suppose that
the ga s first expands from v to a without doing any work and
without recei v ing any heat and is then heated at constant volu m e
to t It is found that the specific heat at constant volume is
nearly the same whatever the vol u me so that according to Mayer s
assumption the first operation of expanding from v to without
doing any work ought to leave the gas at its initial temperature t
a
d thus the heat required for the final operation would be equal to
0 (t
the same as for merely heating from t to t at constant
volume Accordi g to Mayer s assumption we should therefore
e xpect that on allowing a gas to expand without do i ng any work
i ts temperature would not change
G y Luss a c a d later J oule tried experiments to test this point
Tw large vessels Of about equal volumes were connected by
a pipe co taini g a cock
O e vessel was exhausted and the other
contained air at a high pressure On Ope ing the cock the air
e xpanded to twice its volume without doing any work on the
whole and it was found that when the whole apparatus was
x
10
7
.
’
n
,
’
.
1
2
.
,
,
,
.
’
,
,
,
,
1
,
n
7 72
2
,
1
2
’
n
.
.
a
n
-
,
.
,
o
n
n
.
n
.
n
P
II
A
R
T
[
E
202
H AT
immersed in water in a calorimeter no appreciable change of
temperature took place More exac t expe iments made later by
a di fferent method have shown that there is really a slight coolin g
e ffect when air expands without doing any external work This
e ffect however is very small S O that it appears that Mayer s
assumption is almost exactly true
When a gas is compressed it gets hotter so that to keep it t
a constant temperature heat must be removed
from it It follows from the truth of Mayer s
assumption that this heat must be an amou t Of
energy equal to the work do e on the gas in compressing it I f
no heat is removed from the gas then since it gets hotter it follow s
that its pressure increases mo e rapidly as its volume is diminishe d
than it would if the temperature were kept co stant When th e
state of a substance is changed in any way and no heat is allo we d
to enter or leave it duri g the cha ge the change is called n
a di a b a ti c change
If the temperature is kept constant during
a change then the change is called an i o th e m l change
It is di fficult in practice to prevent heat from enteri g or
leaving a substance when its volume and pressure are cha ged
r
.
.
’
.
a
’
.
n
n
.
,
,
r
n
n
n
.
a
,
.
s
,
r
a
.
n
n
.
F i g 24
.
.
We may however imagine the substance to be enclosed in a vessel
the walls of which are perfect non conductors Of h eat and we can
discuss theoretically the properties a substance would have u der
such conditions In practice if the change i the state of th e
substa ce is made very rapidly there may not be time for an
appreciable amount of heat to enter or leave the substance so that
very rapid changes of state are Often practically adiabatic changes
-
,
n
.
n
n
,
,
,
.
P
AR
II
T
[
E
204
H AT
S uppose
the pressure and the volume of the gas are represented
by the point M and that the gas is enclosed in a cylinder and piston
made f a perfect non conductor of heat If the piston is forced
down so as to diminish the volume of the gas it will get hotter so
that its pressure wi ll rise more rapidly than if its temperature
were kept constant Th e relation between the pressure and
O
-
.
.
1 0 00
3 0 00
40 00
volume wil l therefore be represented by a curve like the dotted
line L MN which is steeper tha the isothermal curves A curve
like LMN which represents the relation between the pressure
and the volume when no heat enters or leaves the gas is called an
Th e temperature corresponding to any point on
a di a b ti c c
ve
an adiabatic curve is the temperature of the isothermal curve
n
a
ur
.
.
CH x
.
]
THE C O
NVE RS I O N
OF H
E AT I NTO W O RK
passing through the same poin t Thus the temperatur corre
and
t
o M it is 0
2
to
is
s o di
L
7
3
C
p
g
Th e isothermal curves shown in Fi g 25 also represe t the
relation betwee the pressure d v olume of any other gas for
which p v = p v ( 1 + t) provided the volume of the other gas
taken is equal to that Of one gram of air at the same pressure and
te mperature
A di agram showing the relatio s between the pressure and the
volume of any body is called an i di c a to di a g m In Fi g 26 let
e
.
n
°
°
n
n
.
an
n
t t
a
o o
,
.
n
n
F i g 26
.
ra
r
.
.
.
be a curve S howi g the relation between p and v for some
quantity of any substance If the substance changes from the
state represented by P to that represented by a point P close to
P then the i ncreas e Of volume is represented by N N
If P and
P are ve y close together the press ure at P will be practically
equal to that at P so that since the work don e by the substa ce
when it expa ds from P to P is equal to p (
where
is
its
)
p
pr essure and v v the increase of volume the work done by the
substance in going from P to P will be represented by P N x N N
that is by the area P NN P If on the diagram p is expressed in
dynes per sq cm and the volume in c c the area will represent
AB
n
.
’
’
.
,
’
’
r
n
,
’
n
o,
1
,
,
’
’
’
.
.
.
.
.
c
l
R
P
II
A
T
[
E
206
H AT
the work done in ergs If the pressure is in pounds weight per
square foot and the volume in cubic feet the area will represent
the work do e in foot pou ds If the substance goes from the
state represented by A t the state represented by B alo g the
curve A P P B then the work done by the substance will be
represented by the area A BCD F the whole curve A B can
be divided into a great many short bits like P P and in the same
way as for P P the work corresponding to each bit is represented
by the area Of the vertical strip below it like P N N P Bu t all the
strips make up the area AB CD so that this area represents the
total work done in going through the states represented by the
curve between A and B
A heat engine is a machine for converting heat into mechanical
We are now in a position to consider the
work
i
H t E
elementary part of the theory of such engines
Th e most important kind of heat engine is the steam engine We
.
,
n
n
-
.
o
n
’
,
or
.
’
’
’
’
.
.
.
ea
n
g
n es
.
.
.
F i g 27
.
.
shall first consider a simple form of heat engine the theory Of which
can be easily worked out This engine is called C o t s ideal
heat engine It works on the same general principles as real
engines but is much simpler It cannot be realised in practice ,
but in real heat engines an attempt is made to approximate to
the theoretically perfect conditions under which C o t s ideal
engine is imagined to work Fi g 27 S hows C
t s engine
A is a cylinder and piston Th e side walls of the cylinder and the
arn
.
’
.
.
’
a rn
’
.
.
.
a rn o
.
E
208
[
H AT
PART 1 1
body to keep the temperature Of the whole of the substance fro m
falli g below t
Le t the piston rise slowly till the volume and pressure
represented by B and let AB represent the relation between th e
pressure and volume duri g the expa sion so that A B is an
isothe mal curve for the temperature t Le t the amou t of heat
absorbed from the hot body duri g this operation be denote d
by H
If the cylinder contains water and water vapour A B will b e
a horizontal line and the heat absorbed will be the latent heat of
n
z.
are
,
n
n
r
g
,
n
.
.
n
2.
,
F i g 28
.
.
evaporatio of the water which evaporates when the piston rises
N o w let the cylinder and piston be removed from the hot body B
and put on th e non conductor O Th e substance in the cylinder
is then completely surrou ded by non conductors of heat so that
no heat can enter or leave it Le t the piston now be allowed to
again rise S lowly Th e substance the does more work on the
piston and its temperature falls Le t the expansion co tinue till
the temperature has gone down to t that of the cold body D
B C is then an adiabatic
S uppose B 0 represents this expansio
d
Th e cylinder and piston are now put on the cold body
Curve
the piston is pushed down very slowly Work is then done on the
substance and it tends to get hotter but if the motion of the piston
is slow enough the cold body will remove heat from it fast enough
n
.
-
.
n
-
.
n
.
n
.
,
.
,
n
.
an
.
.
,
CH x
.
]
TH E C O N
V E RS I O N
OF H
E AT I NTO W ORK
209
to prevent its temperature f i i in g above t L t OD represent
this isothermal compressio
at t Th e compression is stopped
when the poi t D which is on the adiabatic curve through A is
r eached
Le t the amount of heat given up t the cold body be
denoted by H Th cylinder and piston are n e w again put on the
conductor C and the piston is very slowly forced down u til
the temperature of the substance h rise to its original value
This adiabatic compression is represented by D A because D
t
was chosen S as t be on the adiabatic cur v e through A Th e
substa ce has w been brought back exactly to its original state
represented by the p int As so that the amount of energy in it
must be the sa me as at the start Th e work done by the
sub tance duri g the expa nsion along AB C is represented by
the area AB OC A Th e work done o the substance during th e
compression along ODA is represented by the area
Th difference between the work done by the substance on the
piston and the work done on the substance by the piston is
therefore represe ted by the area AB CD Th e series of opera
tions described is called a C o t s cycle because it was in v ented
by Carnot and brin gs the substance back t its initial state Th e
result f the cycle is that a total amount of work represented by
the a ea A B CD has been done by the substance and an amou t
f heat H has been taken from the hot body and an amount of
heat H h as been given to the cold body L t W den te the work
represented by AB CD A amount of work W has been do e and
an amount Of heat H — H has disappeared We therefore con
elude that
ron
n
t
r s
,
l
.
e
.
n
o
.
,
n on
e
.
n
-
n
as
,
.
o
O
.
n o
n
o
.
n
s
’
’
n
.
'
e
n
.
.
’
a rn
o
.
O
r
n
O
2
1
.
.
2
e
n
o
n
1
W
.
J ( H2
where J denotes as usual the mechanical equivalent Of heat Th e
e fficiency of a heat engine may be defined to be the amount f
work it does di v ided by the mecha ical equivalent of the heat
which it receives from the source Of heat used to drive it Th e
e fficiency E of C
t s engine is therefore given by the equation
.
O
n
.
’
a rn o
W
us now consider what would happen if we worked the
engin e backwards S tart at A and expand adiabatically to D
w P
14
Le t
.
.
.
P
II
A
R
T
[
E
21 0
H AT
and then isothermally on the cold body to G ; then compress
adiabatically to B and isothermally o the hot body back to A
If these operatio s were done extremely slowly the substance
would go backwards rou d exactly the same cycle f pressures and
volumes represented by A DOB as it previously went round forwards
If the cycles were not done very slowly the pressures and
volumes could not be the same when goi g backwards as when
going round forwards Co sider the expansion from A to B If
this were done quickly the temperature would fall below t duri g
it so that the actual relatio between the pressure and the volume
would be a curve j oini g the poi ts A and B but lyi g below the
isothermal cur ve A B D uring the backwards compression from B
to A the temperature would ise above 25 so that the actual relation
between the pressure and the volume would be repr esented by
a curve j oi ing B to A but lying above the isothermal curve BA
Only if the cycle is gone through extremely lowly can it be
represented by the sa m e curves whether done backwards or
forwards A cycle which is gone through in such a way that it
can be represented on the i dicator diagram by the same curves
whether gon e through backwards or forwards is called a reversible
cycle C
t s engi e is supposed to work in a reversible cycle
and it is therefore called a perfectly reversible engine Th result
f the cycle whe do e backwards is that a qua tity f heat H
has been taken from the cold body a quantity H given to the hot
body d a total amou t of work W represented by the area ABCD
has been done on the substance As before W J ( H H )
It is now necessary to consider a pri ciple called the e c o d la w
m i c s which is based on experience
i
It
h
e m o dy
s found that
t
o
f
heat does o t pass from cold to h t bodies by itsel f It can only
be made to do S O by the expe diture f work Heat flows by
conductio from hot to cold bodies but ever from cold to hot
bodies When C o t s engi e is worked backwards heat is taken
from the cold body and more heat is given to the hot body but
amount of work W has to be done If we could devise a process
for making heat go f om cold to hot bodies witho t the expenditure
of work we should be able to concentrate heat by means f it into
any desired body and so should not need to burn coal or other fuel
to get high temperatures N othing i our experience j ustifies us
in supposing that anything of the sort is possible
n
.
n
n
O
.
n
n
.
.
n
2
n
,
n
n
n
.
r
2
n
.
S
.
n
’
n
arn o
.
,
e
.
n
O
n
O
n
2
,
n
an
:
.
.
n
o
.
O
n
n
.
.
n
s
n a
-
1
2
n
r
1
.
n
’
a rn
n
,
.
u
r
O
.
n
.
an
R
P
A
T
II
[
E
21 2
H AT
it is impossible for any perfectly reversible engine using the same
hot and cold bodies to be less efficie t tha C o t s e gine
Hence all perfectly reversible engines using the same hot and cold
bodies or worki g between the same temperatures t and t hav e
equal efficie cies Th e efficiency Of a perfectly reversible engi e
is therefore i depe dent of the nature f the working substa ce in
it and depends only on the temperatures between which it works
If the e fficiency f a perfectly reversible e gi e working between
th
temperatures t and t is E then we have
’
n
n
n
n
arn
n
.
1
2
n
.
n
n
n
O
.
O
e
n
2
1
n
,
E =f ( tl tz)
,
,
where f ( t t ) denotes some function Of t and t only
We have then in E a quantity depe ding o ly on temperature s
Th e
d not on the special properties Of pa ticular substa ces
scale of temperature given by the constant volume hydrogen
thermometer is an arbitrary scale for it depe ds on the properties
f hydrogen
Th e scale depending
the expansion of a liquid like
me cury is equally arbitrary Lord Kelvin proposed to define
a scale of temperature by mea s of the e l ti o E =f ( t t ) and
so to get a scale of temperature indepe dent of the properties f
particular
substa
ce
y
T do this all that is necessary is to choose a particular form
for the function f If t = t it is easy t see that E must equal zero
for then the two isothermal curves in the Carnot cycle coi cide s o
that the area AB CD which represents W is zero This co dition
is satisfied if f ( t t ) = A (t — t ) where A is a quantity to be
determined
S uppose then that
,
?
,
1
2
n
.
n
n
r
an
.
n
,
O
.
on
.
r
.
n
r
a
n
1
2
,
,
l
O
n
an
n
.
o
1
.
o
2
,
n
.
,
,
2
g
l
,
n
,
.
E = A ( A A)
.
If E were equal to u ity all the heat taken from the hot b dy
w uld be converted i to work Le t the temperature Of the cold
body for which E 1 be taken as the zero of the new scale of
temperature We then have 15 0 when E 1 so that
n
n
o
.
o
.
1
:
1
te
Hence
o
CH x
.
]
TH E C O N
VE RS I ON
I NTO W O RK
E
O F H AT
This
21 3
relation defines a new scale f temperature which is called
the absolute s c a l because it does not depend on the properties of
an
particular
substance
y
Th e size of the degrees on this new scale are chosen so that
there are 1 00 degrees between the freezing point of ice and the
boiling point of water at 7 6 cms of mercury pressure We might
if it were possible determ i ne the e fficiency Of a perfectly reversible
e gine working between these two temperatures ; it would be
found to be equal to nearly 02 6 8 1 We should then have
O
e,
.
.
.
,
,
n
.
1 00
6
02 8 1
Hence
ts
and therefore t = 27 3 Thus the freezing point
on the absolute scale and the boiling point of
O f ice is at 27 3
water at 3 7 3 when the interval between these two temperatures
Practically it is not possible to co st r uct
is made equal to
a perfectly reversible engine so that the absolute scale ca ot be
realised in this way However we can S how that it must coincide
with the scale of temperature given b y a constant volume gas
thermometer containing an ideal gas the temperature f which
remains constant when it expands without doing work Hydrogen
gas very nearly satisfies this condition so that it follows that the
scale of temperature given by the standard hydrogen thermometer
is nearly the same as the ab olute scale
Fi g 29 shows an indi c ator diagram for a Carnot e gi ne
the ideal gas j ust mentioned When expanding
c onta i ing
isothermally from A to B the work done by the gas is represented
’
by the area ABB A Bu t for such an ideal gas the heat absorbed
is mecha ically equivalent to this work Hence J H is represented
by the area ABB A Th e efficiency of the engine is therefore
ven
by
g
W
area AB CD
area A BB A
JH
If AB and CD are very near together that is
early equal we have
area AB CD P Q P R Q R
area A BB A PB
PR
t2 = 3 7 3
1
.
°
°
n
n n
,
.
O
.
,
s
.
n
.
n
.
’
.
n
2
.
'
’
.
°
l
’
”
2
,
n
,
’
Hence
’
,
P
II
A
R
T
[
E
21 4
H AT
and Q R represent the pressures of the gas at the
temperatures t and t and the constant volume OR It appears
therefore that the pressure of the gas at constant volume is
proportional to the absolute temperature
Th e pressure p f the hydrogen in the standard hydrogen
thermometer is therefore very nearly proportional to the absolut e
temperature At the m lting point Of ice p 1 00 e m s of merc ry
and at the boiling poi t f water p 1 3 6 6 7 m s of mercury
If t denotes the melting point of ice on the absolute scale and
if we take the boiling point of water equal to t + l 00 so as t
Bu t P R
1
2
.
.
O
e
.
O
n
u
.
e
.
,
.
o
o
o
F i g 29
.
.
have 1 00 degrees between the melting point of ice and
boiling point of water we have
th e
,
to + l 00
1 3 6 67
to
1 00
which gives t = 27 2 7 or nearly
Hence 0 C is equal t
nearly 27 3 on the absolute scale and so t C is equal to nearly
t + 27 3
on the absolute scale
°
°
°
o
.
°
°
,
°
.
.
o
T
P
II
AR
[
E
21 6
H AT
If m is taken equal to the molecular weight of the gas in grams
this becomes
O
O
m
2
calories
)
(
Th e following table gives some values of O 0 m and of
,
'
”
p
.
p
,
0
,
,
01 07"
0
( p
Ga s
C,
01 )
Ai r
02 3 8 9
O
( p
m
( 70 ) m
0 1 70
Hy dr o ge n
20 1 6
i id
0 1 50
0 1 95 2
It will b seen that ( O
is very nearly equal to
2 calories for these gases
W have seen that the efficiency f a perfectly reve sible
engine is equal to ( t t )/t where t and t are
i
the
temperatures
between
which
it
works
on
the
t i
H
absolute scale T get a high efficiency it is clear
that the hot body or source of heat should be as hot as possible
and the cold body as cold as possible In actual heat engines such
as steam engi es the conde ser where the steam is condensed after
doing work in the engine corresponds to the cold body and the hot
water in the boiler may be taken to correspond to the hot body
Th condenser ca not be kept below about 1 5 C to 20 C so that
to increase the theoretically possible efficiency of a steam engi e
the only practical plan is to raise the temperature of the water in
the boiler Th higher the temperature of the water the higher
the pressure of the steam It is difficult to make large boilers
which will work safely at pressures much above 400 pou ds weight
per sq inch Th temperature of the water at which its vapour
pressure is e qual to 400 pounds per sq inch is about 23 0 C Th
theoretically possible e ffi ciency of a heat engine working between
23 0 C and 1 5 C is equal to
Ca r b o n d
ox
e
e
p
.
,
E ffi c
ea
en c
r
O
e
o
y
En g
1
2
2,
1
f
n es
.
o
.
.
n
,
n
,
,
.
e
°
n
°
.
.
,
n
e
.
.
n
e
.
.
°
.
e
.
°
°
.
.
( 23 0
27 3 )
(1 5
23 0
27 3
27 3 )
S uch
a heat engine therefore cannot convert into work more
than 43 of the heat which gets into the water in the boiler At
least 5 7 / of this heat must be given up to the condenser and
cannot be converted into work In practice this theoretically
possible efficiency cannot be obtained Th e temperature of the
.
0
O
.
.
CH x
.
]
THE C ON
VE RS I O N
E
OF H AT
I NTO
WORK
21 7
urnace of an engi ne may be 1 5 00 C If the engine could be
made to work bet ween this temperature and 1 5 C its possible
e fficiency would be
°
f
.
°
.
o r
about
84
F
N
RE ERE CES
Th eo ry
H
e
a t, C
f
o
Hea t Po y n
,
i
t
n
g
an
l
e r k-
d
Ma x we
J J Th
.
.
ll
.
om son
.
C HAP T ER
THE K I
XI
Y
N ETI C
THEOR
OF G AS ES
chief properties of gases can be very completely explained
by the theory according to which a gas consists of an imme s e
number of i dependent molecules moving about in the space
occupied by the gas This theory is called the ki etic theory of
gases In a gas at 0 C and 7 6 cms Of mercury pressure there are
about 2 6 x 1 0 molecules per cubic centimetre Th e diameter
of each molecule is about 3 x
cms so that the total volum e
of all the molecules in o e c c is only about
TH E
n
n
n
.
°
.
.
.
19
.
.
,
’
n
26
x
10
19
x
.
14
.
x
3 6
x
Thus
c c
.
only about four parts in ten thousand f the total volume i s
actually occupied by the molecules in a gas at 0 C and 7 6 cms
pressure Th e molecules collide with each other d with th e
walls of the vessel containing the gas Th e collisions between
the molecules continually change their velocities in direction and
in magnitude and produce a certain average distribution Of
velocities amo g the molecules Th pressure of the gas on th e
walls f the vessel containi g it is due to the impacts f th e
molecules Th e molecules are s u pposed to be perfectly elastic s o
that when they hit the walls of the vessel they bounce Off
without loss of velocity
Consider a plane area of o e square e m on the wall of th e
vessel containing the gas S uppose the gas near this are a
contains molecules per c c each of mass m that have velocity
components towards the area equal to c Th e number of these
molecules striki g the area per second will be n o Each of these
molecules wh e it strikes the surface and bounces o ff will com
m n i c te an amount of momentum 2m v to the surface for th e
momentum of the molecule in the direction towards the surface
O
°
.
.
an
.
.
n
e
.
O
O
n
.
,
.
n
.
.
.
n
.
,
.
n
.
n
u
a
,
P
II
A
R
T
[
E
220
H AT
squares of the velocities of the molecules Th e following table
gives the values of p and U for several gases t 0 C and 7 6 cms
of mercury pressure calculated in this way To calculate U it is
necessary to express p i dynes per sq cm and p in grams per c c
7 6 cms of mercury 1 0 1 4 1 0 dy e s /c m
D ity t 0 C d 7 6 m
.
.
°
a
.
.
.
n
.
.
6
en s
Ga s
.
°
a
an
s cm 3
/
0 0 0009
Ox y g e n
0 0 0 1 429
N tr o g e n
Ca r b o n m o n
0 0 0 1 25 4
id
s
.
.
Hy dr o ge n
ox
c
.
gr a m
i
.
2
n
.
.
0 0 01 25 1
e
li m
0 0 00 1 7 8 7
of a gas is equal
Th e kinetic energy of the molecules in one
3
to p U or since p = 4p U the ki etic energy is equal to 3 p /2
This is the energy of the translatio al motion of the molecules
only d does not i clude any energy they may have due to
rotation or other forms of internal motion
When a gas is heated so that its temperature rises the energy
of the molecules is increased Th total energy required to raise
the temperature f o
of a gas at constant volume from t to
Th
t)
part of this which goes to
t C is equal to J p C ( t
increase the velocities of the molecules is equal to
He
u
c c
2
2
~
2
n
,
,
.
.
n
n
an
,
.
,
e
.
O
n e c c
x
°
2
v
.
g( p
No w J
2
2
( see
O
°
1
.
—
1
P
“
I
e
.
)
3
900!
2
page
t
(z
t1 )
o
In this equation we may put
H
because
is
independent
of
ence
the
energy
p
p /p
p
p
required to increase the velocities of the molecules is equal to
—
3
f
h
O
ratio
this
to
the
total
energy
T
O
J
C
t
t
)
)(
3p (
required to heat the gas at constant volume is therefore
0
,
—
,
p
v
z
l
.
e
.
(
7
Q
C”
2
_
F
where
gases like helium argon and mercury
1)
1 which shows
vapour it is found that y = so that 3
that all the energy required to raise the temperature of these
gases goes to increase the velocities of the molecules Fo air
2
=
Fo these
1
so that % ( y
hydrogen and nitrogen y
gases therefore g of the energy required to b e a t them at constant
volume goes to increase the internal energy f the mol ecules
or
,
f
7
,
r
.
e
5
r
)
O
.
,
CH
xI
.
]
THE
K I NE TI C
E
TH O R
Consider
Y
OF
221
G AS ES
a cylinder and p s ton cont ining a gas Le t the
piston move down with a un i form vel city 7) S O as to compress the
gas S uppose a mol e cule having an upward velocity collides
with the piston Th velocity f the molecule relati v e to the
After the impact its v elocity
piston before the impact is +
will be di rected downwards but will still be
1 relative to the
piston Th e v elocity after the impact is therefore equal to
Every molecule which collides with the piston h as its velocity
ormal to the piston increas ed by 21} in the same way so that the
motion of the piston increases the average velocity Of the
molecules and therefore makes the gas hotter In the same way
i f the piston moves up so that the gas expands the velocities of
the m lecules e decreased and the gas gets colder If the gas
i allowed to expand int a v acuum the velocities of the molecules
are not changed so that the average tempe ature of the gas does
not alter
If the number of molecules in one
of a gas is and each
one h a s a m a ss m then p the density of the ga s is equal to
m hence
a
i
.
o
u
.
O
e
.
u
)
a
,
u
.
n
.
,
,
ar
o
.
s
o
r
,
.
c c
n
.
,
n
,
p
g
n
m
U
2
.
If we have ano ther gas at the same pressure and temperature
containing molecules per c c each of m a ss m then
n
’
’
.
.
,
—
fi
2
rm
:
where U denotes the average value of the squares Of the vel o cities
f the molecules of the second gas
If we suppose that the
average ki etic energi es Of the molecules of di fferent gases are all
equal at the same temperature then we have
’2
o
.
n
,
m
4
U =
2
’
’
z
Am U
.
With the previous equation this gives = so that if the
assumption j ust made is true all gases S hould contai equal
numbers of molecules per c c when they are all at the same
pressure and temperature Their de n sities S hould therefore be
proportional to their molecular weights which is found to be the
ca se This shows that all gaseous molecul e s have nearly equal
ave rage kinetic energies of translation at any gi v en temperature
REF EREN CES
H t P y ti g
d J J Th m
H
J
J
Ki ti Th
G
y f
'
n
u
,
n
.
.
.
,
.
.
ea
,
o
n
n
an
.
.
o
so n
.
ne
c
eo r
o
a ses ,
.
.
ea n s
.
P AR T I I I
S OU N D
C HAP T ER
P ROD U CTI O
N
AND
I
VEL OCI TY
OF S OU
ND
sensation of sound is produced by a disturbance trans
Th e
m i tte d from sounding bodies through the air to the ear
study of the processes taking place in the ear and brain belongs
to psychology and physiology while the study of the process taking
place outside the head belongs to the branch of physics known as
d
S ound is produced by rapidly vibrating bodies and by any
so
sudden disturbance of the air F example when a gun is fired
THE
.
,
un
.
or
.
Fi g 1
.
,
.
a large volume of gas is suddenly emitted by the g so that the
surroundi g air is v iolently pushed away from the muzzle of the
If
one
end
of
a
short
steel
spring
is
held
in
a
vice
and
the
g
other end pulled sideways and le t go the spring vibrates rapidly
backwards and forwards and gives out a sound Fi g 1 shows
a tuning fork which is a symmetrical steel fork with two prongs
If the prongs are pressed towards each
A and B and a ha dle 0
other and then let go they vibrate and give out a sound Th e
fork can also be set vibrating by striki g one of the prongs with
a light wooden hammer covered with felt Th e felt is i tended to
un
,
n
un
.
,
.
.
,
n
.
,
.
n
.
n
224
SO
UND
P
T
1
1
A
R
1
[
where 1) denotes the average velocity of the sound betwee
the gun and the observer
Th e velocity of sound in d y air at 0 C is about 3 3 200 c m / e c
or 1 090 feet per second Thus if the bserver is 1 0900 feet away
from the g u the i terval between the flash and the report is abou t
To find the velocity of sound in air accurately is very
1 0 seco ds
di fficult Th e velocity is affected by wi d and depends on th e
temperature and humidity of th e air Th e distance used must b e
large to get a time interval which can be measured exactly and i t
is di ffi cult to get the temperature and humidity exactly over a larg e
distance
Th e velocity has been found fairly exactly by selecting tw
stations at an exactly known distance say 1 0 miles apart on hill
with a v alley between At each station a cannon s placed and
an observer provided with a stop watch or other form of chrono
graph to measure the time interval Le t the two stations b e
denoted by A and B Th e observer at A fires his cannon and the
observer at B measures the time bet ween seeing the flash and
hearing the report from A Th e observer at B also fires his can o
as soon as he sees the flash at A and the observer at A measures th
time between seeing the flash and hearing the report from B
Th e velocity is got by di viding the distance between A and B b y
the mean of the two times Th e temperature and humidity
observed at A and B and possibly at intermediate points If th
wind is blowi g from A to B with a velocity a then the velocity o f
the sound from A to B will be 2) u and from B to A it will b
Hence
7)
s
t
/,
n
.
°
r
n
.
s
.
O
.
n
s
.
n
,
.
n
.
.
,
.
o
,
,
in
,
s
,
i
.
-
.
.
n
n
e
.
are
.
e
.
n
,
e
1)
14
.
8
U
u
”
where t is the interval measured at B t that measured at A a
Th e mean of 15 and t is therefore
s the distance from A to B
,
,
2
1
.
n
d
2
il + ie
2
—
v
u
2
2
If the velocity of the wind is small
"
with v so that
u
,
or
v
?
,
it
'
can be neglected compare d
I]
CH
.
PROD U CTI O N
AN D
VEL OC I TY
OF S O
U ND
225
If the wi nd is blowing across the direction fro m A to B then it
carries the so u nd waves sideways so that the sound takes longer to
go either from A to B or from B to A than it would if there were
no wind In this cas e therefore the e ffect of the wind is not got
rid of by taki ng the mean of the two time intervals Th e best plan
is to make mea surements only on very calm days when there is little
or no wind and when y wind there may b e is blowing from A to B
or from B to A It is foun d that very powerful sounds travel quicker
than ordin ary sounds Thus close to the cannon the velocity is
greater than at a distance This e ffect however is too small to
produce an appreciable error when the velocity is measured over
a dis tan ce of several miles Except in the case of unusually
intense sounds it is fo d that all sounds travel with the same
v elocity ne rly 3 3 200 e m s /se c in dry air at 0 C At a temperature
t C the velocity is equal to 3 3 200
6 l t e m s /s e c provided t i small
Th e velocity is found to be independent of the pressure of the air
It is the same at the sea level as between the tops of high moun
tains where the pressure is lower
Th e velocity of sound in water has been found by striking
a bell un der the water in a large lake A gun was fired above
the bell by the same action that struck the bell An observer
at a great distance measured the time between seeing the flash of
the gun and hearing the sou d of the bell in the water Th e
v elocity was found to be 1 43 500 e m s /s e c which is more than four
t i mes that in air
,
.
.
an
.
.
.
.
un
,
°
a
.
.
°
.
s
.
.
.
.
.
.
n
.
.
.
F
N
d J J Th
RE ERE CE
Po y n
W
.
P
.
ti
n
g
an
.
.
om so n
.
C HAP T ER I I
WAVE
M OTI O
N
D I S TU RBAN CE produced at a point in a medium like air is
propagated in all directions with a definite velocity In a similar
wa y a disturbance produced at a point on the surface of water is
propagated in all directions over the surface Fo example if a
small stone is thrown into a pond waves spread out from the
point where the stone enters the water in the form of circles the
radii of which increase at a nearly uniform rate with the time In
such cases a movement travels through the medium but after it
has passed by the medium is left in its original position or very
near to it Th e propagation of such moveme ts through a medium
is called wave motion If we fix our attention on a particular
particle in the medium then this particle moves while the waves
are passing over it but is left in or very near to its original position
when the waves have gone beyond it When the motion of the
particle is parallel to the direction in which the waves are moving
the waves are called longitudinal waves and when the motion
of the particle is in a plane perpendicular to the direction
i n which the waves are travelling the waves are called transverse
waves It is found that sound in air consists of longitudinal
waves
A powerful electric spark from a battery of Leyden j ars pro
duces a sound wave of great intensity Th e spark very suddenly
heats the air so that it expands and its pressure is suddenly
increased In this way an intense wave is started which moves
outwards from the spark in the form of a thin hollow sphere with
the spark at its centre Th e radius of the sphere is equal to vt
where v is the velocity of sound in air and t the time since the
spark occurred In the wave that is close to the surface of the
A
.
.
r
,
,
.
,
,
n
.
.
,
.
,
.
.
.
.
.
.
228
SO
wby long cords
P
III
T
AR
[
UN D
etween each S phere and the next a spiral
S pring is fixed
If the first sphere is suddenly pushed towards the
next one the spri g is com pressed and pushes the second sphere
and S O compresses the second spring which moves the third sphere
and so on Each sphere moves forward and compresses the next
S pring but is brought to rest in doing this
Th e compression
can be seen to move along the row of spheres with a uniform
velocity
Co sider a long cylinder 00 F i g 4 full of air with a piston P
at one end S uppose the piston is suddenly pushed in from A to
B
Th e air close to the piston is then suddenly compressed so
that its pressure rises and it is also gi ven a velocity in the direction
AB
This layer of compressed air is brought to rest by the
backward force exerted o it by the air in front of it but at the
same time it compresses and sets in motion this air which is in
turn brought to rest by the next layer of air This next layer is
'
ro
B
.
.
n
,
,
,
.
.
.
n
.
,
,
,
.
.
,
.
n
,
,
.
Fi g 4
.
.
also set moving and compressed i the same way and i turn passes
on its m o ti o
d compressio to the next layer and so on Thus
the motion and compression are handed on from o n e layer to th e
next and so travel along the cylinder as a longitudinal wave in
which the pressure is higher than in the undisturbed air After
the wave has passed the pressure is the same as before but the air
has been moved along the cylinde through a distance equal to
AB
Th wave travels more than 1 000 feet in one second while
the distance AB may be only a small fraction of an inch Th e
motion of the air in the cylinder produced by suddenly moving the
piston from A to B is represented in Fi g 5 Th e horizo tal rows
of circles represent air particles situated along a line parallel to the
le gth of the cylinder These particles are taken at equal distances
apart in the undisturbed air before the piston is moved Th e top
row is supposed to represent N air particles numbered 1 to 1 7
n
n a n
n
,
n
.
,
.
,
r
.
e
.
.
n
n
.
.
.
CH
.
II ]
W AVE
I
229
M OT ON
with the surface of the piston at A at the instant when the piston
begins to move Th e second row represents the piston and the
same 1 7 particles at a short time after the piston beg ins to
move Th e piston has moved forward but the particles shown
have not yet moved Th e following rows S how the positions of
the pa rticl e s and the piston at the ends of success ive equal
.
1
.
.
Fi g 5
.
i ntervals
.
of time Thus distances measured vertically downwards
from the top row are proportional to the time elapsed S ince the
piston began to move and di s tances measured hori onta lly from
left to right from the vertical line through A are proportional t
t h e di sta nces from the original position of the piston
Th wave
produced by the motion of the piston travels with a velocity 1) so
7
.
,
z
o
.
e
SO
P
III
R
T
A
[
UND
that the air particles at a greater distance than vt from A have
not yet been disturbed Le t the original distances between the
particles shown be each equal to d and suppose that in the time
the sound wave travels a distance d so that d 11 Then all the
particles represented by circles which are above the line AD are in
their original positio s During the i n terval 0 to 3 the piston
moves with uniform velocity if from left to right and its motion is
represented by the line AB It then remains at rest as shown by
the vertical line BF Th e air in co tact with the piston therefore
comes to rest at B at the time 3 and if we draw a line B G parallel
to AD all the air particles represented by circles below this line will
be at rest for the motion communicated to the air by the piston moves
along with velocity v Th e particles represented by circles between
AD and B G are movi g from left to right with a velocity equal to
the velocity of the piston during its motion from A to B Th e
lines like 1 1 22 3 3 etc drawn through the circles which represent
successive positions of the same particle S how the motion of each
particle Fo example particle N o 1 0 remains at rest till the
time 1 0 and then moves forward a distance equal to the distance
which the piston moved in the interval 0 to 3
N O 1 0 moves
forward during the i terval 1 0 to 1 3 and then remains at rest
In the same way N o 3 moves duri ng the interval 3 to 6 and
then remains at rest Thus each particle moves in the same way
as the piston but not at the same time Th e time when each
particle moves is later than the time when the piston moves by /v
where is the distance of the particle from the original position
of the piston and v the velocity of sound in air Where the
particles are moving they are closer together so that in the wave
the air is compressed Th e distance between the particles in the
wave in Fi g 5 is about half that between the undisturbed particles
In ordinary sound waves the compression is very small but in very
intense waves it may be large F i g 5 shows an extremely intense
wave in which the motion of the air is large so that it can be clearly
seen on the diagram
Fi g 6 is a similar diagram S howing the motion of the air
particles when the piston moves from A to B during the interval
a d th e n
moves back to its original position during the
0 to 3
interval 3 to 6
.
7
7
.
‘
n
7
.
.
n
.
7
,
.
n
.
,
,
.
,
r
.
.
,
7
7
n
.
.
1
7
.
7
.
1
.
.
x
,
as
.
.
.
.
,
.
.
.
~
.
.
7
n
7
.
7
.
23 2
SO
U ND
A
R
T
P
III
[
piston so that its distance from No 1 0 is d minished from d m
to (v
Th e rar efaction is produced in the same way by the
backward motion Th e distance between the particles in the rare
factio is
Th e diagra m s S hown describe the motion of the air particl es
produced by the motion of the piston but they do not explain why
the air moves in the way described Le t us consider a particular
particle N 9 say and S how that it is acted on by the forces
required to give it th e motion described Up to the ti m e 9 N 9
remains at rest and the pressure is the same on both sides of it so
that there is o force tending to move it At the insta t 9 the
front f the compression arrives at No 9 so that j ust then the
pressure is greater behind it than in front ; it therefore receives an
impulse and starts moving forwards While it is in the com
pression that is from 9 to 1 2 the pressure is the same on both
sides of it and so its velocity remains co stant When the back of
the compression reaches it the rarefaction is behind it and the
compression in front so that there is a backward impulse on it
which converts its forward velocity into a backward velocity From
1 2 to 1 5 it is in the rarefaction and moves with uniform velocity
backwards and when the back of the rarefaction reaches it it gets
a forward impulse which j ust stops it and it then remain s at rest
in its original position
In Fi g 7 let t h ere be a wave of compression between the
planes AB and CD and let it be moving from left to right with
’
velocity
Le t the pressure in the wave be p and let that in the
surrounding undisturbed air be p At the plane CD there is an
unbalanced force per unit area equal to p p Bu t the plane CD
is advancing with velocity 7) so that if the velocity of the air in the
wave is v the amount of momentum given to the air per seco d
per unit area of CD is equal to pvv where p is the density of the
undisturbed air
i
.
,
-
r
.
n
.
o
,
.
,
o
7
.
n
n
.
O
.
.
1
,
.
,
7
7
,
,
n
,
.
,
.
7
1
,
,
.
.
,
,
.
’
.
'
n
’
,
.
Hence
p
'
p
'
v
v
p ,
nce force is equal to rate of change of momentum In a short
time t the plane CD moves forward a distance vt and the air at
CD at the beginning of the time 5 moves forward a distance v t
so that a volume of air vt outside the wave is compressed into
Si
.
,
'
7
,
CH
.
II ]
WAVE MOTI O N
23 3
a volume ( v — v ) t in the wave Th e change of volume per unit
volume of the air is therefore equal to
’
’
.
If E denotes the volume elasticity of the air we have therefore
w
I
w
Hence
E
-
,
,
,
.
—
1)
or
’
E
'
p vv
'
1)
It appears therefore that the velocity with which longitudinal
so u nd waves advance through air or any
other fluid is equal to the square root of
the quotient of the bulk modulus of elas
ti c i ty by the density of the fluid
When sound waves pass through a gas
the chan ges of pressure take place so quickly
that there is no time for any heat to enter
or leave any portion of the gas If the gas
is compressed when a wave passes over it
it therefore gets hotter which makes its
pres sure rise more than if its temperature
had remained constant Th e bulk modulus
of elasticity is defined by the equation
.
.
,
,
.
V
'
P
=— E
P
-
—V
V
Fi g 7
where V is the volume at pressure p and V
the volume at pre s sure p and p is supposed to be only very slightly
greater than p If the temperature of the gas is supposed constant
we have for a gas that obeys Boyle s law
’
’
.
.
'
,
.
’
’
’
= V
V
P
P
(p
Hence
’
E
v
ar
’
V
V
-
p
.
If p and p are nearly equal which is the case in sound waves
of ordinary intensity we may put E = p Th e bulk modulus of
’
,
,
.
23 4
SO
U ND
P
III
A
T
R
[
elasticity of a gas at constant temperature is therefore equal to its
pressure This bulk modulus is called the isothermal elasticity of
the gas
.
.
N ewton
who first obtained the formula
,
in the year
1)
tried to calculate the velocity of sound in air by putting
E =p
Th e result he obtained did not agree with the observed
velocity Laplace a d Poisson in 1 8 07 poi ted out that the
temperature of the air should not be supposed to be co stant
because when the gas is compressed in a sound wave its tempera
ture must rise Instead of the isothermal elasticity the elasticity
when no heat enters or leaves the gas should be used This is
called the adiabatic elasticity of the gas
To calculate the adiabatic elasticity of a gas we may suppose
that the work done on the gas when it is compressed is converted
into heat which raises its temperature Le t V denote the volume
of one gram of the gas at pressure p If it is compressed to
’
a volume V very S lightly less than V the work done o it is
because
for
a
small
cha
ge
in
V
the
change
in
can
be
V
p (
p
neglected in comparison with p Le t the rise of temperature be
from absolute temperature 9 to 9 so that
1 7 26 ,
.
n
.
n
n
,
.
.
.
.
.
n
n
.
’
V
P(
where 02is the specific heat of the gas t constant volume and J
=R
the m echanical equivalent of heat We have also J ( C
see
page
where
is
the
gas
constant
for
one
gram
of
the
R
(
gas and p V = R9 Also p V R9 where p is the pressure when
’
the volume is V We suppose p and p are nearly equal Le t
’
—
= d so that
=
V c a dp
V
p
a
,
'
p
.
’
'
’
’
.
'
.
.
’
n
,
=
V
d
R
9
V
c
d
c
+
+
+
p
p
'
product c d can be neglected because both 0 and d are very small
so that S ince p V = R9 we get
Th e
,
,
c
p
We have
so that
’
=
R (9
V
d
+
a
p
Vd
J C., ( 9
’
J Op ( 9
,
23 6
This
Th e
SO
P
III
A
R
T
[
UND
formula agrees well with the observed velocities in air
velocity of sound in other gases
be calculated by the
formula
.
Th e
0
of sound in liquids
Fo r
.
v
equation 0
water we have p
V2 x
10
10
1 40000
also gives the velocity
1
and E
c m s’
,
S OC
.
which agrees fairly well with the value found
“
.
2x
hence
C HAP T ER III
WAV E
TRAI
NS
WE have seen that whe n a sound wave
produced in a long
cylinder by suddenly movin g a piston at one end of the cy linder
each particle of air moves in the same way as the p i ston but at
a tim e t later than the piston given by w/v = t where is the
distance of the particle from the piston and v is the velocity of the
sound S uppose now that the piston is made to move with a
simple harmonic motion of amplitude A and period T In Fi g 8
is
,
a:
,
.
.
.
let P BA be a circle of
A a n d let P move round the circle
with a uniform velocity such that it goes once round in the time
T Le t AB be a fi ed diame ter and draw P N perpe n di cular to
Then N describes a simple harm o ic motion of amplitude A
AB
,
x
.
.
,
n
23 8
SO
and period
P
III
A
R
T
[
UND
the angle P OB be denoted by
so that
where
is
the
time
measured
from
the
instant
when
t
P
T
2
t
6
/
9
was at B Fo r P goes once round in T seconds so that the
angular velocity of OP is 2 /T We may take the motion of the
piston to be the same as the motion of N so that ON is equal to
the displacement of the piston from the middle point of its
7r
T
Le t
.
,
,
.
7r
.
,
oscillations
that
No w ON
.
OP
A cos
cos I
y
27 ?
()
A cos
1
Le t ON
y
,
so
2 15
5
This
equation gives the distance y of the piston from its mean
position at any time t when it is moving with a S imple harmonic
motion of amplitude A and period T N o w consider a particle of
air in the cylinder at a distance fro m the mean position of the
piston It will move in the same way as the piston but at a time
.
.
as
,
.
later
onsequently
if
the
displacement
of
the
air
C
/
particle at the time t will be the same as that of the piston at
the time t Th e displacement of the air particle at t is there
fore given by the equation
x
v
.
’
'
.
y
=A
cos
27r t
A
—
T
271
COS
—
T
(
f
a:
displacement of the air particle at any time
given by the equation
Th e
y
=A
(
t
c os
,
y
cos 2
7
( f)
—
t
m
is therefore
x
If the number of vibrations per second is n then
=A
t
so that
a
.
variation of y with given by this equation at the particular
instant t = 0 is shown in Fi g 9 curve A
A
0 v/
When
4U/ etc we get 3
A
v/2n 3 0 /2
etc
we get y
When
When
v/4
5 0 /4m etc
we get y 0
Th e curve showing the relation between y and w is what is
called a cosine curve
Th e
a:
.
as
:
’
u,
,
a:
23
n
,
:
.
,
«
n
,
,
,
.
n
,
.
,
.
,
.
,
1
.
.
.
240
'
SO
U ND
P
AR
T
I
I
I
[
particles are represented by the curves 1 1 22 3 3 e tc We see
that a series of compressions with rarefactions between them start
from the piston and move along the cylinder Th e period of the
~
,
,
,
.
.
Fi g 1 0
.
.
oscillation is 8 and two complete oscillations of the piston are
At the ti m e 1 6 two complete waves have been formed
S hown
In the curve represented by the equation
7
7
.
.
=A
y
cos 2 m
7
(
t
’
U
the values of y are repeated when the angle
increased by 2
If
w is increased by v/n
7r .
.
t
(
27 m it
is constant then y is repeated when
Th e distance t / is called the w a ve
,
a
CH 1 1 1
.
W AVE
]
I
241
TRA N S
length of the sound If A denotes this wave length then
A = v/ or 0
one second the piston makes complete
I
Mt
vibrations and therefore gives out complete waves which extend
a distance so that 11 A Th e equation
.
n
'
:
,
11
n
.
n
0,
n
:
.
(
2
n
cos
t
y
0
may therefore be replaced by
21
—
= A cos
v
t
(
)
y
A
=
A
7r
x
.
or
by
=
A
y
for chan ging vt
Th e equation
x
to
vt
a:
=A
y
27 r
(
T
008
—
d}
vt
)
,
does not change
271
c os
displacement y
.
“
(
T
—
w vt
)
shows clearly that the waves advance with velocity v for if is
i creased by d to m + d and t at the same time incre a sed from
x
,
n
t
,
to
t+
d
5
,
the value of y remain s unchanged
.
A
series of waves
like that coming from the piston is called a train of waves
If instead of a piston vibrating at the end of a cylinder we
consider a body vibrating in the open air such as a tuning fork
then a series of sound wa v es spreads out from it in all directio n s
with the velocity v Th e waves are spherical and as the radius of
a wave gets bigger the amplitude in it gets less A train of
spherical waves radiating outwards in all directions from a point
can be represe ted by the equation
.
,
,
.
,
.
n
A
y
7
7
271
'
cos 7
(r
)
vt
,
where the amplitude at a dista nce from the source is equal to
A / and y is the displacement of an air particle in the direction
of th radius
Th e maximum velocity of a particle describing a simple
harmoni c motion is equal to 2 A so that its energy is propor
l to A
Thus we see that the
or to A /k since
ti o
energy pe
in a train of waves is proportional to A /M In
the case of spherical waves the energy which starts from the
sou r ce is spread over a sphere of surface equal to 4m so that the
WP
16
r
r
e
r
.
7r
z
n
n a
2
Q
n
,
2
g
r
.
2
‘
,
.
.
242
SO
P
III
AR
T
[
UND
energy per
in the waves must fall o ff inversely as the square
of the dista ce from the source Th e amplitude therefore falls
off inversely as the distance from the source This is assumi g
that no energy is bsorbed by the medium as the waves pass
through it If some is absorbed the amplitude falls o ff more
rapidly than inversely S th e dista ce
Th e propagation of a train of longitudinal waves can be
If the first sphere
S hown with the model described on page 227
is moved backwards and forwards with a simple harmonic motion
the compressions and rarefactions can be seen to follow each other
along the row of spheres and each sphere can be seen to move
backwards and forwards like the first one
A cos
Th e equation y
w
gives
the
displacement
v
t
)
(
P
of the air at any point a train of sound waves
T
w
T i
Le t p denote the pressure in the undisturbed air
and p that in the train of waves We have
n
.
n
.
a
.
a
n
‘
.
.
,
.
h
ressu re
e
ra
n
Of
in
a ve s
in
a
.
'
’
.
P
where
—
=— E
P
V
V
’
V
is the volume of any quantity of the air when undis
tu b e d and V the volume of the same quantity at the point in
the wave train where the pressure is p Consider two planes at
distances w and
from the origin Le t the longitudinal dis
placement o f the air at be y and at let it be y We have
V
’
r
’
.
cc
’
.
’
x
y
93
.
,
=
'
2T
:
A COS
%
(
.
—
w vt
)
’
w
(
)
vt
.
volume of the air when undisturbed between the two planes
is equal to — per unit area of the planes When the air
is disturbed by the train of waves this volume becomes
because
the
air
which
was
at
is
displaced
to
5
+
+
)
)
(
(
y
y
is displaced to w y I f the
+ y and the air which was at
two planes are taken very near together we may put V = — w
=
V
and V = (
so
that
V
(
y)
y),
y
y
Th e
’
x
x
.
'
a:
9
a:
,
’
'
as
x
,
’
.
’
x
’
Hence
’
as
’
-
’
’
x
.
p
’
—
p
= —
a
E
(I
f
—
—
y
.
cc
244
SO
.
y
so that
p
’—
A
=
P
p
U ND
cos
(
(
c os
x
P
III
A
T
R
[
)
vt
—
w vt
)
phase of the pressure variation is therefore
Th e
°
90
or
W
ahead of
that of the displacement If the pressure curve were moved back
W
e
A
and
made
equal
to
the
two
curves
would
coincide
P
4
M
see from Fi g 1 1 that the pressure is a maximum where the
.
.
.
?
3
l
l
Fi g 1 1
.
.
displacement in the forward direction is increasing most rapidly
and a minimum where it is dimini shing most rapidly for as
the displacement curve moves along from left to right the dis
placement a t a fixed point rises where it S lopes downwards and
falls where it S lopes upwards Th e same thing may be seen
by studying Fi g 1 0 ( page
,
.
.
C HAP T ER
IV
N OTES
N U S I CAL
I
sound produced by a body vibrat i ng with a simple harmon i c
motion is a musical note What is called the pitch depends on the
number of vibrations per second or the frequency When the
frequency is great the pitch is high and when the frequency is
small the pitch of the note is low It is found that the sound
produced by a vibr ating body like a tu ing fork is not audible
unless the frequency lies between certain limits These limits are
Fo r most people they are about 3 0
di fferent for di fferent people
to
V ibrations per second A tuni ng fork or other vibrating
body vibrating more than
times per second produces no
sound audible by most people and a fork making less than 3 0
vibrations per second is also inaudi ble Vibrating bodies with
frequencies outside the limits of audibility can be shown to pro
duce trains of waves in the air ; but the trai ns do not affect the ear
Th e motion of the air at any point produced by a body like
a tun ing fork movi ng i a S imple harmo i c motion is also a simple
ha monic motion of the same frequency as the fork A simple
harmo i c motion is completely determi ed when we know its
amp litude frequency and phas e If the motion is represented
by the equation
y A cos ( 2 t a )
so that when t 0 y A cos a then the angle is called the phase
of th e vibration Th e phase is usually of no importance when we
are dealing with only one vibration but when the result of ad di g
two or more vibrations together has to be considered the phases
may be important When dealing with only one vibration we
can always reckon the time from an instant when y = A so that
TH E
.
.
,
,
.
n
.
.
.
.
.
n
n
r
.
n
n
,
.
7r u
,
,
,
a
,
.
n
,
.
,
246
SO
P
A
III
R
T
[
U ND
cos = 1 and the phase is zero Th e loudness or intensity of
a musical note depe ds on the amplitude of the V ibration and
the pitch on the frequency Th e relative loudness of notes of
di fferent frequencies is difficult to estimate and we do not know
the relative amplitudes of notes of di fferent pitch which seem
equally loud All we can say is that if a note seems louder than
another of equal frequency then it has the greater amplitude
Th e energy per cubic cm in the air due to a train of waves
is proportional to A where A is the amplitude and the
frequency
If a series of tun ng forks having frequencies proportional to
the numbers 1 2 3 4 5 6 etc are sounded together then it is
found that the sound produced seems to have the pitch of th e fork
of lowest frequency but the quality of the sou d depends on the
relative intensities of the sounds due to the different forks Th e
sounds of the di fferent forks ble d together and see m to consist of
only one musical note Fo example if forks with frequencies 1 28
25 6 3 8 4 and 5 1 2 are sounded together the sound produced would
be considered by a musician to have the same pitch as the sound
produced by the 1 28 fork sounding by itsel f It is found that the
sounds produced by most musical instruments co sist of the sum of
a series of V ibrations with frequencies proportional to 1 2 3 4 etc
Th e vibration of lo w
est frequency is called the fundamental tone
and determines the pitch Th e other Vibrations are called the
harmonics of the fundamental to e Th e vibration with frequency
equal to twice that of the fundamental is called the first harmonic
that with three times the seco d harmonic and so on Th e quality
of the sounds emitted by musical instruments depends on the
relative intensities of the harmonics present A musical no te like
that produced by a tuning fork which co sists of the fundamental
alone is sometimes called a pure tone or a S imple tone Th e
loudness or intensity pitch and quality of a musical note are
therefore determined respectively by the amplitude frequency of
the fundamental and relative intensities of the fundamental and
harmonics It is found that the relative phases of the fundamental
and harmoni e s make no di fference to the quality of the sound
In later chapters we shall consider the notes emitted by various
d the harmon i cs which they contain
m usical instruments
a
.
n
,
.
.
.
.
2
7
9
n
,
.
i
,
,
,
,
,
.
,
,
n
,
.
n
r
.
,
,
,
,
.
n
,
,
,
,
.
.
n
.
“
,
n
.
,
.
,
n
.
,
,
,
,
,
.
.
,
an
.
248
SO
AR
T
P
III
[
UND
given by the S iren has the same pitch Th e speed is then kept
constant and the number N of revolutions in a known time t is
.
,
fou d
n
Th e
.
f equency is then equal to
r
n
N
7
frequency of a tunin g fork can be found with the apparatus
1
T
S hown i n Fi
3
h
e tuning fork F is supported by a pillar H
g
fixed to a wooden base A glass plate P P can S lide along the
base between two guides A light metal pointer S is fastened to
one prong of the fork and just touches the glass plate Th e plate
is coated with lamp black by holding it in a smoky flame and if it
is made to S lide along the base while the fork is vibrating the
Th e
.
.
.
.
.
,
Fi g
pointer
.
13
.
traces a wavy line in the lamp black Another light
o m te
T
also
j
ust
touches
the
plate
close
to
the
pointer
T
S
h
e
p
pointer T has a small piece of iron attached to it which is attracted
by a small electromagnet M when a current is passed through the
mag net On e of the wires from the magnet leads to the top of
the pendulum G of the clock 0 and the o the r wire is conn ected to
a battery B and a spring K which the lower end of the pendulum
j ust touches when at its lowest point Th e clock pendulum has a
period of two seconds so that it touches K once eve y second This
makes the pointer T move suddenly sideways once every second
When the glass plate is pulled along between its guides the
S
.
.
r
.
.
.
,
r
.
.
CH I v
.
]
M
U S I C AL N OTES
249
v ibrating tuning fork makes a wavy line and close beside this
the pointer T makes a straight line with kinks in it Th time
from one kink to the next is one second It is easy to count the
number of waves in the wavy line made by S between two k i ks
made by T which is equal to the number of vibrations made by
the fork in one second By doing this several times and taking
the mean of the results the frequency of the fork
be found
accurately
experiments on sound it is often convenient to be able
I
This
di ly vibrating
to
keep
a
tuning
fork
stea
l t ll
M i t
d
can
be
done
by
means
of
an
electr
cal
dev
ce
i F
shown in Fi g 1 4 Th fork F is supported by
a pillar R fixed to a wooden base B A short electromagnet M
is fixed between the prongs of the fork but does not touch them
,
e
.
.
n
,
.
can
.
n
E
.
ri c a
ec
a
Tu n
n
a
n
g
in
y
o
i
e
o rk s
i
'
.
e
.
.
.
Fi g 1 4
.
.
prong has a platinum wire P fixed to it by a small screw S
This wire proj ects slightly from the fork and j ust touches the face
of the disk 0 c arried by a screw which passes through a pillar A
fixed to the base Th e face of the disk 0 is made of platinum
O n e of the wires from the magnet leads to a binding screw N and
the other to the pillar A A wire leads from the pillar R to the
other bin ding screw N If N and N are con ected to two or three
dry cells t h cu rent passes from N to R and hence to S and P
From P it flows to O and then through the magnet to N
Th e
magnet draws the prongs of the fork together d so the W ire P
breaks its contact with the disk 0 This stops the current so that
the magnet stops attracting the prongs of the fork which move
back so that P and 0 again make contact I this way the fork is
kept vibrating steadily Th e electric current in the wires leading
t N and N is an inte mittent current consisting of as m any flows
of electri city per second as the fork makes complete Vibrations
On
e
.
.
.
.
n
e
’
r
.
.
an
.
,
,
.
n
.
o
’
r
.
250
SO
P
III
A
R
T
[
UND
intermittent current c be used to dri v e a Simple kind of
electric motor called a pho ic wheel by means of which the
frequency of the fork can be very c Cu a te ly fou d
Fi g 1 5 S hows a phonic wheel
It consists of an iron wheel 0
mounted on a shaft carried by bearings Th wheel has a number
of equidistant proj ections or cogs on it Two small electro m agnets
A and B are supported at opposite ends of a diameter of the wheel
so that the cogs almost touch the magnets whe the wheel rotates
If the intermittent current from an electrically maintai ed fork
and battery are passed through the magnets of the phonic wheel
Th e
an
n
,
.
n
r
a
.
.
e
.
.
n
.
n
Fi g 1 5
.
.
and the wheel started at the proper speed the magnets keep the
wheel running S uppose the fork makes 200 vibrations per second
then the magnets of the phonic wheel will be excited 200 times
per second If the wheel is started
magnet per second then as each cog is c
will be attracted for a moment and so the
Th shaft of the wheel drives a revolution counter so that
number N of revolutions it makes in a k own time t can be f
,
.
,
.
,
,
e
-
,
o
n
If the wheel has m cogs then the frequency of the fork is fi
ll
,
Th e
wheel can be kept going for a long time
,
r
25 2
SO
P
III
AR
T
[
UND
imple relations bet ween their frequencies Th e ratio of the
frequencies is called the i te v l between the notes Th e interval
is called an octave Fo example a note of frequency 5 00 is
called the octave of a note of frequency 250 Th e following table
gives the names and frequency ratios of other intervals used in
music
S
.
n
r
a
.
r
.
.
musical scale of notes called the diatonic scale is obtained
by starting with a definite note called the key note and choosing a
series of notes between which and the key note the above ratios
hold
Three notes with frequencies proportional to 4 5 and 6 form
what is called a major chord A minor chord consists of three
notes with frequencies proportio al to 1 0 1 2 and 1 5
If the key note of the diato i c scale has a frequency
B etween
its octave is 2
and 277 in the diatonic scale
are six notes with frequencies 3 g g 53 3 and ls
next octave higher from 2 to 4 contains six notes betw s
and 477 with frequencies double those between n and 2 and
for the other octaves Th e frequency of the key note
taken to be 25 6 for scientific purposes In musical
is generally taken to be somewhat higher Th e dia
only used with instruments like the violin which can
any frequency Th e scale used on the piano contains eleven no
between and 2 with freque cies adj usted so that the ratio
any note to the next is the same in all cases Th e value of t
9
ratio is therefore 2
Th e ratios of the frequencies on this s o
are not exactly those of small whole numbers and the music
the piano is less pleasing to musicians than music
diatonic scale
Th e
.
,
.
n
.
,
77 ,
n
77
71
.
.
77 ,
a
n
,
n
72,
77 ,
,
i
77.
77 ,
.
.
.
.
77
n
n
.
5
.
,
.
n
.
CH
.
I V]
U I L N OTE S
25 3
M S CA
In scientific work it is best to specify musical notes by stating
thei frequencies and not by giving the letters used to designate
them by musicians Th middle C on the piano usually has a
frequency rather greater than 25 6
r
e
.
.
F
N
RE ERE CES
S oun d
a n
d
l
l
d J J Th m
Mu s i c , S e d e y Ta y
i
S o u n d , Po y n t
n
g
an
.
.
or
o
.
so n
.
C HAP T ER
F
RE L EX I O
N
,
F
N TERF EREN CE OF S OU N D
P ERPEN DI CU L AR V I B RATI O N S
RE RACTI O
COMP OS I TI O
N
OF
V
N
,
I
AN D
WH E N a sound wave in air comes to the surface of a solid
body like a w ll it is reflected from the surface Fi g 1 6 represents
a photograph of a spherical sound wave produced by an electric
spark being reflected by a plane surface Th e spark occurred at S
and WW is the S pherical wave spreading out with S as centre
a
.
.
.
.
Fi g 1 6
.
is a plane metal plate and B B is the part of the wave which
has been reflected B B is part of a sphere with its ce tre at S
Th e line S S is perpendicular to A B and the distance from A B
to S is equal to the distance from AB to S Th e dotted line shows
where the wave wo u ld have been if the plane had not been there
AB
n
.
’
.
’
’
.
.
SO
UND
A
P
R
T
1
1
1
[
plane S heet of metal is put up betwee the two mirrors it can be
shown that it stops the sound from getting to the second mirror
If the sheet of metal is placed in a vertical plane i clined to the
direction of the beam of sound the beam is reflected from it and
can be detected by means of the flame and second mirror if these
are moved round the S heet until the place where the flame is
stro gly affected is found
Thus if the sheet is placed so that it makes an a gle of 45
with the beam
first mirror
reflected beam will
n
.
n
n
.
n
along
re
fl ex i o n
direction
right angles
°
that
.
beam before e fl e i o may be called the incident beam of
sound It is found that the incident and reflected beams are
equally inclined to the reflecting surface
Th e velocity of sound in a heavy gas like carbon dioxide is
less than in air ; in a light gas like hydrogen it is
i f
R f
greater Th e formula 7)
g ves the following
velocities at 0 C
Th e
r
x
n
.
.
e ra c
t on
o
.
°
.
i
V]
CH
.
I
E
R F RACT O N O F S O
U ND
25 7
V l ity
e oc
9
3 3 240
em s
/
sec
.
26 000
1 28 6 00
smaller velocity in carbon dioxide can be shown by means
of the apparatus represented in Fi g 1 8 AB is a metal ri g about
two feet in diameter to which two very thin sheets of i di
bb e
are fastened Carbon dioxide gas is pumped i to the space between
the rubber S heets so that they are blow out as S hown d fo m
a lens —shaped body W is the whistle and F the se sitive flame
Th spherical sound w ve S like CD spreadi g
t from W pass
through the carbon dioxide lens In the le s the waves travel
more slowly than in air so that at the middle of the lens where
it is thickest the waves are retarded relatively to the parts of
Th e
.
n
.
n
a ru
r
n
.
an
n
,
n
.
n
a
e
r
.
ou
n
.
,
Fi g 1 8
.
.
the waves which are further from the middle Th e result is that
when the relati v e positions of W AB d F are properly adj usted
the convex waves become nearly plane inside the lens and on
coming out on the other side become c ncave like C D and
converge to a focus at F Th e change of di rection of the sound
waves i passi g through the lens is called refraction Th theory
of e fl e i and refractio will be more fully discussed in the part
of th i s book dealing with Light I stead f the le s made with
rubber sheets a large soap bubble filled with carbon dioxide can be
used
When two sou d wa v es are passi g over a point the dis
placement of the air at the point is the resultan t
m
f
of the displ ceme ts due to the two waves Th e
change of pressure at the point is the sum of the changes due
w P
17
.
an
,
,
’
o
.
n
n
.
x on
r
e
n
n
.
o
n
.
n
I
n
e r fe r e n c e
°
a
.
.
n
.
’
25 8
SO
P
III
T
A
R
[
UN D
to the two waves S uppose there are two trains of waves passing
through the air Le t the pressure variatio s due to them at a
fixed point be given by the equations
.
n
.
277
'
p
and
p
Th e
—
P
p
Si n
3?
2m
= P
p
s1 n
”
A
resulting pressure variation will be given by
p
Le t
—
7
v
X
7
77
_
p +p
and
_
=P
p
7)
"
A
vt + P
f
i
7
si n
so that
n
'
77
’
and
sin
'
vt
f
i
—
7
.
,
are the number of
77,
vibrations per second in the wa ve trains Also let P = P so that
—
i
2
P
sin
2
S
t
t
)
(
p
p
p
p
2P sin wt ( +
cos 1 (
Th e last expression may be regarded as representing a
pressure variation of amplitude 2B cos wt (
and frequency
’
.
’
”
7 r 7i
’
n
n
7 r 77
”
’
'
n
77
n
’
77
77.
2
77
Fo r
2
example if
20 1 ,
n
’
= 202
and
200
we have
’
n
so that the resultant has a frequency
2
77
,
201
and
its amplitude is proportional to cos 2 i Its amplitude therefore
is equal to zero twice every second and has a maximum value
ositive
or
negative
twice
every
second
p
When two tuning forks of nearly equal frequencies are sounded
together the sound produced rises and falls in i te sity Each
maximum of intensity is called a beat We see from the foregoi g
calculation that the number of beats per second is equal to the
di fference between the frequencies of the two forks Th e beats
are said to be due to interference between the two sounds Th e
way in which the beats are produced may be explained S imply as
follows When the pressure changes due to both forks are in the
same direction the resultant pressure changes are greater tha
when they are in opposite directions If one fork makes
vibrations per second and the other
the the first gains a
71
.
.
n
n
.
n
.
.
.
.
n
n
.
n
772,
whole V ibration on the other in
1
77
m
second
.
Th e
time between
a maximum of sound and the next minimum is the time in which
26 0
SO
P
III
A
R
T
[
UND
If we have a series of forks each one hav ing a slightly higher
frequency than the one before it in the series d extendi g o v er
an octave the freque cies of all the forks c be found by counti g
the beats betwee each pair of adj ace t fo ks in the series F
example suppose there are 3 3 forks in the series and each one
gives 4 beats per second with the one with the next higher
frequency ; also let the last e give the octave of the first one
The if the frequency of the first fork is
that of the second
that of the third and so on we have the 3 2 equations
,
,
n
,
an
n
an
n
n
n
r
or
.
,
on
.
n
77 1
77 3
772 ,
,
,
Bu t
2 so that
these up we get
25 6
1 28 and
When a train of waves is reflected from a plane surface
perpendicular to the direction of motion of the waves that is
parallel to the waves themselves we get an i teresting case of
i terference of sound waves A train of waves travelling lo g
a t be may be reflected from a closed or from open end of the
tube d then the waves travel back alo g the tube in the opposite
direction and interfere with the waves coming towards the end of
the tube
S uppose we have a train of waves the displacements in which
are given by the equation
Adding
7 71
7 733
:
77
77 3 3
:
33
:
77 1
.
,
n
,
n
,
a
.
n
an
u
.
,
n
an
.
A
cos
2
;
(
a:
)
at
,
which represents waves travelling from right to left If these are
reflected so that they go back fro m left to right the displacements
i the reflected train may be represented by
.
,
n
’
’
=A
y
cos
27 7
(
7
7
a;
vt + c
)
,
for the velocity of the waves has been cha ged from 7) to v
Th e constant c is added because the phase of the reflected waves
at any point need not be the same as the phase of the waves
.
.
n
-
.
I NTE RF ERE NCE
v]
CH
.
or
SO
U ND W AVE S
26 1
before e fl i
Le t the reflecting surface be at m = 0
Then
0 the air particles are in contact with the reflecting surface
at
which we shall suppose to be rigid so that at w 0 th e resultant
longitudinal displacement of the air must be zero
Th e resultant displacement at any point is y + y so that at
0 we have
r
ex on
.
.
a:
,
.
’
,
a
":
cos
27
7715
’
A
A
cos
27 1
(c
A
)
at
This
equation must be true whatever the value of t so that it
’ —
is clear that A
A and
0
We
have
therefore
at
any
point
p
at which E is greater than 0
,
.
,
y + 7/
cos
=A
27 7
“
—
7
27 1
271
7
T
'
2A S I D
.
.
This
o
f
%
-
ac
(
)
'
shows that at any point there is a simple harmonic v ibration
amplitude
2r
cos
—
x
vt
= s in
2A
sin
If
77 7r
=0
.
Thus
where
x
1, 2
,
etc then
.
,
at the reflecting surface and at any
plane distant from it by a whole number of half wave lengths the
a
ir
remains at rest
If
.
as
:
2
77
(
I
)
;
C
then 31 + y
g
;
2A s i n
'
that then there is a vibration of the maximum amplitude 2A
Th e pla es where the air remains at rest are called the nodes
of the trai n of waves reverses the direction of
Th e e fl i
the displacements in it and also the direction of propagation
A displacement in the direction of the propagation remains a
displacement in the d i rection of propagation after the e fl e i o
At a point half a wave length from the reflecting surface the
reflected waves have travelled a whole wave length further than
the incident wa v es so that if th e direction of the displacement
had not been reversed the two wave trains would agree in phase
the
n d rei force each other
efl e io
Owing to the re v ersal o
two displacements at half a wave length from the surface are
equal and op posite and so al ways annul each other Th e motion
of the air particles when a train of waves is reflected by a rigid
surface parallel to the waves is shown in Fi g 20 Th e re flectin g
.
n
r
.
ex on
.
r
x
,
a
n
n
.
.
.
.
r
x
n
n
.
26 2
SO
U ND
1
P
A
T
1
1
R
[
surface is at A and a row Of 1 5 equidistant air particles numbered
1 to 1 5 is shown Th top row shows the positions of the particle s
when t = 0 and they are in their undisturbed positions Th e
following 1 2 horizontal rows S how the same 1 5 particles at the
Th e time of
ends of 1 2 successi v e equal intervals of time
a complete vibration is equal to 1 2 and the wave length is
equal to twelve times the distance d between the successive
particles when undisturbed
e
.
.
7
7
.
,
.
A
s o
r =1
2
o
o
g
0
7 0 0
0
Q CD O
O
O
o
o
o
o
o
o
o
o
0
0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0
0
0
0
O
0
0 0 00 0
0
0
0
O OCI DO
0
0
0
0
0
0
0 00 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
O
14
o
13
o
0 0 0
0
O
12
O
11
o
10
0
0
0
0
5
o
9
o
s
o
7
o
0
0
10 00 0
11 0 0
0
0
8 00 0
0
0
0
6 0
o
O
o
o
=
t l
4
o
3
o
6
O
1 5 =x
O
0
0
particles 6 and 1 2 and the reflecti g surface remain at
rest Th e particles 3 9 and 1 5 oscillate with the greates t
amplitude At t = 3 the air is compressed near 6 and rarefied
ear 1 2 At t = 9 it is rarefied near 6 and compressed near 1 2
We see that the maximu m pressure variation occurs at 6 1 2 and
at the reflecting surface where the air remains at rest Th e place s
of zero pressure variation are 3 9 and 1 5 where the motion is
greatest Th e curves 22 3 3 44 are roughly parallel showing
that little compression or rarefaction occurs between the particles
Th e
n
.
,
.
n
.
.
,
.
,
.
,
,
,
,
26 4
the
SO
U ND
P
T
III
AR
[
aves reflected f om P coincide in phase wi th those pro
d the air in the tube is set into
d e d by the motio of P
a stationary vibration of great intensity Wh e e tl e air remai s
at rest the l y copodium powder is not disturbed but where the
air vibrates the powder is blown about Th e result of this is
that most of the powder soon collects i to little heaps at the
points where the air remains at rest There is a h eap close to
the cork and one close to the disk and a series of equidistant
heap s between these h lf a wave length apart If the dis tance
bet wee the e d heaps is measured and divided by the number of
half w ve intervals we get half the wave length of the ote emitted
A where v is the velocity of
by the V ibrati g disk We have
sound in air the freque cy and A the wa v e le gth so that can
be calculated If the tube is filled with another gas instead of
i the wave length can be found in the same way
Le t it be A
"
Then 77
is the velocity of sound i the gas We have
A where
w
r
an
n
uc
’
r
.
i
n
,
.
n
.
,
a
,
n
.
n
a
n
,
n
,
77
.
n
77
77
:
,
n
,
77
,
.
'
a r,
.
’
77
’
so that by findi g
n
be determined
n
7
and
A
A
.
I
3
77
A
A
the velocity of sound in the gas
equation
Th e
.
’
.
can
enables the ratio of th e
7)
specific heat t constant pressure to that at constant volume for
any gas to be calculated when 7) is known K dt s apparatus is
often used to find the v alue of y for a gas
When a train of wa v es travelling alo g a tube comes to
open end of the tube the wa v es are reflected from the open end
d go back into the tube so that very little of the sound escapes
If the cork at T i the K dt s apparatus is removed the stationary
waves and heaps of powder can still be obtai ed if the distance
betwee the disk d the end of the tube is adj usted properly
Th e heap nearest to the open e d is found to be o ly about
a quarter of a wave length from it instead of half a wa v e length
as it is when the cork is used Th spaces bet ween the heaps
are half wave lengths as before
L t the pressure change in a train of waves travelli g along
a pipe from right to left be represented by the eq u ation
a
’
.
r
un
.
n
an
,
an
.
’
’
un
n
n
n
an
.
n
n
e
.
.
n
e
p
’
—
=P
p
g
c os
>
I NTERFE REN C E
v]
CH
.
OF S O
UND W AVE S
26 5
Le t the
and suppose the end of the pipe at
0 and is open
reflected train of wave goi g from left to right in the pipe be
represented by
is
x
.
n
s
r
—
p
w
(
the open e d the air is not co fined by the walls of the
tube so that it is much freer t move than in the tube Th e
pressure at the open end will therefore di ffer very little from the
normal pressure p so that at
0 we have approximately
At
n
n
o
,
.
x
,
0 =p —p +p
'
Hence
—
c o s g—
{
7
=P
p
'
vt
— —
and
He ce at any point in the tube
’
P =
P
—
c
+
’
P
2
c os
-
F
(
7
—
vi
c
)
.
= 0
.
n
p
'
—
p +p
-
=P
p
2P
-
This
27 m
SID
T
—
cos
27
m
(
—
w vt
)
A
equation is exactly similar to the equation
y y
2A
2771 73
8 1 11
27
T
k
0 15
,
which was found to represent the displacements in the closed tube
It appears that the pressure variation is zero at the open end of
the tube and at a series of equidistant planes half a wave length
part Half way between these planes of zero press ure variation
there are the planes of zero displacement There is therefore
a plane of zero displacement at a distance A/4 from the open end
L A etc
d a series of others at 52
O
from
the
pen
end
A E
A
j
E
Fi g 20 may b e taken to represent the motion of the air particles
in a tube with an open end if the open end is supposed to be at
the particle 1 5 Th e planes of zero displacement are at 1 2 6
and 0 and the planes of zero pressure variation at 1 5 9 and 3
S tationary vibrations can also be obtain ed and examined wi th
the apparatus shown in Fi g 22 MM is a large concave reflector
which reflects the sound f om a high pitched whistle W so as to
form a parallel beam of sound Th e bea m is reflected from a plane
metal sheet P P so that it returns along its path S tationary
vibrations are produced between the mirror and plane P P by the
.
a
.
.
a n
7
,
-
,
,
.
.
.
.
,
,
,
.
.
r
.
.
.
26 6
SO
A
P
R
III
T
[
UND
incide t d reflected waves Th e positions of the planes of zero
displaceme t can be found with a sensitive flame FF which is t
affected at these planes but roars whe at a plane of maximu m
displacement and e o p e s e variation In this way the wav e
length of the sou d produced by the whistle can be determined
an
n
.
n
n o
,
n
‘
z
r
r
s ur
.
n
.
Fi g 22
.
.
If a high pitched whistle is used as a source of sound and
a
se
sitive
flame
as
a
detector
then
it
is
found
S
d hdw
that an obstacle prevents the sound reaching the
flame if it is cut by the straight li e j oining the whistle to th e
base of the flame Th e obstacle casts a sound shadow j ust like
the shadows produced by a source of light With sounds of lower
frequency shadows can also be obtained provided the size of the
bstacle is increased i the same proportion as the wave le gth of
the sound ;
Fo example a cannon fired on one side of a hill may o t be
audible on the other side although it can be heard at much greater
distances in other directions To get a hadow the
be large compared with the wave le gth of the
theory of the formation of shadows will be m
in the chapters on light which like sound is a fo
If a point mo v es with a S imple harmonic
n
ou n
a
s
o
s
,
.
n
.
.
n
o
r
n
n
,
.
S
.
n
,
,
,
26 8
SO
A
R
T
P
III
[
U ND
S uppose we
wish to get a curve described by a point moving with
two simple harmonic motions at right angles of equal frequencies
and amplitudes Le t the poi t start at the intersection of any
ho izontal with y vertical li e as at P It will then move to
the next vertical line in the same time that it moves to the ext
horizontal line because the two frequencies are equal It will
n
.
an
r
n
.
n
.
F i g 23
.
.
therefore move round the curve drawn through P which always
passes from o e corner of a rectangle to the opposite corner If
we suppose the point starts at A then in the same way we see
that it will go round the circle A CBD If it starts at O it will
move backwards and forwards along a diagonal of the square
S uppose we wish to get a curve describe d by a point moving
with two simple harmonic motions at right angles of equal
n
.
,
.
.
P I I
COM OS T O N O F
Fi g 24
.
.
V I B RATI O NS
26 9
27 0
SO
P
III
A
R
T
[
UND
amplitudes but with the period of the vertical motion half that
of the horizontal motion Then while the point moves from one
vertical li e to the next it will move from a horizo tal line to the
next horizontal line but one That is it covers two vertical spaces
to one horizontal one S uch a curve is shown passi g through the
point Q d another passing through the points E O and F If
the ratio of the two frequencies desired is 77 to 777 then the curve
must cover horizontal spaces while it covers vertical spaces
It is easy to draw curves for any S imple ratio like 1 3
etc Fi g 24 sho ws such curves for several frequency ratios
.
n
n
.
n
.
an
.
,
,
77
777
.
:
.
.
.
.
Fi g 25
.
.
composition of two S imple harmonic m o ti o s t right angles
may be obtaine d experimentally in a number of di fferent ways
O e of the S implest known as Blackburn s pendulum is S hown in
i
Th e
n
a
.
n
’
,
,
27 2
SO
UN D
II
T
I
P
A
R
[
screen If the fork F is e t vibrating P oscillates in a horizon tal
line and if the fo k F is set V ibra ti g P moves i a vertical line
If both F d F are set vibrating the motion of P is the resul t
of compoundi g together the two perpendicular imple harmo i c
motio s If the two fo ks have exactly equal freq encies th e
curve desc ibed by P is in general ellipse If the freque cie s
are very nearly but not exactly equal the S hape of the ellips e
S lowly changes and passes successively through shapes like thos e
shown i Fi g 24 for the ratio
Th S hape of the ellips e
depends on the phase difference of the two vibrations which varie s
slowly when the freque cies are not exactly equal With forks
havi g frequencies nearly but not exactly in a simple ratio lik e
we get the correspo ding curves which slowly chang
1 2 or
in shape as the phase di fference varies This arrangement w s
first described by Lissaj ous and the curves obtai ed are sometime s
called Lissaj ous figures
’
s
.
r
,
n
n
’
an
n
S
n
n
.
u
r
.
an
r
n
.
,
n
e
.
,
n
,
.
n
n
e
a
.
n
’
.
C HAP T ER
RES ONA
VI
N CE
W HE N a tunin g fork or a pendulum is set vibrating it oscillates
with a definite frequency This frequency is sometimes called the
f equency of free vibration and the time of one V ibration is called
the period of free vibration or the free period Th e frequency of
free vibration is obtained when the tuni g fork or other body is
allowed to vibrate freely with no external forces acting on it It
is possible to make a body v ibrate with a frequency di ffere t from
'
.
r
,
.
n
.
n
its atural freque cy of f ee vibration by allowi g suitable forces
to act on it S ch a vibration is called a forced vibratio A an
example of a forced v ibration consider the case of a simple pendulum
suspended from the end of a ho izontal sp i g which can v ibrate
in a horizontal pla e S uch
arra gement is sho w in Fi g 27
V is a vice fixed to a table i which a steel or hard brass strip S S
is firm ly clamped This strip may be 2 cms broad 05 mm thick
wP
18
n
n
.
n
r
n
u
r n
r
n
an
.
,
.
.
.
s
.
n
n
.
n
.
,
.
.
27 4
SO
P
III
AR
T
[
UN D
and about 5 0 cms long A weight Wc slide along the strip d
be clamped on it in any desired positio A imple pendulum OP
having a very light bob is hu g from the free end of the spri g
If the S pri g is pulled to one side and let go it vibrates in a
horizontal plane so that 0 moves with a S imple harmonic motio
perpe dicular to the plane of the paper Le t the period of this
vibration be T a d let the len gth of the pendulum be l so that
if 0 was a fixed point the free period T of the pendulum wo ld be
T
T
F
give by T 2 VJT
irst
suppose
is
greater
than
then
it
g
is found that the pendulum swi gs as shown in Fi g 28 Th e
end of the spring oscil lates betwee 0 and O and the pendulum
bob between P and P with the period T Th e point A in the
string remains at rest Th e direction of the displacement of P
an
an
.
.
n
S
.
,
n
,
n
.
n
n
,
n
.
’
n
,
u
’
“
n
77
,
n
.
.
’
n
’
'
.
.
0
1
0
,
F i g 29
.
.
is opposite to that of O S ince A remains at rest the pendulum
swings like a pendulum of length AP l so that T = 2 W79
Th e amplitude of swing of the pendulum bob is equal to
.
’
’
,
0
0
8
-
71
1
.
’
AP
l
AO
7— 7
where (7 % OO is the amplitude of the vibration of the end of
the spring
’
S ince
T 2 9 75
and T 2
7
'
:
.
:
7
71
’
we have
T
l
We see from this that if
T
2
’
T
—
T
becomes early equal to
n
T
the
27 6
SO
P
III
A
R
T
[
UND
body will execute a forced V ibratio of frequency
displacement be given by the equatio
A cos 2
t
Th resultant force on the body at any time t is
’
F
P cos 2v t
m
so that its acceleration a is given by the equation
P cos 2v 77 t
Mu
cos 2 t /A
n
x
77
.
n
a;
7 r 77
’
.
e
'
r 77
’
r
7r 77
so that
u
,
a
;
u
,
’
,
x
,
M( 7
x
,
.
Thus M u
which is equal to the resultant force on the body is
proportional to
and the period T of vibration of the body
must be given by the equation
’
x,
which gives
477
u
,
47 r
u
,
77
111
2
M77
'
2
I
so that finally
2
2
,
)
47 7 1” ( 77
2
2
When the frequency of free vibration of the body is greater
than the frequency of the applied force then at any instant the
displacement of the body is in the same direction as the force F
s
positive
so
that
and
both
have
the
same
sign
for
A
P
i
When is greater than A and P are of opposite signs S O that
the displaceme t of the body is in the opposite direction to the
periodic force F When is nearly equal to A becomes very
large compared with P If
then A becomes infinite
theoretically but in practice owing to friction and other causes
the amplitude cannot rise above a certain finite value In Fi g 3 0
of the
the variation of the amplitude A with the frequency
is very S mall
applied force F is shown graphically When
A
A has a co stant small positive value i depende t of
approaches the free frequency
at
A rises and becomes
77
,
'
77
2
.
,
77
’
77 ,
,
n
’
.
77
’
77 ,
77 ,
.
,
.
.
’
77
’
’
77
.
’
n
n
77,
s 77
n
00
V I]
CH
.
R ES O NAN C
E
27 7
When
is greater than and very large A becomes
zero If is dimi ished as it approaches A becomes appreciable
Thus
and is negati v e When becomes equal to A
at
A changes from
to
In practice A of course
does not rise to infi ity but to a large value represented at B and
then changes to an equal value of opposite ign represented at O
Th e experiment described at the beginning of this chapter
serves a s a good illustration of the theory j ust gi v en Th e strin g
exerts a periodic force on the pendulum bob This experiment
'
77
71
,
’
77
.
n
77
.
77
77 ,
,
’
co
77 ,
’
77,
co
co
n
,
S
.
.
.
Fi g 3 0
.
.
may be done more S imply by holding the string in the hand and
moving it backwards d forwards with di fferent frequencies A
stri king experiment illustrating resonance can be shown with a
wheel mounted in bearings If a sti ff spring is attached to one
of the be ri gs a n d then the wheel set rapidly rotating as the
wheel g radually lows down a time comes when the period of
revolution of the W heel is equal to the period of vibration of the
spring When this occurs the spring begins to vibrate violently
an
.
.
a
n
,
s
.
,
.
27 8
SO
T
P
III
AR
[
UN D
its bearings slightly with a period equal to its
pe iod of revolution
If a weight is hu g up by a spiral spring it can oscillate
up d down with a defi ite period If tw equal spri gs and
weights are hung from the same support then if one f them is
set vibrati g it exerts a p eriodic force on the support so that this
mov es lightly if it is n o t too rigid Th e motion of the S upport
causes a small periodic force to act on the other spring and weight
which gradually sets it vib ating with a large amplitude because
the period of the force acting o it is nearly equal to its own free
period
A small periodic force if Continued l ng e ough may set a
very large body vibrating violently if the free pe iod of th body
coincides with the period of the force S oldiers when marchi g
over a bridge always break step because if they did not the
periodic forces due to their regular marching might set the bridge
vibrati g dangerously if the free period of the bridge happened
to be eq al to the period of vibration of the soldiers legs
Large bridges have bee destroy ed by soldiers marching over
them without breaki g step
When the frequencies lie within the limits of audibility
resonance can be observed by means of the sound produced If
the handle of a vibrati g tuning fork is allowed to touch a body
with a nearly equal free period the body will be set vibrating and
give out sou d
A resonato is a box or pipe usually closed at one e d and
made of such a size that the air
it has a definite period of
Vibration Th e vibration of the air in such pipes will be dis
cussed more fully in a later chapter If the free period of the
air in a resonator is equal to the free period of a t ning fork
then when the fork is sounded near the resonator the air in the
resonator is set vibrating If the fork is mounted on the top of
the resonator then sounding the fork sets the air in the resonator
vibrating strongly and a loud sound is produced Tuning forks
mounted on resonators are often useful in experiments on sound
Fi g 3 1
They give out a strong sound of definite frequency
shows a fork mounted on a wooden resonator
If two forks of exactly equal frequencies both mounted on
Th e wh e e l s h a k e s
'
r
.
n
,
n
an
o
.
n
o
,
n
S
.
r
,
n
.
o
n
e
r
n
.
,
n
’
u
.
n
n
.
.
n
,
n
.
r
n
In
.
,
.
u
,
.
,
.
,
.
.
.
.
C HAP T ER
V I B RATI O N
VI I
OF S TRI
N GS
a long flexible cord is fixed at one end n d the other end is
passed over a pulley and a weight attached to it then on pulling
the cord near the pulley to one side and letting it go a transverse
disturbance or wave can be seen to run along the cord When the
wave reaches the fixed end it is reflected and comes back in the
Opposite directio
A cotton rope about 05 cm in diameter and
20 metres long stretched by a ten pound weight may be used for
this experiment Th e motion of tra sverse waves along a flexible
cord can also be S hown with a piece of rope hung vertically from
a fixed point If the lower end of the rope is suddenly moved
to one S ide a wave runs up the rope and is reflected at the top
IF
a
,
.
n
.
.
n
.
.
.
,
Fi g 3 2
.
.
velocity with which a transverse wave travels alo g
a flexible string can be easily calculated L e t the tension in the
stri g be P and the mass of the string per unit le gth be
In
Fi g 3 2 AB is the string in its undisturbed position
S uppose
now that A is moved upwards perpendicularly to AB with uniform
’
After a time
A will have reached A and AA
velocity
If the velocity with which a transverse wa v e moves along the
string is then the string will have begun to move upwards like A
for a distance A C 71 but beyond 0 it will still be undisturbed
Th e string between A and O will be moving up with the velocity
communicated to it by the motion of A Th e momentum give
to th e string in the time is therefore equal to
Th e force
Th e
n
.
n
.
n
777
.
.
,
’
7
:
,
7
.
.
7
n
CH
.
V II ]
I
V I ERATIp N
28 ]
O F S TR N G S
required to move A with velocity v is equal and pposite to the
component of the tensio T in A O along A A This compone t
is equal to
’
o
’
’
n
n
.
’
AA
’
AO
'
this force must be equal to the momentum communicated
the string per second so that
Bu t
I
P
77
5
This
= m wfi
result may also be obtai ed i another way as follows
Imagi e the stri ng to be pa s sed through a smooth tube OD part
of which between A and B ( Fi g 3 3 ) is bent into a curve of any
n
n
.
n
.
Fi g 3 3
.
.
shape Th e stri g presses against the tube with a force equal to
P / per u i t length where is the radi us of curvature of the tube at
the poi t considered To prove this consider two points N and N
n
.
n
r
r
,
’
n
.
Fi g
.
34
.
very near together o the string ( Fi g
D raw N O
d
perpendicular to the st ing Then ON =
Le t R Q and R Q
represent the forces exerted by the rest of the string on the part
n
an
.
r
.
r
’
.
28 2
P
III
R
T
A
[
UND
SO
of it between N and N These forces are both equal to the
tensio P i the stri g Complete the parallel gram Q RQ R S O
that RR represents the resultant force on N N Th tria gle
R Q B is imilar to the triangle N ON so that
’
.
n
n
n
o
.
’
’
e
.
’
n
’
S
’
BR
NN
resultant force
’
NO
BQ
Th e
’
’
therefore equal to
IS
P
NN
A
L e t N ON = 9
NO
N
U f
JJ
’
i
so that
9 and P
P
P9
NN
Th e force per unit length on the string is therefore equal to
7
7
P
P9
TO
and
.
“
7
directed towards 0
If the string slides through the tube with a velocity the force
per unit length on it required to keep it on its curved path is
equal to
since 777 is the mass of unit length If v is increased
u til
is
.
,
,
.
n
P
7
'
7
or
7)
the the resultant force P / o
portion
of
the
string
due
to
the
y
te sion in it will j ust be that required to make it move
its
curved path so that the stri g will then not press on the tube at
n
n
r
an
n
in
n
,
all
.
When
f the tube can be taken away and the strin g
’
7)
V
777
will retain its shape
.
It appears therefore that a wave in the
string moves relatively to the string with the velocity
If P is expressed in dynes and in grams per cm the velocity will
be expressed i e m s per sec Th e unit of tensi n or force is ML T
where M L and T denote the units of mass length and time
respectively
777
n
.
.
.
‘
o
,
,
.
Th e
equation
7)
7)
may therefore be written
L
L
T
T
which shows that both sides of it represent a velocity
.
2
28 4
SO
P
III
R
T
A
[
UND
goes to B d is again reflected d moves back to wards B After
the two reflexions the displaceme ts in the wave are in the same
directio as at the start When the wa v e arrives at P from B it
has gone a distance P A AB BP = 2l Th e other wave goes
from P to B then from B to A and then from A to P so that both
waves arri ve at P at the same time after travelling the dista ce 2l
an
a n
.
n
n
.
.
,
n
a:
Fi g 3 6
.
.
and their displacements are then the same as that at the start
Th e motion of the string therefore repeats itself after th e time 2l/v
.
where
Th e
7)
fundamental note in the sound emitted by
the stri g is therefore of frequency
n
“
27
If the middle point of the string is held fixed each half can
,
vibrate with frequency
j ust like the wh ole stri g
n
.
If the
string is held fixed at a series of equidistant points which divide
it into parts each of length We where k is a whole number then each
,
part has a frequency of vibration
When the string is only
a
fixed at each end it can vibrate with any or all of the frequencies
represented by
77
where
27
k
is equal to
1
or
2
or
3
or any
other whole number F example if we hold the string at a point
one third of l from one end and pluck the string half way between
this point d the nearer end this part of the string is set V ibrating
.
or
,
-
an
,
with the fundamental frequency
2
7
Th e
other two equal
parts have equal frequencies so that they are set vibrating by the
small periodic disturbances which get from the first third of the
string to the rest Thus all three thirds of the stri g are set
V ibrating but the two dividing points remain at rest When
.
n
.
CH
.
V II ]
V I B RATI O N
I
28 5
O F S TR N G S
a string is struck or plucked it is usually set vibrating so that
many of the possible vibrations with frequencies
7)
26
27)
3 7)
47)
2l
25
25
are present Thus we get a fundamental of frequency /2i and
a series of harmonics with freque cies which are exact multiples of
the fundamental frequency
vibration of stretched strings can be studied with the
Th
apparatus shown i Fi g 3 7 which is called a monocho d BB is
a wooden box about 1 20 cms long A wire or string P WQ is
stre tched along the top of the box over two knife edges at P and Q
Th e end of the wire e a P is attached to a spring balance S which
serves to measure the tension in the wire Th e other end is
wrapped round a conical plug R which fits rather tightly into
a hole in a wooden block fastened to the end of the box A move
able k i fe edge W S lightly higher than those at P and Q c
77l
.
n
.
e
n
.
r
,
.
‘
.
.
.
n
r
.
.
n
an
Fi g 3 7
.
.
be put under the wire at any desired point A millimetre scale
fixed to the box parallel to the wire between P and Q serves to
measure the length of any part of the wi e If the wire is struck
or plucked it vibrates and sets the box vibrating so that an easily
audible sound is produced Th W ire alo e would produce very
little sound because its surface is so small To verify the formula
.
r
.
.
e
n
.
we may vary the length and tension i the wire until
n
2]
it gives a note of the sa me frequency as a tuning fork of known
freque cy Th e le gth is varied by movi g the bridge W and the
tension by turnin g the plug R Th e wire p esses against the
knife edges so that they hold it fixed when it v ibrates up and
down A good way to tell when the wire and fork have equal
frequencies is by means of resonance If the frequencies are equal
d the handle of the fork is held against the box when the fork
n
.
n
n
r
.
.
.
an
28 6
SO
U ND
III
T
P
A
R
[
is sounding the wire will be made to vibrate strongly A small
piece of paper be t into a V shape may be put on the wire near
the middle of the part which vib ates and if the frequencies
are equal touching the box wi th the fork will make the paper
j ump ff
In this way it can be S hown that when the frequency is
constant l is proportional to VP By s m g di ffere t forks it a
be shown that is proportio al to VP when l is con stant and
inversely proportional to l whe P is constant By usi g wires of
different thicknesses d m aterials and known masses it can be
shown that when I and P are kept constant varies inversely
as ME and does not depend on the material of which the wire is
.
,
n
r
o
.
c
n
u
.
n
n
77
n
n
.
an
77
7
made
F
.
inally if
P
,
and
l
7
2
are all measured
m
B can be
calculated and will be found to agree approximately with the
frequency observed
Th e vibration of the wire with frequencies hi g her than its
fundamental can be easily sho w
Le t the bridge W be put at
a point 20 cms from e end of the wire the whole length l of
which is 1 00 cms Then place paper riders at points 3 0 40 50
the
wire
is
now
6 0 7 0 8 0 and 9 0 cms from the same end
If
plucked half way between the bridge and the end near it the riders
at 3 0 50 7 0 and 9 0 cms will j ump ff while those at 40 60 and
8 0 cms will not be disturbed
Thus the wire vibrates in five equal
sections d the points 20 40 6 0 and 8 0 cms from the end remain
at rest Th e note emitted can be recognised as having a frequency
five times that of the fundame tal note given by the whole wire
If the bridge is put 25 or 3 3 3 cms from one end the wire can be
S hown in the same way to vibrate i
4 or 3 sections with freque cies
4 or 3 times that of the fu damental
Th e points which remain
at rest when the wire is vibrating with one of its higher frequencies
are called the nodes
Th e modes of V ibration of a stretched string can be very
beautifully shown by a method due to Melde Th e apparatus used
in Me lde experiment is shown in Fi g 3 8 F is a large tuning
fork mounted vertically A string is faste ed to one of the pr ngs
of the fork at A and passes over a pulley P to an adj ustable
weight W If the weight W and the distance AP are adj usted so
.
n
on
.
,
.
,
,
,
,
,
.
.
[
,
,
o
.
,
.
,
,
.
an
,
.
,
.
n
.
.
,
n
n
n
.
,
,
.
.
’
s
.
.
.
.
n
o
28 8
SO
P
III
R
T
A
[
UN D
convenient sou ce of light Th e microscope is focused on the
bead so that whe the string is at rest and the fork vib ati g the
image of the bead as seen i the micr scope vibrates pa allel to
the string and so looks like a straight line If the stri g is bowed
r
.
n
,
n
r
n
o
r
n
.
Fi g 3 9
.
.
so that it vibrates perpe dicularly to the axis of the microscope
bet ween S and S the motion of the image of the bead is due to
the composition of the simple harmo ic motion of the fork and the
perpendicular vibratio of the stri g If the te sion of the
stri g is adj usted S O that its fundamental frequency is equal
to the freque cy of the fork the image of the h e d i the
microscope appears to be a curve like o e of those shown i
n
’
,
n
n
n
n
.
n
n
a
,
n
n
n
Fi g
.
40
.
F i g 40
.
Curves
.
like these c
be obtai ed in the following way
D escribe a circle on AB ( Fi g 41 ) as diameter and draw a
e
p p
di c l diameter OD
Divide each of the four arcs OB B D DA
and A C into a number of equal parts say 6 Through the dividing
points draw horizontal lines Then divide AB into 1 2 equal parts
an
n
.
er
.
u ar
.
,
,
.
.
,
n
CH
.
V II ]
V I B RATI O N
I
28 9
O F S TR N G S
and t h rough the dividing points draw vertical lines If curves are
d awn so that they j oi opp site cor ers of successive recta gles
we get cu ves as sho w which are like those seen in the vibration
microscope when the stri ng is bowed These curves represent the
composition of a simple harmonic motion with a perpendicular
vibration in which the velocity is constant in magnitude but changes
ign at the end of each swing
Th e motion of a point on a violin string while it is bowed is
therefore a uniform velocity reversed in directi n at regular
intervals Th e velocity in o e direction may be different from that
in the opposite d i rection
.
r
o
n
n
r
n
n
,
.
S
.
o
n
.
.
Fi g
41
.
F
.
N
RE ERE CES
Th e Dy n a m i c a l Th eo ry
S o u n d, Po y
W
.
P
.
n
ti
n
g
an
d
S o u n d, Ho r a c e La m b
f
o
J J
.
.
h
T
o m son
.
.
C HAP T E R
V I B RATI O N
OF A I R I N O PE
WE have seen in
V III
N
AN D C L OS ED P I PES
Chapter V
that a train of waves travelling
along through the air in a pipe is reflected from either a closed or
an open end of the pipe A train of waves in a pipe is therefore
reflected up and down the p ipe between the ends If the different
parts of the train which are travelling in the sa m e direction at any
fixed point in the pipe reinforce each other then we get a stationary
vibration of the i in the pipe S uppose we have a pipe open at
both ends and that a tuning fork is kept vibrating near one end
A feeble train of waves from the fork enters the pipe and is
reflected from the ends up and down the pipe Th e length of the
trai in the pipe may be many times greater than the length of
the pipe S that at y point in the pipe there may be many
superposed parts of the train half travelli g one way and half the
opposite way If these parts all reinforce each other the air in the
pipe is thrown into a powerful stationary vibration but if they do
not agree in phase they destroy each other by interference so
that the air vibrates only very feebly Fo the fork to set the
air vibrating strongly it is necessary that the frequency of the
fork should coincide with the frequency of a possible statio ary
vibration of the air in the pipe In a stationary vibration the open
e ds are planes of maximum displaceme t and the distance from
such a plane to a node or place of zero displacement is one quarter
of the wave length A Between the Open ends therefore there
must be a whole number of half wave le gths because pla es
of maximum displaceme t and nodes follow each other alternately
If l is the length o f the pipe we have therefore
.
.
,
,
,
a r
.
.
.
n
,
an
O
n
.
,
r
.
n
.
n
n
-
.
n
n
.
l
A
2
777
where
i s any whole number
Hence
A 2l/
777
n
’
.
777
.
29 2
SO
where
1 , 2, 3 , 4,
777
e tc
;
and for pipes closed at one end
’
U
7)
X 47
“
where
A
R
T
P
III
[
UND
m
2
(
etc
Th e fundamental frequency of open pipes is v/2l and the higher
possible frequencies include all the harmonics of this fundamental
ote Th e fundame tal freq u ency of a pipe closed at one end is
n
1 , 2, 3 ,
777
.
n
.
F i g 43
.
.
or half that of an open pipe of the same length Th e high er
possible frequencies include o ly the harmonics having frequencies
which are odd multiples of that of the fundamental note
Th e air in the pipes used in organs and other musical
instruments is sometimes made to vibrate by blowing a current
of air from a S lit across one of the open ends of the pipe S uch
an organ pipe is S hown in Fi g 44
Th e air enters at A and blows through a narrow slit towards
a sharp edge on the other side of the opening at B Th e end
Th e air in the pipe betwee B
0 may be either open or closed
and O is thrown into a state of stationary vibration which includes
all the possible states so that the pipe produces its fundamental
note together with a series of harmonics If the end 0 is open we
get harm ics with freque cies 2 3 4 5 etc times that of the funda
me tal and if 0 is closed we get a fundamental of half the frequency
4
l
/
77
.
n
.
.
.
.
.
’
n
.
,
.
on
n
,
n
,
,
,
,
.
V III ] V I BRATI ON
CH
.
I N O PEN
O F AI R
L ED P I PE S
AND C OS
29 3
and harmonics with frequenc es 3 5 7 9 etc times that of th e
fundamental Th e quality of the sound emitted by an open pipe
is quite different from that of a pipe closed at one end Th e fact
that there may be a node near the middle of an open pipe can be
shown with a vertical glass pipe by means of a light paper box
containing some sand Th e box is hung up by a thread and
lowered into the pipe Near the top or bottom of the pipe the paper
vibrates up a n d down and shakes the sand about while near the
middle of the pipe it remains at rest Th e box tends to prevent
the air vibrating so th t i ts presence at any point in the pipe
favours the production of those stationary Vibrations which have
i
,
,
,
,
.
.
.
.
.
,
.
a
.
F i g 44
.
.
nodes near W here it is and more or less completely stops the others
Th e box will al ways S how a node near the middle of the pipe and
also usually nodes half way between the middle and either end
corresponding to the first harmonic
It is found that if an open organ pipe is blown with air at
more than a certain pressure it ceases to produce its fundamental
note and gives only the harmoni cs so that the fundamental f e
e
of
the
sound
emitted
is
doubled
W
hen
it
is
sound
i
ng
i
q
y
this way the pa per box and sand may S how the two nodes corre
s o di
to
the
first
harmonic
and
no
node
at
the
middle
of
the
p
g
pipe but usually the presence of the box near the middle is
su fficient to start the funda m e tal vibration and more or less
completely to stop the first harmonic
.
.
r
,
u
n c
.
n
n
,
n
.
n
P ART I V
LI G HT
C HAP T ER I
S OU RCES
OF
Y
L I G HT, P HOTOM ETR
I G HT is emitted by very hot bodies Th most important
source of light is the sun the tempera ture of
which is estimated to be about 6 OOO O If the
temperature of a solid body in a dark room is gradually raised it
begins to emit light and so becomes visible at about 4OO O At
this temperat re it appears dull red At about 1 000 C a solid
body is bright red hot and at 1 5 00 C it is white hot and emits
a bright light Artificial light is almost always o b tained from hot
solid bodies Candle and coal gas flames co tain an immense
number of minute particles of solid carbon which at the tem
t
1
of
the
flame
about
emit
nearly
white
light
500
C
p
Incandescent electric lamps contain a thin wire made of carbon
tungsten or some other solid body which is kept at a very
high temperature by passi g a current of electricity through it
Mercury arc lamps consist of a glass or quartz tube in which
a current of electricity is passed through mercury vapour In
these lamps the light is emitted by the mercury vapour which
is not very hot so that they form an exception to the rule that
light is usually obtained from hot solid bodies Ordinary electric
arc lamps consist of two carbon or magnetite rods with their ends
near together A current of electricity is passed fro m one to the
other across the gap between them and the ends of the rods become
very hot and emit light
L
e
.
,
°
.
,
°
.
°
u
.
.
°
,
.
,
.
n
.
era
°
ure
.
,
.
,
,
n
.
.
,
,
.
.
,
.
L I GHT
29 6
P
A
T
I
V
R
[
and not perfectly sharply defined It appears therefore that the
light which passes close to B deviates S lightly from a perfectly
straight path This deviation is called di ffraction and will be
discussed in a later chapter
F
many purposes light may be regarded as travelling i
straight lines b e c s e th e deviation of the light which passes
close to the surface of a body is so small that it c usuall y
be neglected a d the light which does not pass close to the
surface is not de v iated at all It is found that light travels
through a vacuum j ust as it does through air Fo example the
bulbs of incandescent electric lamps are very perfectly exhausted
of air so that the light has to pass across a vacuum before it c
get out of the bulb S ubstances like glass and water which allow
light to pass through them are said to be transparent while
substances like metals through which light does not pass are said
to be opaque A vacuum is perfectly transparent but all sub
stances even air and pure water absorb some light so that they
are not perfectly transparent
A line drawn from a source of light so that it everywhere
coincides
with
the
direction
in
which
the
light
R
is travelling is called a ray of light Th e rays
fro m a small source in air or a vacuum are straight lines
If the light from
di ti g out from the source in all directions
the small source falls on an opaque screen with a small hole in it
then the arrow beam of light which passes through the hole will
follow the path of a ray of the light If a white screen is put up
perpendicular to the narro w b eam the centre of the bright spot
i t marks the positio of the ray which passed through the centre
of the hole S uch a narro w beam of light is sometimes said
to consist of a bundle of rays of light
Th e way in which a source of light produces sh dows of bodies
h ows that light is somethi g which starts
near
it
s
N t
f
ht
from the source and moves from it in all directio s
It is found that light travels through a vacuum with a velocity
of
miles or 3 x 1 0 cms per second It therefore takes
5 00 seconds to come from the sun to the earth a distance of
x 1 0 em s
When light falls on a black body it is absorbed
and the body gets hotter which shows that light has energy
.
.
.
or
n
’
au
,
an
n
.
r
.
,
an
,
.
,
.
,
,
,
,
.
a
y
s
.
.
ra
n
a
.
,
n
.
on
,
n
.
.
a
n
a
u re o
Li g
n
.
10
.
.
,
13
.
,
.
CH
.
I]
SO
U RC ES
L I G HT P HOTOM ETRY
OF
29 7
,
hot b o dy like an incandescent lamp filament c emit large
quantities of light for a long time without any loss of weight
Bu t energy has to be supplied to the filament while it is
emitting light to keep its temperature constant This energy
is supplied by the electric current passing through the filament
S uch a filament receives electric energy which is converted into
heat in the filament and partly emitted as light Th e emission
of light by the filament is analogous to the emission of sound by
an electrica lly maintai ned tunin g fork Th e amount of matter i
the fork remains constan t but it receives electrical energy and
emits some of it in the form of sound waves which travel away
from the fork through the surrounding air Th e sound waves
have energy ; they are not a form of matter but merely a wave
motion i the air Light is believed to be a wave —motion in
a medium called the ether which fills all space including the
spaces between the atoms of material bodies N O way of removing
the ether from any portion of space is known What we call a
vacuum is full of ether Th e actions between bodies which seem
to take place across empty space like the attractions between the
earth and other bodies or the attraction between a magnet and a
piece of iron are believed to be transmitted through the ether
Th e ether i s believed to exist because such actio s are found to
take place and because light travels through empty space and
is found to have all the properties of a wa v e motion t avelling
through a medi um Th e frequency and wave length of t i n s f
light wa v es can be determined experimentally j ust as in the c a s e
of trains of sound waves in air Th e methods by which this can
be done will be discussed in later chapters
A immense number of well established fact s can be explained
by the theory that light is a form of wave motion travelling
thro gh a medium which fills all space N o facts are known
which are inconsistent with this theory which is called the m
du l t y theory of light and is universally accepted among
physicists A small source of light is a centre of disturbances
in the ether which are propagated from it in all directio s in
the form of spherical light waves j ust as spherical sound waves
spread out from a body V ibrating in the air Light waves are
transverse waves not longitudi nal waves like sound waves in air
A
an
.
.
.
,
.
n
.
,
.
n
.
,
.
.
.
,
.
,
n
,
-
r
‘
ra
-
.
o
,
.
.
n
-
-
-
u
.
0
u
,
a
or
.
n
,
.
.
,
.
L I G HT
29 8
P
I
A
R
T
v
[
If a light wave starts from a small source at a certain instant
the after a time t it will be a sphere of radius such that
7
where denotes the velocity of light In a vacuum
,
n
7
7)
'
“
7
77
,
.
v=
3
x
10
In air and other gases the velocity of light is very slightly less
than i a vacuum but the differe ce is so small that it can be
neglected i most cases
Th e illuminating power of a source of light is taken to be
proportional to the am ou t of energy in the form of light which
it emits in unit time Th nit of illuminating power used for
practical purposes is that of what is called a standard ca ndle At
one time the illuminati g power of a spermaceti candle inch in
diameter burning away 1 20 grains per hour was used as the unit
of illumi ating power of sources of light It is fou d that s c h
candles do t always give the same amount of light so that
they do not provide a very satisfactory unit for exact work Th
unit adopted by the International Co gress of 1 8 90 is one twentieth
part of the illuminating power of o e square centimetre of liquid
platinum at the melting point of platinum This unit is nearly
equal to one average standard spermaceti candle and is called
a Decimal Candle F making rough measureme ts ordi ary
paraffin candles may be used and assumed to be sources of unit
illuminating power Th e illuminating power of a source of light
expressed in terms of that of the standard candle as unit is called
the candle power of the source Th e light waves from
source spread out into the surrounding space in the
spheres Th e area of the surface of a sphere is
the square of its radius so the energy in the li g h
through unit area in unit time from a small source is inversely
proportional to the square of the distance from the source This
is true when none of the light is absorbed in the space surround
ing the source Light travels through air without appreciable
absorption so that the energy in the light waves from a candle or
other small source in air passing through unit area in unit time
varies inversely as the square of the distance of the area from the
source
n
n
,
n
n
e
.
u
.
n
n
u
n
.
.
n o
,
e
.
n
-
n
.
n
n
or
.
.
.
.
,
.
.
,
.
L I G HT
3 00
R
T
A
P
IV
[
by the lamp I f C denotes the candle power of the candle and K
that of the lamp we have then
.
,
K
0
r
’2 ’
"
where is the distance from the candle to the screen and 7 that
from the lamp to the screen If we suppose O 1 we have
7
'
.
K
With this apparatus it can be shown experimentally that the
intensity of illumi ation due to a small source varies inversely as
the square of the distance T do this first find and using
one candle in the way j ust described Then place four candles
close together in the position previously occupied by the one
ca dle S O that they produce only one shadow and move the lamp
nearer to the screen until the two shadows are again of equal
intensity It will be found that the distance of the lamp
from the screen is half that found with one candle If nine
candles a e used the lamp will have to be placed at one third of
the distance This shows that the inte sity of illumination of the
screen by the lamp varies inversely as the square of the distance
between them
Another simple for m of photometer is called B unsen s grease
spot photometer It consists simply of a piece of paper with a
grease spot on it which can be made b y melting a small piece
of paraffin wax on the paper with a hot iron Th e grease spot is
more transparent than the rest of the paper so that when it
is illuminated on one side the grease spot appears brighter than
the rest of the paper when viewed from the other side and less
bright when viewed from the illuminated side If both sides
are equally illuminated the grease spot is scarcely visible and
both sides appear alike To compare the candle powers of two
sources they are placed one o each side of the piece of paper
on a line perpendicular to its plane Th e distances between the
sources and the paper are then adj usted until the sides of the
paper are equally illuminated so that the grease spot becomes
almost invisible and both sides appear alike Th e candle powers
of the sources are then proportional to the squares of their
distanc es from the grease spot This photometer is more con
n
7
o
.
'
.
,
n
,
,
.
.
r
-
n
.
.
’
.
-
,
-
.
,
-
.
-
.
n
.
-
,
.
-
.
CH
.
I]
SO
URC E S
L I G HT P HOTOM E TRY
OF
3 01
,
than the shadow ph oto meter It can also be used to
prove that the intensity of illumination due to a source varies
inversely as the square of the distance by comparing the source
with one four and nine can dles all at the same distance from
the grease spot
Th e intensity of illumination due to a source may not be the
s me at the same distance in all di rectio s from the source Th e
candle po wer is then different i different directions Th e average
value of the candle power for all directions is called the mean
S pherical candle power
ve n
i en t
.
,
-
.
n
a
.
n
.
.
F
N
RE ERE CES
Th e Th eo r y
Th e Th eo r y
f
o
f
o
L i g h t, P r e s
t
on
Op ti c s , Dr u de
.
.
C H AP T ER II
F
REFLEX I ON AN D RE RACTI O
N
F
N
AT P L A E S U R AC ES
W HE N light falls on a body it is usually partly absorbed by the
body and partly scattered about in all directio s
i
R
from the surface Each point of the illuminated
surface scatters light in all directions so that the surface is like a
source of light and can be seen from any direction If the surface
on which the light falls is made of very smoothly polished silver
very little light is absorbed and very little scattered about but
nearly all is reflected in a definite direction Most metals wh e n
polished reflect light i this way but not so completely as silver
A clean surface of liquid mercury reflects light almost as well as
polished silver A smooth polished surface which reflects nearly all
the light that falls on it and scatters or absorbs very little is called
a mirror
Th e e fl e i o of light from a plane mirror can be studied with
the apparatus show i Fi g 3 which is called an optical disk and
can be used to illustrate clearly many of the properties of light
i
Th e optical disk consists of a circular disk about 3 0 m
diameter painted white with its circumference graduated into
degrees Th e disk is mounted o a horizontal axis at its ce tre
about which it
be rotated and clamped in any position P LQ
is an opaque screen also mounted i ndependently on the same axis
as the disk At L there is a horizontal slit about one millimetre
w i de the length of which is perpendicular to the plane of the
’
disk A pla e mirror MM can be screwed to the disk so that the
axis of rotation of the disk lies along the surface of the mirror
If a source of light is placed at S a narrow beam passes
through the slit L and falls on the mirror at the centre of the
disk Th source is placed slightly in front of the plane of the
disk so that it illumi ates a narrow strip of the surface of the
n
efl ex o n
.
.
,
.
,
,
i
.
n
.
,
.
.
r
x
n
n
n
.
,
,
.
e
s
n
.
,
,
n
.
c an
n
.
.
,
n
.
.
.
e
n
L I G HT
3 04
P
A
R
T
I
v
[
so that 2NAL = L AB 29 We see therefore that whe a ray of
light is reflected f om a plane mirror and the mirror is turned
through y angle the reflected ray turns through double the
angle turned through by the mirror
If we look at a plane mirror we see an image of the obj ects in
front of the mirror by mea s of light reflected from the mirror
Th e way in which
Th e image appears to be beh i nd the mirror
the image is formed is shown in Fi g 4 CD represents a plane
\
n
.
r
an
.
n
.
.
.
Fi g
.
.
4
.
mirror the plane of which is perpendicular to the pla e of
paper S is a small obj ect in front of the mirror A y e at
EE sees an image of S in the mirror by means of rays of light
like S AE and S BE which are reflected from the mirror From S
draw S N perpendicular to the surface of the mirror and produce
it to S so that N S S N Join ES and E S cutting CD at A
n
,
n
.
.
'
’
.
’
’
'
.
’ ’
e
CH
.
and
II ]
REFL Ex I O N AN D R
EF RACTI ON
AT
PL ANE S U RFAC E S
3 05
respectively Join S A and S B Dra w AA and BB pe pe
di c l
to the plane on Then ASN = AS N AS N = A AS d
Therefore a ray of light
Hence
travelling along S A is reflected long AE In the same way it
may be sho wn that the ray S B is reflected along BE Thus the
rays reflected from the mi rror into the eye seem to c m e from
S
Th e image of a point formed by a plane mirror is therefore
’
B
'
r
.
.
’
'
u ar
.
,
a
,
n
an
.
’
.
o
’
a line through the point perpendicular to the plane of the
mirror and at a distance b ehind the mirror equal to the distance
from the point to the mirror Th e rays of light reflected from
the mirror seem to come from the image but of course do not
actually do so S uch an image is called a virtual image
I n stead of cons idering the e fl e i o of the rays of light from
a po i nt on the obj ect we may consider the spherical waves which
di verge from it Th e rays are normal to the waves In Fi g 5
w P
20
on
.
,
.
.
r
.
.
.
x
n
.
L I G HT
3 06
P
I
V
A
R
T
[
represents a source of light and A B a plane mirror Th
co centric circles round S represent successive positions of a light
wave diverging from S Th e dotted parts of the circles indicate
where the wave would have been if the mirror had not been
present Th e dotted part marked L is actually in the position
marked B such that NL = N B for when the wave reaches the
mirror it is reflected back towards S In the same way N B NL
and N B NL Th e reflected wave is a sphere with centre S at
a distance NS from the mirror equal to NS and on a line S NS
perpe dicular to the plane of the mirror
When a ray of light in air or a vacuum meets the surface of a
transparent
substance
like
glass
or
water
it
is
R
ti
partly reflected fro m the surface but part of it
enters the transparent substance Th e incident ray is split up
S
e
.
n
.
.
,
’
’
.
’
”
”
.
’
’
n
.
.
e fr a c
on
.
,
.
Fi g 6
.
.
into two rays the reflected ray and the ray in the transparent
substance which is called the refracted ray Th e direction of the
refracted ray is not in general the same as that of the incide t
ray Th e e fl i o and refraction of a arrow beam of light
at a plane glass surface can be examined with the optical disk
previously described Instead of the plane mirror a semicircular
glass plate with polished sides is fa tened to the disk as show in
Fi g 6
B OD is
S is the source of light and L the slit as before
the glass block which should be about one cm thick Th e block is
fastened to the disk so that the centre of the r e BB C is at the
,
.
n
r
.
ex
n
n
.
‘
.
n
s
.
.
.
.
.
a
L I GHT
3 08
T
P
I
V
AR
[
greatest possible value of can easily be found with
the optical disk by turni g the disk so that B C is parallel to L A
Le t us now consider what happens when a ray of light
travelling inside a transparent substance like glass meets the
surface separating the glass from air T exami e this case with
the optical disk it is only necessary to turn the disk so that the
light from the sli t L falls O the curved surface of the block of
glass B OD as shown in Fi g 7 Th e beam of light S LA is
perpendicular to the curved surface BDO so that it is t de viated
at this surface At the plane surface B O it is partly reflected
41
This
°
7
'
n
.
n
o
.
n
.
.
n o
,
.
Fi g 7
.
.
along AB and partly refracted along AP Th e angle of incidence
7 = LA
M is equal to the angle M AB Th e angle of refraction
d the ratio of
N AP is greater than the angle of incidence
"
i 7 to sin
is found to be constant as in the previous case
.
’
.
7
7
’
an
,
’
s n
r
.
’
s n
W
th
i1
respect to glass and it is found to be equal to the reciprocal of
the refractive index of glass with respect to air
If as before denotes the refractive ndex of glass with
respect to air then in the prese t
1
sin
Th e
rat o
i
s n
,
In
th s case
i
is
called the refract ve ndex of
i
i
ai r
i
,
.
i
u.
,
c a se
n
,
”
7
u.
"
is
7
,
si n
7
’
7
"
"
1
always greater than and if is increased until sin
then
so that the refracted ray AP becomes parallel to
7
7
u.
,
7
,
II ]
CH
.
R
EFLEX I ON
EF RACTI O N
PL ANE S U RFAC E S
AT
AN D R
If is made greater than the value for which
the equation
AB
p
7
.
1
Si n
u
si u r
7
si n
3 09
’
i =1
,
’
cannot be satisfied It is found that then there is no refracted
This is called total
v and all the light is reflected at A
If the optical disk is slowly turned round so as to
efle i
increase the angle M AL then the angle NAP slowly increases
until it is equal to
and then the refracted ray disappears and
the reflected ray suddenly g ets brighter Th e angle of incidence
"
If
1 is sometimes called the critical angle
for which S i
Th e
then the critical value of i is equal to 41
equation
.
ra
.
r
x on
.
,
.
.
u
,
n
7
.
’
Sin
°
7
Sin
7
which applies when the incident light is in the air and the equation
"
1
sin
sin
which applies when the incident light is in the gl a ss S how that
the path of a ray of light through the surface of the glass is not
changed when its direction of motion is reversed for we have
7
7
u.
,
'
,
S in
'
Sin
7
’
7
"
sin
for all possible values of i and hence if i = then
When a ray of light falls on a parallel plate of glass it is
refracted at both surfaces of the plate and emerges parallel to its
origi al direction In Fi g 8 let AB be a ray of light incident on
a plate of glas s at B Le t 7 and be the angles of incidence and
refraction at B Le t the refracted ray meet the second surface at
"
and be the angles of incidence and refraction at C
C and let
Also
If the sides of the plate are parallel then
1
sin i sin
s
and
sm
Sin
7
7
7
n
.
.
7
.
.
,
7
7
’
.
7
,
in
.
u
,
:
7
7
7
s1 n
7
Si n 7
hence 7
so that AB and OD are parallel If we look at an
obj ect through a thin parallel gl a ss plate like a window the
bj ect does not appear to be seriously distorted because the
directions of the rays from the obj ect are not altered Th e side
ways displacement o f the rays produces some distortion especially
.
,
o
.
,
L I G HT
310
P
A
R
T
T
V
[
when the obj ect is seen in a direction inclined to the normal to
the plate
When an obj ect is looked at through a thick glass plate in a
direction normal to the plate it appears to be nearer to the eye
than if the plate were not present In Fi g 9 let AB CD be a
thick glass plate with parallel sides and 0 an obj ect which is
viewed by an eye at E Th e ray OE which is perpendicular to
AB and CD passes through the plate without deviation
.
.
.
.
.
Fi g 8
.
.
emerges along Q E which
like OP is refracted at P and Q
parallel to OP If Q E is produced backwards it meets OE at O
so that the rays entering the eye seem to come from O and the
obj ect 0 seems to be at O O is the position of the virtual
image of O seen by the eye Th e angle P ON is equal to the
angle Q O M and is the angle of incidence of the ray OP at P
Denote thi s angle by 7 and suppose it is very small so that
sin 7 = t n i = i Then we have
’
’
’
,
.
’
’
'
.
.
'
.
,
a
.
QM
’
:
i MO
PN
7
t,
L I G HT
312
means
P
A
T
I
V
R
[
this equation and the relations
sin 7 s i
sin
7
S
the value of 03 can be computed when 7 9 and are known If
"
9 is a small angle and also small the
and 7 are all small so
that approximately
Hence
By
of
n
7
7
in
u
,
,
.
’
'
7
7
n
,
,
7
,
u.
,
deviation produced by a small angled prism is therefore
independent of the angle of incidence on it provided this is small
Th e
.
Fi g 1 0
.
.
important practical case is when P Q is equally inclined
"
and therefore
In this case 9 2
AP and A Q so that
and gt 2 9 so that
An
7
,
7
’
7
7
°
7
,
9
5
+
0
)
2 7
sin 49
Another
9
:
7
’
an
important case is when
"
9 so that
d gt 7
7
:
so that 7
0
,
,
Sin
sin
similar case s when
and qt 7 9 so that
A
i
,
"
7
"
7
sin ( 4
9)
)
Sin
so that
9
7,
0
,
In
this case
9=7
CH
.
II ]
EF RACTI ON
REFLEx I ON AN D R
AT
PL AN E S U RF AC ES
313
as in the preceding case These two cases are shown in Fi g 1 1
If the ray passes through the prism then the angle cannot be
gr eater than the critical angle which is about 41 8 for crown
glass If C denote s the critical angle then if is greater tha
C the ray is totally reflected at the surface A C of the prism as
.
.
7
,
°
.
’
’
’
.
7
,
n
,
Fi g 1 1
.
.
sho wn in Fi g 1 2 and so does not pa ss out through A C at all If
the refracting angle 9 is g reater than 20 then a ray which enters
through AB for which therefore is less than 0 c nnot pass
out through A C beca use 9
so that must be greater than
or else no
C F
crown glass 9 must therefore be less than 8 2
rays c a pass through both AB and A C
Th e pas sage of a narrow beam of light through a pri sm can be
ex mined with the optical di sk previously described If the prism
is fastened to the d i sk so that its ref acting angle is close to th
centre of the dis k then the deviation of the beam by the prism
can be easily measured for any angle of incidence It is found
that if we start with a large angle of incidence and turn the disk
slowly so as to diminish this angle then the deviation diminishes
quickly at first and then slowly and then stops dimini shing and
begins to increase again Th e deviation is a minimum when the
p
so
that
and
ra
as ses symmetrically through the prism
y
If the mini mum value of the deviation is observed then
.
.
,
,
a
7
’
7
7
°
or
.
n
.
a
.
e
r
,
.
,
.
,
7
,
L I G HT
314
the refractive index of the prism
P
I
V
A
R
T
[
be calculated by the formula
c an
AM
P 7 9)
-
S in
9
%
where (j is the minimum value of the deviation and 9 the
refracting angle of the prism
A right angled prism of crown glas s is often used to reflect
light through an angle equal or nearly equal to a right angle as
)
.
-
,
Fi g 1 4
.
.
shown in Fi g 1 3 Th e ray is perpendicular to AB so that its
angle of incidence on A C is
which is greater than the critical
angle so it is totally reflected through B O S uch a prism can
also be used to invert rays coming from an obj ect as shown in
Th e rays are totally reflected on the base and emerge
Fi g 1 4
parallel to their original directions
.
.
,
.
,
.
.
.
L I G HT
316
P
AR
T
I
v
[
the mirror at A A ray of light from S travelling along S A will be
reflected back along AS because RA is perpendicular to the surface
of the mirror at A N o w take any point P on the mirror and
j oin B P A ray S P will be reflected at P so that the reflected ray
is in the same plane as S P and B P and makes an angle with B P
equal to the angle S P B Le t P I be this reflected ray Th e two
rays from S reflected at A and P respectively meet at I
[ PE
We have
PIA
P EA
P EA RPS
P SA ;
therefore P SA P I A I PB P EA P EA RPS
or since
P SA P I A
2P EA
If S N is very small so that OA is small and if P is very near
to A then these angles will all be small so that their tangents
may be substituted for them without serious error Hence
.
.
.
.
.
.
,
A
.
,
,
.
PA
AS
AS = u
,
+
2P A
AI
AR
AI =
1
then
PA
v,
1
and
'
AR =
7
,
2
°
u
77
S ince S N
7
is supposed very small
and 7) are equal to the
distances of S and I from the centre of the mirror 0 This shows
that the position of I is independent of the position of P so that
all rays from S which fall on the mirror near to A and 0 will be
reflected so that they pass through I Th e rays are said to come
to a focus at I and there is said to be a real image of S at I
S ince the distance of I from O is independe t of S N so long as
S N is small it follows that the image of N is at M and M I is the
image of the line N S for each point on N S has an image on MI
Th e ray S O is reflected along OI and S O
M OI hence
N
77
,
.
.
.
n
,
,
,
SN
ON
u
IM
OM
77
minus S ign is required because N S is upwards and M I down
wards Th e height o f the image is therefore to the height of the
source in the ratio of the distance of the image from the mirror to
Th e
.
CH
.
III ]
S
PHE RI CAL
I
317
M RRORS
the distance of the source from the mirror
obser v ed i s inverted Th e equation
.
Th e
image it will be
.
1
1
2
u
v
7
gives the distance of the image from the mirror correspond i ng to
position
of
the
source
so
long
as
the
source
is
near
the
axis
of
an
y
the m i ror If
we get
so that
Hence an obj ect
near the axis at a d i stance from the mirror equal to the radius of
curvatu e of the mirror gives an image at the same dista ce from
the mirror This image is inverted and equal to the obj ect This
is shown in Fi g 1 7 Th e image of N is at N but the image of S
i at I and
r
u
.
7
7
77
u
77
.
n
r
'
.
.
.
.
s
,
Fi g 1 7
.
.
If the so u rce is nearer to the mirror than the centre of curvature
so that
we get 7)
In this c a s e the image is further from the
mirror than the source If
2
we
get
so
that
becomes
0
/
i ndefin i tely large
This means that the reflected rays are parallel
so that they ne v er come to a focus This case is shown in Fi g 1 8
u
7
7
,
.
.
u
7
,
7)
.
.
Fi g 1 8
.
.
.
.
L I G HT
318
P
I
V
A
R
T
[
If the source is at a very great distance from the mirror we have
In this case the rays falling on the mirror
1/
0 so that 77
are parallel and are brought to a focus at a distance from the
O e half the radius of curvature s often
mirror equal to
called the focal length of the mirror If is less than
then the
formula
u
n
i
-
u
.
1
1
u
2
1)
7
‘
makes 7) negative This i n dicates that there is no real image but
that the rays after e fl e i seem to come from a point behind the
mirror at a distance from it equal to
There is said to be a
.
r
x on
77
Fi g
Virtual image at this point
have
19
This
.
RPQ
P SA
Also
.
.
.
case is shown
P fA
PE
A
P EA
S PE
E
SE
i n Fi g 1 9
.
.
We
,
.
EPQ
.
’
P SA
I A 2P EA
ce
As before we suppose N S and OP to be very small so that if we
AS or ON =
d OB =
put AI or OM =
we get
Hen
— P
.
,
u an
1
1
7
,
2
'
u
77
7
reflected rays like AB and P Q when produced backwards
meet at the virtual image of S that is at I Th e image is erect
and larger than the obj ect
Th e
,
.
.
L I G HT
3 20
P
A
R
I
T
v
[
It appears that the equations
MI
7)
apply without modification to all cases of e fl e i o from either
concave or co vex mirrors provided that only rays of light near the
axis are considered Th e distances denoted by 7) and must be
reckoned positive when they are in front of the mirror and negative
when they are behind it
Th e positio of the real image of a bright source of light like
an electric lamp formed by a concave mirror can
i
t
w it h
l Mi
S h i
be eas ly found by putt ng up a wh te screen n the
reflected light and moving it about un til the place is found where
there is a sharply defined image of the source Th e equations
r
x
n
n
u,
.
7
.
n
Ex p er m
p
en
,
s
er c a
,
rro rs
.
i
i
i
i
.
1
1
2
u
7)
7
NS
u
MI
7)
can be verified by measuring and and the S ize of the image
with the source at di fferent distances from the mirror
A plane white screen with a hole at its centre covered with
wire gauze may be used to find the radius of curvature of a concave
mirror Th e screen is put up i fro t of the mirror so that it is
perpendicular to the axis and the hole is close to the axis Th e
hole is illuminated from behind by an electric lamp d the
screen is moved al o g the axis of the mirror until a sharp image of
the gauze is formed on the scree close to the h ole in it Th e
distance from a point half w y between the image and the hole to
the mirror is then equal to th e radius of the mirror Th e radius
of curvature of a convex mirror is best found by methods
i volving the use of lenses which will be described in the followi g
chapter
S far we have considered only incident and reflected rays of
light
making
very
small
angles
with
the
axis
of
the
F l i
mirror and near to it and have calculated the size
and position of the image formed by such rays It is found that
7
77
.
n
.
n
.
an
n
n
.
a
.
n
n
.
o
oc a
L
n es
.
,
.
CH
.
III ]
S
P H E R I C AL
I
M RRO RS
3 21
rays making large angles with the axis do not all pass through the
same point after e fl e i o so that a defin i te image is not produced
by a spherical mirror whe such rays are employed
In Fi g 21 let M CM be a spherical mi rror with centre of
curvature at O and S a small source of ligh t Le t S M F F and
’
be rays from S J oin OM and OM and produce S O
S M Fn
cutting the reflected rays at F and F S ince S O is part of a
di ameter of the spherical surface it is easy to see that all rays from
All the reflected rays therefore cut
S will cut S O after e fl e i o
the line F F If a whit e screen is put up at F F we get a line
of light on it This li e is called the second focal line Also if
r
n
x
,
n
.
’
.
1
.
2
’
’
.
’
2
x
r
n
Z
.
.
’
’
2
2
2
.
2
n
.
.
F i g 21
.
.
we imagine the plane S M MF to be rotated about OS as axis we
see that the di stance of F from OS will not be altered so that all
the reflected rays pass through a line at F perpendicular to the
plane of the paper This line is called the first focal line Th e
distances of these focal lines from th mirror can be easily
calculated Le t
'
2
1
1
.
.
e
.
CO
7
,
CS
CF1
u,
7
Also
1
CF2 = 7 2
.
,
let the angle of incidence OCS 7
M OM
We have
0 723
0 717 s
.
’
M 0M
’
77 37 7
'
0 777
1
:
OEF,
MP , M
MS
O
0 777 s
'
W
.
P
.
7
0 7 71
,
.
’
;
L I G HT
3 22
Adding
P
I
V
A
R
T
[
up these equations We get
MSM
277 677
M 23 77
’
.
If we suppose that the diameter MM of the mirror is small then
these angles are small so that they may be replaced by their
tangents N o w
’
MS M
tan M OM
t
MM cos i /
tan MF M MM cos
’
,
.
’
’
an
’
’
1
2
u,
I
1
hence
If
77
7
0
th s becomes
i
7 COS 7
1
2
1
5
7I
i
7
agreement W th the result
In
i
previously obtained for rays making small angles with the axis
we have the area of the triangle S CF equal to
To determine
the sum of the areas of S 0 0 and OCF Hence
sin 27
7 sin 7
i 7
1
1
2 cos i
wh ch gi ves
.
2
Q
.
7 27 S n
u
,
i
u
7
sin 2
'
7
If
i =0
7
2
2
cos 7 sin 7
.
2
this also becomes
.
2
When
i
is small the two
lines are very short and near together and when 7 0 they coincide
and form a si gle point the image of S Th e way in which the
reflected rays pass through the two focal lines when the angle of
incidence is not very large is shown approximately in Fi g 22 AB
:
n
.
,
.
F i g 22
.
.
.
C HAP T ER
IV
N
L E S ES
following construction e ables the refracted ray to be
draw
when
a
ray
of
light
in
air
meets
a
trans
n
g’ q
f iiég
parent sphere L t AD ( Fi g 23 ) be a sphere of
glass or other transparent substance the refractive index of which
Le t P A be a ray of light in air meeting
with respect to air is
THE
IQ e r a c t i
S l
s rr f
n
n
a
t a
.
e
.
.
,
u
,
.
F i g 23
.
.
the sphere at A Le t the radius of the glass sphere be and let its
centre be at O With centre 0 describe tw spheres EC and FB
havi g radii equal to / and 1 respectively Produce P A to
meet the sphere of radius at B Join OB cutting the sphere of
7
.
o
.
n
7
77
u.
,
u.7
,
.
.
CH
LE NS E S
]
Iv
.
radius
at 0 Join AC
along A C We have
Th e ray P A
OA
OB
.
.
3 25
is refracted at A and travels
.
d
so that the triangles OAC and OBA are similar triangles and
the efore the angle CAO is equal to the angle A BO B t
sin BA 0 OB
r
.
Mn AB O
and therefore
u
fi
‘
OA
MD BA O
2
7
sin CA 0
which shows that A C is the refracted ray corresponding to P A
Th i s co stru ction shows that all rays in the air like P A which
when produced p a ss through the point B are refracted at the
s rface of the glas s sphere s that they pas s through the po int C
If there were a small source of light at C ins ide the sphere then
a ray from C like CA would be refracted at the surface of the
sphere and would travel along AP Thus all rays from C like CA
a fter refraction appear to come from B so that B is the virtual
image of C formed by refraction at the spherical surface
A le n s is a c i rcular disk of glass the surfaces of which are
S pherical in shape
Th e centres of curvature of
the two surfaces and the middle point of the lens
should be in a straight lin e which is called the axis of the lens
A lens forms an image of an obj ect near to its axis which may be
real or virtual as with spheric l mirrors A thin lens is one in
which the distance between the spherical surfaces at the axis is
very small compared with their di i of curvature Th e p sition
of the image formed by a thin lens can be easily calculated In
Fi g 24 let AB be a thin lens 0 and O the centres of curvature
.
n
u
o
.
,
.
.
‘
.
L en
s es
.
.
a
.
ra
.
o
.
’
.
,
F i g 24
.
.
L I G HT
3 26
v
P
I
A
R
T
[
of its surfaces and OCO its axis Le t S be a small source of light
very near to the axis Th e ray S CI meeting the lens at its centre
is not deviated by the lens because at the centre the two surfaces
of the lens are parallel and a thin parallel plate does not deviate
a ray of light as we have seen Take any point P in the lens near
to C Th e ray S P is deviated by the lens and meets the ray S CI
at I Produce I P to Q Th e angle Q P S is equal to the angl e
through which the ray S P ] is deviated b y the lens At P the
angle between the two surfaces of the lens is equal to the sum of
’
the angles P OC and P O C because OP and O P are perpendicular
to the surfaces at P When a ray passes through a small angled
prism the angle between the two surfaces of which is 9 then the
ray is turned through an angle equal to ( u 1 ) 9 where 7 is the
refractive index of the prism Hence the angle Q P S is equal to
’
.
.
,
.
,
.
.
.
.
’
-
.
,
7
.
(
u.
,
1
P
0
6
4
(
)
where is the refractive index of the lens with respect to air
Bu t the angle Q P S is equal to
u
.
,
.
A
1 ) (P O
so that
P SC P I C = ( a
C + P OC)
S ince P C and S N are very small we can replace these
their tangents without serious error Le t
and
Then we get
’
.
.
CB
(M
1
This
CB
1
equation S hows that 7) is independent of the position of P so
long as P C and S N are small so that all rays from S pass through
I which is therefore the image of S
Th e lengths u
and are all measured fro m the centre of
the lens C and are reckoned negative when on the opposite side of
the lens to the source and positive when on the same side as the
source Th e quantity
,
,
,
.
,
’
7
7
.
0
4
—
1)
L I G HT
3 28
P
A
R
T
I
v
[
1 7 5 so that the i m age is on the same side of the
which gives 7)
lens as the source and the image is therefore a virtual one If
we get
,
.
1
1
1
7
7)
5
1 7 5 which S hows that the image is on the
which gives 7)
opposite S ide of the lens to the obj ect and so is a real image
‘
.
F i g 25
.
.
concave lens can give only a virtual image
= l 0 and
= — 1 0 we get
f
A
u
1
1
1
10
7)
10
Fo r
.
example if
so that v = + 5 Th e image is therefore on the same side as the
obj ect and is a virtual o e This case is shown in Fi g 26 In all
cases it is easy to see that
.
n
.
.
F i g 26
.
.
.
find graphically the position of the image of a point formed
by a lens of given focal length the simplest way is to draw a ray
from the point through the centre of the lens a d another ray
To
,
n
F i g 27
.
.
LE NS E S
]
CH I v
.
3 29
parallel to the axis I n Fi g 27 let AB be a thin lens with centre
Le t S be the point of which the image
C and let N C be its axis
is required Draw S C and produce it Draw S P parallel to N C
From N C produced if necessary cut off CF equal to the focal length
so that F is on the opposite side of the lens to S if the focal length
is po sitive d on the same side if negative Join P F and produce
it if necessary to meet S C at 1 which is the image of S Fi g 28
shows the same thing when the focal length is negative
.
.
.
.
.
.
,
,
,
an
,
.
.
.
,
,
.
F i g 28
.
Fo r
a crown glass lens
.
152
u
,
about so that
,
G 3)
7
.
If both sides of the lens are convex and of equal curvature then
I
7
‘
or f
;
1
Th e focal length of a thin convex lens can be found by using
a bright source d forming a real image of it on a white screen
by means of the lens As source a piece of wire gauze strongly
illum i nated from behind by
electric lamp may be used Th e
gauze i s put up perpendicular to the axis of the lens at a distance
from the lens not less than f A white screen is put up on the
other side of the lens and moved alo g until the position is foun d
at which there is a sharp image of the gauze on it If the di stance
from the gauze to the middle of th e lens is 3 0 cms and from th e
lens to the screen 5 0 cms then
= 3 0 and
1
so that
05 2
,
an
.
an
.
.
n
.
.
.
,
u
1
30
f
=
1
50
f
’
em s
.
L I G HT
330
P
A
R
T
I
v
[
If the distance between the gauze and screen is greater than
then there are two positions of the lens between them at which
it gi ves a S harp image of the gauze on the screen A good way
of finding f is to measure the distance d between the gauze and
the screen and also the distance 0 between these two positions of
the lens
Then it is easy to see that
.
.
—
d c
d+
2
2
d+
c
—
d c
c
2
2
exa m ple if the distance from the gauze to the screen is
1 20 m s and the distance between the two positions of the lens is
20 cms then
Fo r
‘
e
.
.
,
20
1 20
2
20
1 20
2
which give
or we may take
f
=
em s
1 20
.
20
2
which give
29 1 7
f
as before If the distance between the gauze
the screen is
made gradually smaller then becomes smaller until there is only
one position of the lens at which it gives a distinct i m age on the
screen Th lens is then half way between the gauze and screen
so that
and therefore
:
em s
.
.
0
,
e
.
-
77
1
1
1
u
u
f
in this case d 4f: If d is made less than
image can be obtained on the screen
Hence
.
no distinct
L I G HT
332
A
v
R
T
I
P
[
light falling on the seco d lens comes from this image
which is at a distance v d from it Hence the second lens forms
an image at a dista ce fro m itself given by the equation
Th e
n
.
n
1
1
1
"
f
d
7)
v
f /( f
u
so that
1
1
1
"
7)
f
7
f
If d
0
’
this reduces to
1
1
1
1
__
f f
u
If F de o te s the focal length of the combination we have t h ere
fore when the two lenses are in contact so that d 0
n
,
,
Fo r
example if f
1
F
f f
and f
’
20
1
+ "
1
40
1
1
1
F
we get
20
40
’
which gives F
40 so that the combination acts like a convex
lens with focal length 40 If f f then F becomes ndefinitely
large which shows that the combination acts like a piece of plane
glass
Th e equation
,
i
.
,
.
j
—
r
— l
(A
)
enables the refractive index of the lens to be calculated when f
’
and are known Th e radii and 7 can be measured with a
spherometer and f found as above described A simple way of
finding the ra dii when a spherometer is not available is the
following shown in Fi g 3 0 Th e lens is made to float on a small
quantity of mercury con tained in a shallow glass vessel It may
be fixed in position with some wax if necessary This co verts
the lower surface of the lens into a mirror A pin NS is put up
above the lens so that its point S lies on the axis of the lens Th e
pin is moved up and d wn until the image of the point of the
,
’
7
7
.
.
,
.
.
.
n
.
.
.
o
7
,
CH
.
I V]
LE NS ES
333
pin formed by e fl i at the lower surface
with the p i t itsel f Th e image
of the pi n is then at S i ll When
the pin and image coincide their
apparent relative positions do not
change when the eye looking at
them is moved about Th rays
from S return to S after being
reflected so that their angle of
incidence on the lower s urface of
the lens must be zero Th e rays
in the lens if produced therefore
meet at the ce tre of c u rvature O
of the lower surface Le t this radius
be denoted by and the distance
from S to the centre of the lens by
then we have
the le s coincides
n
ex on
r
o n
.
.
e
.
,
.
,
,
n
.
7
Fl s 3 0
.
77 ,
1
1
1
j
7
77
"
where f is the focal le gth of the lens 0 is the virtual image
of S formed by the lens because if the mercury were removed the
rays passing through the lens would seem to come from O
15
df = 3 0 we get
example if
F
n
.
.
an
u
or
1
1
I
15
7
30
so that
Th e other radius
can be found i the same way
30
by tur ing the lens over S uppose
3 0 then we have
7
7
.
n
’
n
’
7
.
(M
1
)
,
1
1
(m
—
)
30
5
which gives
150
S far we have considered only rays of light very near to the
axis of the lens and have calculated the position of the image
fo r med by such rays We have also supposed the lens to be very
thin
In Fi g 3 1 let LMN be a large lens having one side plane and
the ther con v ex Le t B B B B B B B B be parallel
.
u
,
.
o
.
.
.
o
’
.
,
,
’
’
3
L I G HT
334
P
A
I
R
T
v
[
rays of light fall in g on the plane side of the lens and perpendicular
to it Th e rays B and B which are near to the axis are brought to
a focus at F Th e rays R and B meet at F which is further from
the lens than F Th e rays B and B meet at F and B and B
at F S uch a lens does not form a distinct image of an obj ect
because all the rays from a point on the obj ect do not meet at
a si gle point after passing through the lens Th e rays further
from the axis meet further from the lens than those nearer the axis
A distinct image can be obtained by covering the lens with a screen
having a small aperture in it at the centre of the lens so that only
rays close to the axis are able to get through
Th e distance between the point F at which parallel rays near
the axis meet aft r passing through the lens and a po i n t li k e F at
which parallel rays a distance from the axis meet is called the
’
.
'
1
I
I
.
,
’
3
2
2
.
’
2,
3
3
.
n
.
.
,
.
‘
e
3
7
Fi g 3 1
.
.
aberration of the lens at a distance from its axis Lenses
intended to form distinct images have to be designed so as to
diminish the spherical aberration to a very small amou t This
can be done by giving the surfaces suitable curvatures and if
necessary using a combination of several lenses instead of one Th e
spherical aberration of a lens is less when it is arranged so that both
surfaces deviate any ray equally Fo example the lens S hown in
Fi g 3 1 has much less spherical aberration if its cur ved surface is
turned towards the parallel rays
When the light from a small source passes through a lens in
a direction inclined to the axis of the lens then the rays pass
through two perpendicular focal lines as with a concave mirror
These lines are shorter and nearer toget h er the smaller the inclina
tion to the axis
h
er i c a l
p
7
.
n
.
.
.
r
.
.
,
.
.
L I G HT
336
P
v
AR
T
I
[
the slit A plan of the same apparatus is shown in Fi g 3 3
When the prism is put up the image of the slit C moves from I
t P Q and is drawn out in a direction perpendicular to the length
of the S lit i to a band of differe t colours which is called the
spectrum of the white light Th e colours of the band in order
starting at P are red orange yellow green blue and violet AS
we pass along the sp e c t u m f m P to Q the colour g adually
changes in quality so that the red changes into orange and yellow
without any sudden variation and the yellow cha ges gradually
into greenish yellow and then green and so on Thus there is an
indefinitely large number of shades of colour in the spectru m A
.
.
.
o
n
n
.
,
,
r
,
l
.
,
ro
r
n
,
.
.
similar experiment was tried by N ewton who supposed that
ordinary white light is a mixture of different coloured lights which
are separated from each other by the prism because lights of
di fferent colours are deviated through di fferent angles Th e red is
least deviated and the violet most This shows that the refractive
index of the prism with respect to air is greater for violet light
than for red light Th e formula
sin 4( d 9)
sin A9
shows that the m i ni mum deviation } is greater when is g reater
Th e spreading o u t of white light into a spectrum by refraction is
called dispersion
,
.
.
.
)
c;
.
u
,
.
CH
.
D I S P ERS I O N
v]
337
If the different coloured lights which make up the spectrum
are mixed up agai they produce white light This can be sh w
by pu tting a convex lens between the prism d the screen i
such a p iti n that it th ows
image of the face of the p ism
from which the light issues up the screen All the di fferent
col u ed lights are then c centrated on to this image which
appears white A other way is t place a large concave mirror at
F i nstead of the screen and let the light reflected from it fall on a
suitably placed screen It is f u d that the mirror d screen
can be arranged so that the whole S pectru m is concentrated into
a patch of white light which is an image of the face of the prism E
If a arr w slit is made in the screen F it can be arranged so
that light of any desired colour alone passes through it S uppose
the slit is put so that it lies across the yellow par t f the spectrum
the we get a beam of yellow light on the other side of the screen
If this beam f yellow light i focused on another scree with a
c n v ex le s we get a yellow image of the slit in F If a prism is
put up in front of the lens the beam of yellow light is deviated by
the prism but it is t drawn out into a spectru m Th e yellow
image f the slit is m ved t a new position but its a ppeara ce is
u cha ged By moving the lit in F along the spectrum this
experiment can be tried with all the di fferent colours It is found
that the result is the same with all of them Th e prism deviates
the violet much m re than the red but it does not split either u p
i to new colours
It is found that the wave le gths of the di fferent coloured
lights in the spectrum of white light are different As we pass
alo g the spectrum from vi let t red the wave le gth increases
Th f llowi g table gives the wave le gths of the di ff erent colours
in millionths of a millimetre
n
o
.
an
r
on
o
n
an
r
o
os
n
.
on
r
n
.
o
n
o
.
an
.
,
n
o
.
o
,
n
.
o
o
n
s
n
.
n o
o
n
n
.
n
o
o
S
.
.
.
o
n
.
n
.
n
e
o
o
n
.
P
.
n
n
.
W
o
.
L I G HT
338
v
P
A
R
T
I
[
methods by which these wave lengths have been deter
mined will b described i a later chapter Th e light at any
point i a spectrum co sists f trains of waves in the ether hav ing
a definite wave length (A) and frequency
We have = A
where v de otes the velocity of light Th e velocity is the same
i
a vacuum for light of any wave length In glass and other
tra sparent substa ces the velocity depends on the wa v e length
Th
velocity usually but by no means always is smaller the
smaller the wave length When a t rain of light waves passes
from o e medium into another in which its velocity is di fferent
the wave le gth is changed but the f equency remains the same
If the velocity and wave length in the first medium are denoted
by and A and i the second medium by
d A then we have
v = A and
A
Th e
n
e
o
n
n
.
77
7)
n
7)
.
n
.
n
n
.
e
.
n
n
r
.
’
n
7)
an
77
77
so that
,
’
,
'
v
v
A
A
where denotes the freque cy or number of vibrations per second
at any fixed point while the t ain of waves is passing over it It
is found that the colour of light does not depend on its intensity
According t the wave theory the i tensity of light depe ds on
the amplitude of the vibration i the ether and the colour on the
wave length or frequency
Th only difference between violet light and red light is that
the wave le gth of violet light is about half that of red light
Th e light emitted by very hot solid bodies gives a spectrum like
t hat j ust described which contains an indefinitely large number of
di ffer e ntly coloured lights
d so forms a conti uous band of
olour from the red e d to the violet If a S lit is illuminated
with light of only one wave length then the spectru m of this light
obtained with a lens and pris m consists of a s i ngle image of the
Th e many co loured spectrum of white light from a hot solid
S lit
body co sists of a series of images of the slit infi ite in number
Each image of the slit has a certain
o e for each Shade of colour
width so that at any point in the spectrum an indefinitely large
number of these images overlap At any point in the spectrum
there is therefore not light of one wave length but light of many
wave lengths which however all fall between limits near together
n
77
r
.
.
o
n
n
n
.
e
n
.
n
an
n
c
.
,
.
n
n
n
,
.
.
.
L I G HT
3 40
I
P
A
R
T
V
[
can be varied Th e le gth of the slit is perpe dicular to the
plane of the paper This tube is called a col limator Th e colli
ma tor is fixed to a circular table T on which a prism P
be
placed MN is a other brass tube having a convex le s at M
and a small convex lens at N fixe d to a tube which
slide in
and out This tube wi th its lenses is called a telescope Th e
telescope is carried
arm which
be tur ed about axis
at the centre of the table T and perpendicular to its pla e Th e
circumference of the table is graduated into degrees and f actions
n
n
.
.
.
can
n
.
n
can
.
.
on
an
c an
n
an
n
.
r
of a degree
d the arm carrying the telescope has a vernier
attached to it by means of which the angular positio of the
telescope
the graduated circle can be determined Th e prism
P is attached to a circular plate which can also be turned about
an axis at the centre of the table perpendicular to the plane of
the table This plate has a vernier attached to it by means of
which the angular position of the prism
be fou d If a source
of light such as a flame or a mercury arc lamp is put up at S in
fro t of the slit C the rays of light which pass through each
point of the slit d fall on the lens L should be made pa allel
by this lens Th e distance of the slit from the lens L has to be
,
an
n
on
.
.
c an
n
n
.
,
an
.
r
CH
.
D I S P ERS I O N
v]
3 41
adj usted so that it is equal to the focal length of the lens Th
If the
b eams f parallel rays pass through the prism as sh wn
light from any e point on the slit is all of the same colour it
is all deviat ed by the prism to the same extent and so remai s
a parallel beam after passing through the prism If several
di ffere t col urs are present then they are de v iated through
differe t a gles so that we get as many separate parallel beams
as there are colours or wave lengths present i the light If the
telescope is turned so that the axis of the lens M is parallel to one
of these parallel beams th e n the parallel rays pass through the lens
M and are b rought to a focus at I inside the telescope At 1 we
get a real image of the slit C This image n be observed through
the lens N which magnifies it At I a fine o s wire is usually
stretched acr ss the tube so that it coincides with the image I
whe this is in the middle of the tube Th e angular position of
the telescope when it is turned so that the wi e coinci des with an
im ge of the slit can be read o ff If the spectrum of the light
from the source S c t ins many li es then several of them may
be visible in the telescope at once and by turni g the telescope
round they can all be seen in turn Th deviation f the light by
the prism depends on the angle of incidence on the prism It is
best to turn the prism so that the deviation has its minimum
value If the prism is sl wly tu ed round one way and a par
l
ti
line observed in the telescope the deviation of this line
decreases and then stops decreas ing and begins to i creas e It is
ea y to turn the prism so that the deviati n is a minimum Th e
gle of minimum deviation of any particular spectrum line c a
be found by removing the prism d turning the telescope so that
it is in line wi th the collimator and the image of the slit coi cides
with the cross wire Th position of the telescope is then read
ff wi th the vernier
Th e prism is then put up and the telescope is
turned so that the cross wi e coincides with the line at the position
of minimum deviation Th angle which the telesc pe has been
t u rned through is the mi imum angle f de v iatio Th refractive
index of the pri m f the light in question can be calculated from
the angle of minimum de v iatio and the refracting angle of the
prism Th e angle of the prism can be fou d by turni g it so that
its refr c ting angle points towards the collimator Th e parallel
e
.
o
o
.
on
n
.
o
n
n
n
,
n
.
,
.
ca
.
cr
.
s
o
n
.
r
a
.
n
a
on
n
o
e
.
.
o
.
rn
cu ar
n
s
.
o
.
n
an
n
e
.
o
.
r
e
.
o
n
s
n
o
e
.
or
n
n
.
a
n
.
L I G HT
3 42
P
A
I
V
R
T
[
beam from the collimator is then di v ided into two parts by the
edge of the prism and s me of each is reflected from the two sides
of the prism Th e a gle between the two reflected beams is
measured with the telescope and as is easily seen it is twice the
refracting angle of the prism
Th e wave le gths in air of the light of the spectrum li es of
di fferent eleme ts have been determined experimentally and tables
giving the wave le gths of all know lines in the spect a of each
element
be obtained If the minimum deviations of a number
of lines of known wa v e lengths are fou d a curve can be drawn on
squared paper through points the coordi ates of which are the
minimum deviations d correspond i ng wave lengths This cur v e
shows the relation between the minimum deviatio
d the wave
length If the minimum deviation of an unk own line is measured
the corresponding wave length
be found from the curve and
then the tables of w v e lengths c be searched until a line of
equal wave le gth is found There may be many li es of nearly
equal wave length so that it is not always easy to be sure what
element is emitting the light in question B t if the prese ce of
any particular element is suggested then the other lines in its
spectrum can be looked for and if many of them are found the
presence of this element in the source of the light is established
Th e following table gives the wave lengths of some spectrum lines
due to di fferent elements Th e lines selected are the brightest
and most characteristic lines
o
n
.
,
,
.
n
n
n
n
n
can
r
.
n
,
n
an
.
n
an
n
.
c an
a
an
n
n
.
,
u
.
n
,
,
.
.
.
El
e m en
t
Wa ve
l
en
g
of a
th i m illi th
illi t
on
n
m
me
553 5 6 9
45 5 5 44
45 9 3 3 4
5 58 8 96
58 7 5 8 1
3)
Ca dm
58 7 6 1 5
i
um
47 9 9 9 1
5 08 5 8 2
P o ta s s
i
6 43 8 47
um
7 6 6 8 54
re
s
L I G HT
3 44
v
A
R
T
I
P
[
aki g a gle of 6 0 with DA d d op a perpendicular AE on
thi li e then the top of the prism is divided b y D B d A E i to
two similar triangles A ED d DBC havi g a gles of
60
and a t ia gle AF B havi g angles
45
d
d
d
A ray f light like B P S which e ters the prism th ugh AD
is re fracted so that it becomes pa llel to DB meets AB at an
a g le of 45 d is totally reflected through a right angle so that
it is then pa allel to B C and is i cide t on DC at a gle equal
to 3 0 d passes out thro gh DC in a directio at right a gles to
°
m
n
s
n
r
an
n
an
an
,
n
an
r
an
n
n
°
n
,
n
an
,
an
n
r
°
an
ro
ra
°
°
n
o
n
an
n
n
u
an
n
n
original direction before entering the prism S i ce the telescope
and collimator are at right angles t each other the rays formi g
an image of the lit on the cross wire in the telescope must have
passed through the prism like the ray B P S Th e a gle of
refraction at AD is equal to
so if i is the angle of i cidence
i ts
n
.
o
n
,
S
n
.
n
of such a ray on
AD
S in
then
7
.
u
,
2
sin
where
u.
is the
sin 3 0
refractive inde of the prism f light of the p ti l a
By turning the prism round the angle of
length in questio
incidence of the light from the collimator
it c be varied d
so light of any wave length can be made to form an image on the
c o s s wire
Th e prism is mounted s that it can be rotated by
turning a micrometer screw This screw carries a cylinder on
which a spiral scale of wa ve lengths is marked so that the wave
,
x
°
or
n
.
ar
.
c u ar
e
.
,
on
r
,
an
o
.
.
,
,
an
CH
.
D I S PE RS I O N
v]
3 45
length of a line
be read ff di r ec tly if it is brought on to the
c ss wire in the telescope by turning the screw Th e spiral scale
is g aduated b y bservi ng a series of lines of know wa v e lengths
If the le s at the end of the telescope where the spectrum i
observ ed or the eye piece as it is called is removed d a scree
with a slit in it put in the telescope so that the slit is in the
position usually occupied by the
ss wire then the slit allows
only light of the wave length indicated on the drum to pass
If the collimator slit is then illumi ated with white
t h r ough it
light th other slit can be used as a source of light of any desired
wa v e length An instrument which enables light of any desired
wave length to be selected from a spectrum in the way j ust
described is called a monochromatic illuminator
If white light from an i candescent electric lamp or other
source
is
focused
on
the
slit
of
a
spectrometer
o o
A
we get a continuous spectrum which can be
the telescope If a layer of any substa ce is put
o bser v ed i
up i front of the slit so that the white light has to pass through
it then the substance may absorb parts of the white light so that
parts f the sp ectru m may be diminished in intensity co m pletely
absent F example a piece of red glass stops all the light except
some of th red so that the spectrum of white light after passing
through red glass consists of a red band S uch a spectru m is
called an absorption spectrum Th e absorption spectrum of a
v pour which when hot emits light the s pectrum of which is
a line spect r um is found to co sist of the continuous spectrum
cross e d by harp dark li es Th e v ap ur absorbs chiefly light of
defi i te wa v e lengths and all ws that of all the other wave lengths
to pas s freely Th wa v e lengths of the light absorbed are found to
b e the same as the wa v e lengths of the light which the vapour emits
when hot enough F example sodium vapour in a bunsen flame
emits yell w light of wa v e lengths 58 9 0 and 5 8 9 6 millionths of
a millimetre Th e spectr m of this light consists of two yellow
lines v ery close together If white light is p as sed through sodium
vapour it is found that its spectrum contains two dark lines
corresponding to the wave lengths 58 9 0 and 5 8 9 6 This can
be shown by proj ecting the spectrum of an electric e
t
a sc reen with the apparatus described at the beginning of this
o
can
ro
.
n
O
r
.
s
n
,
,
n
an
c ro
,
n
.
e
.
.
n
b
s
rp
ti
S Pe ° tf a
n
°
n
n
.
n
,
or
o
or
.
,
e
,
.
.
a
,
n
,
S
n
o
.
n
o
e
.
.
or
o
u
.
.
.
ar
,
on
o
L I G HT
3 46
P
I
V
A
R
T
[
chapter and placing a large bunsen flame filled with sodium vapour
in front of the prism so that the light has to pass through the
fl ame
Th e spectrum is the seen to be crossed by a dark line in
the yellow Th two dark lines are so near together that they
overlap d appear like one unless a very narrow slit is u ed If
some sodium salt is then dropped into the electric e it gives out
intense yell w sodium light and the dark line across the spectrum
prod ced by absorption in the flame becomes highly illuminated
This shows that the sodium flame absorbs the same light which
sodium v apour emits when hot It is found that in many e s
the po wer of a substance to emit light of any wave length is
proportional to its power of absorbi g the same kind of light
This is called K irchh ff s law
Th e spectrum of su light
be examined with a spectrometer
by
reflecting
a
beam
of
it
on
to
the
collimator
slit
t
S l
through a convex le s so that the lens forms an
image of the sun on the lit It is found that the spectru m is not
co tinuous like that of the light from an electric
It may
be described as a continuous spectrum crossed by dark lines
Th e dark li es S how that certain wave lengths are absent or
of comparatively small intensity These lines were firs t carefully
studied by Fraunhofer in the years 1 8 1 4 1 5 d they are usually
called Frau hofer s li es It was found by Bunsen and Kirchho ff
that many of the wave lengths of the light corresponding to these
dark lines are the same as the wave lengths of the light in the
bright lines in the spectra of the chemical elements known on the
earth F example there is a pair of dark li es in the yellow of
the solar spectrum which exactly coincide with the two yellow
lines in the spectrum of the light emitted by a flame c taini g
sodium vap ur Th e dark lines in the solar spectrum are due to
the absorption of definite wave lengths from the white light
emitted by the more central portions of the sun Th e central
p rtions are surrounded by clouds of vapour which abs rb light
so that the solar spectrum is the absorptio spectrum of these
clouds of vapour By comparing the dark lines in the solar
spectrum with the bright lines in the spectra of known elements
we can tell what elements are present in the clouds of vapour
rou d the ce tral portions of the sun Many of the known
,
n
.
-
e
.
an
s
.
ar
o
u
.
c as
.
n
.
’
o
.
can
n
o
a r
S pec
ru
m
.
n
S
.
n
a re
.
.
n
.
—
an
’
n
.
n
or
.
n
,
on
o
.
.
o
o
n
.
n
n
.
n
L I G HT
3 48
P
A
I
V
R
T
[
a pris m depe ds on the differe ce between its refractive indices
for red and violet light If H denotes the e f c tiV index of
a specime of glass for V iolet light of wave length 3 9 6 7 millio ths
of a mi llimetre A that for light of wave le gth 7 6 6 1 millionths
of a millimetre d P D that for yellow light of wave length 5 8 9 6
millionths of a millimetre then
n
n
"
u
.
ra
r
,
e
n
n
n
u
,
,
an
,
u
,
“A
H
M
1
)
is usually calle dthe dispersive p wer of the glass for the Fra nhofer
lines I I d A Th e dispersive power for the lines F and C is
equal to
o
an
u
.
NF
MO
ND
1
extreme red and the violet parts of the spectrum of white
light are of feeble intensity so that the dispersive power for the
blue line F and orange line C is of much greater practical importance
than that for the lines I I d A
Th e dispersive powers of the three glasses the refractive indices
of which are given above for the lines F and C are as follows
F li t
F li t
C w
I
S
t
i
h
il
I
I
M
V
ff
m
H
I)
(
(
)
(
)
Th e
,
an
.
:
n
ro
e n
”F
“
pp
i
e
n
n
er z
o
an n
”C
0 0 08
00 1 8
0 0 24
1
05 1 5
0 63 3
0 7 04
00 1 5
iv p w
0 0 29
00 3 4
Th dispersive power of flint glass is about double that of
crown glass By combini g together crow glass d flint glass
prisms it is possible to make compound pris m s which produce
dispersion but little deviation and are called direct vision prisms
Compound prisms which produce little or no dispersion but
considerable deviation can also be made and are called achromatic
prisms because they do not separate the colours in white light
Th deviation produced by a small angled prism is equal to
1
its
refracting
angle
where
is
its
refractive
index
and
p
)
d
(
I
S uppose we wish to design a small angled direct V is
made out of th glasses S teinheil III n d Merz V that
deviate D light L t 95 denote the refracting angle of th
glass and gb that of the flint glass component Then we must
have
D s p er s
e
o
er
e
n
.
an
n
.
.
e
.
-
)
()
u.
,
.
'
-
a
e
.
e
e
’
.
05 1 5
(
p
06 3 3 qb
'
0
.
CH
.
D I S PE RS I O N
V]
If a
3 49
this gives 3
This prism does not
i
Fi g 3 6
S h a p ism is show
deviate 1 ) light but it de v iates H light through the a gle
'
°
10
uc
n
n
r
.
.
n
00 6 9
and
A
8 12
x
05 3 1
x
10
05 1 0
x
10
°
012
light through the angle
0 6 22 x 8 1 2
Thus
By
the violet and red rays are separated b y an angle
combini g together three crown glass prisms wi th two of flint
n
Fi g 3 6
.
.
glass direct vision prisms that do not deviate yellow light but
separate V iolet and red rays considerably can be obtained S uch
a compou d prism is shown i Fi g 3 7 These prisms are used in
,
.
n
n
.
.
Fi g 3 7
.
.
direct v isio spec tr oscopes in which th e collimator and telescope
are in the same straight line which is convenient
T desig a small a gled achromatic prism we have to arrange
to neutralize the dispersion We may make the deviatio of the
F light equal to that of the C light
If 4 is the a gle of the
crow glass prism d 3 that of th e flint glass one then for the
glasses S te i nheil III d H ffmann I this requires
n
,
o
n
.
n
-
n
.
n
)
.
’
n
an
,
an
o
05 21 4
)
0 51 3
'
t
c
’
0 6 9 7 ct
‘
,
which gives
If
then gb must be about 3 0
S uch a prism is how in Fi g 3 8
This prism de v iates eithe F
or C light through an angle give appr ximately by the equati s
°
S
n
r
.
.
n
on
o
9F = 05 21
x
30
0 7 21
x
10
90
x
30
06 9 7
x
10
05 1 3
L I G HT
3 50
deviations of H D and
are approximately as follows
Th e
9H = 05 3 1
x
30
9D
x
30
05 1 5
lights produced by this prism
A
,
A
R
T
I
v
P
[
x 10
0 7 04 x 1 0
x 10
It appears that the deviation of the violet I I l i ght is lightly
greater than the red A ligh t that the pr sm not perfectly
achrom atic for the wh le spect um
94
05 1 0
x
30
S
,
SO
r
o
Is
i
.
Fi g 3 8
.
.
We have seen that the focal length of a lens is give by the
equation
n
7
4
(1 5)
>
focal length is therefore not the same for light of differe t
colours Th image of a source of white light formed by a le s
consists therefore of a ser es of coloured images Th violet image
is ea est the le s and the red image furthest from it F
example suppose a convex le s of the fli t glass Ho ffma n I has
—
200 e m s
radii of curvature
and
Its focal lengths
are give by the equatio
Th e
n
n
e
.
i
n
.
n
r
.
n
,
n
7
n
n
f
hich
w
e
followi g values
L i g ht
n
H
G
F
D
n
’
.
or
L I G HT
3 52
If the gla ses are
S teinheil
s
MF
I
—
”F
,
“C
:
/
l g
f
a
1
7
S uppose
=
—
,
III
6
3
7
7
an
d Ho ffman
n I we have
th a t
SO
1
P
R
A
v
T
I
[
.
7
that
5 cms
d
5 m
— 1 5 cms
This lens is
equation then gives
show i Fi g 3 9 Th seco d surface of the
crown glass lens d the first surface of the flint
glass lens are often made with equal radii so that
they fit each other d can be cemented together
if desired
Th collimator and t elescope le ses in spectro
meters al w ays co sist o f achromatic combi ations
so that lights of diffe ent colour are made parallel
by the collimator and focused by the telescope in
the same position
7
:
.
7
’
e
-
s
,
an
7
5
em s
;
the
.
n
n
e
.
.
n
an
,
an
.
n
e
n
n
,
r
.
Fi g
.
39
.
C HAP T ER
VI
C OL OU R
W HE N a source of wh ite light is looked at through a plate of
coloured glass it appears to be coloured Th e coloured glass
absorbs some of the colours in the white light more completely
tha the others Fo example red glass absorbs the yellow
green blue and Violet ray s al m ost completely and is only trans
pare t to the red rays Th e colour of transparent bodies when
seen by transmitted light is due to selective absorption of certain
rays from the incident white light A substance which absorbs
all the rays equally so that their relati v e intensities are not
changed by passing through it is colourless like water
When a beam of light falls on a mirror it is nearly all
reflected in a de fi i te di rection but ordinary rough surfaces
scatter the light which falls on them in all directions Th
scattering or irregular e fl e i o enables bodies to be seen when
they are ill u minated All the light which falls on bodi es is not
scattered much of it is absorbed and co verted into heat in the
body
Th e incident light is partly reflected at the surfaces of the
pa rticles composing the rough surface d emerges from the
surfac e after undergoing a series of reflexions and refrac tions
S me of it is reflected only once and s me may be reflected and
ref a cted se v eral times before emerging
If the particles composing the substance absorb rays of all
colours equally the scattered light will be of the same colour as
the incident light but if the particles absorb some f the colours
more strongly tha thers the scattered light will be coloured
di fferently to the i ncide t light
F
example the particles composi g red paint absor b all the
ther colours more than red so that when red paint is illuminated
w P
23
.
n
r
.
,
,
,
n
.
.
,
.
n
,
.
x
r
e
n
.
n
,
.
an
.
o
o
r
.
,
o
,
n
o
n
or
,
o
.
.
.
n
L I G HT
3 54
P
A
I
V
R
T
[
by white light the scattered light consists of a mixture of white
light and red light so that the paint looks red When two
differe tly coloured paints are mixed the mixture has the absorbing
powers of both paints F example yellow paint absorbs most
of the red blue and violet and scatters chiefly yellow and some
green Blue paint absorbs the red yellow and violet and scatters
blue d some green A mixture of yellow d blue paints there
fore absor b s nearly every colour except green so that it acts as a
green pai t
S ome substances have the power of re fl ecting light of certain
colours more stro gly tha others so that when white light is
incide t on a polished surface of such a substance the reflected
light is coloured F example gold and copper mirrors reflect
red and yellow light better than blue d violet Certain aniline
dyes have this power of selective e fl e i o or surface colour in a
marked degree F example a polished surface of solid cyanine
scarcely reflects green light at all but reflects all other colours
fairly well Th e reflected light from it is purple and is a mixture
of red some yellow blue and violet rays S ubstances which
reflect a particular colour strongly in this way may absorb it
strongly also A substance which selectively reflects green may
appear purple by transmitted light because it absorbs and reflects
the green s that o ly the other colours get through it
It is found that the colour of a mixture of lights of di fferent
wave lengths is not alo e su ffi cient to determine even roughly the
wave lengths present Fo example a mixture of blue light and
yellow light in suitable proportio s appears perfectly white
lthough it contains no red green or V iolet Th apparatus
s hown in Fi g 40 may be used to study the colours got by mixing
lights of different wave lengths
White light is focused on a slit at S and made parallel by a
lens L It then passes through a prism P and lens M which form
a spectrum at a screen K This screen contains an aperture equal
to the spectrum ; it serves to cut o ff any stray light A lens N is
adj usted S that it forms
image of the lens M on a white
screen P This image is a circular patch of white light in which
the di fferent colours separated at K are recombined By coveri g
parts of the aperture at K with suitable strips of metal any
,
.
n
or
.
,
,
.
,
an
an
.
,
n
.
n
n
,
n
or
.
,
an
r
.
or
x
.
n
,
.
,
,
.
.
n
o
.
n
r
.
,
n
a
e
.
,
.
.
.
.
.
O
an
.
.
n
’
C HAP T ER
O P TI CAL I
VI I
N S TRU M EN TS
P HOTOG RAP HI C camera consists essentially of a convex lens
through
which
light
enters
a
box
on
the
opposite
h
hi
T
"
c
ide of which is a screen Th e dist a nce of the
lens from the screen is adj usted S that the image of ex ternal
bj ects formed by the lens falls on the screen Th screen is a plane
and it is necessary that the lens S hall for m a distinct image all
ver the surface of the screen Th e formula
A
h
e
p
a
m
era
o
to g r a p
c
S
.
O
O
.
o
e
.
1
1
1
f
u
7)
which gives the distance of the image from a thin co vex lens
is o ly applicable when the obj ect its image and all the rays
of light passi g through the l ens are very near to the axis of the
lens
hows the shape of the image of a straight line
F i g 41
mple con vex lens Li lI
A BCDE perpendicular to the axis of a
n
n
,
n
.
.
S
/
Si
Fi g 41
.
.
.
image of C is at C that o f A a t A and so on Th e image
a curved line Th e lens used in a camera has to be designed
’
Th e
,
.
’
.
VII ]
CH
.
P I
O T C AL
I NS TRUM EN TS
3 57
that it for m s an image which is straight and perpendicular to the
axis CC so that it lies on the screen in the camera Th e camera
lens also must be an achromatic combination Camera lenses
usually consist of two or more achromatic lenses designed so that
they produce a plane undistorted image In order to let a large
amount of light into the camera i a hort time the diameter of
the le s has to be large This requires rays of light to be used
which do not pass through the lens near to its axis A S imple
co vex lens does not cause rays which pass through it t any
great distance from its axis to com e to a focus at the same poin t
ays close to the axis ; so for this reason also a more complex
as
system than a simple thin le n s is required in a camera Th e
theory of the design of lenses for special purposes is too compli
c te d to be dis c ussed in this book
Th e human eye is in principle not unlike a photographic camera
It
consists
of
a
nearly
spherical
ball
filled
with
E
T
a transparent medium composed chiefly of water
Th e skin f the ball is opaque excep t a circular p t h in front
called the pupil of the eye Light enters the eye through the
pupil and is refracted at the outer spherical surface and within the
eye so that rays coming from a dista t point are brought to a focus
on the inner surface of the ski n at the back of the eye This inner
surface is called the re t ina and the light forms an inverted real
image of outs ide obj ects on it j ust as an i verted image is formed
on the screen at the back of a photographic camera Th e retina
is covered with an immense number of nerve endings from which
nerves lead to the brain Th e sensation of S ight is due to the
action of the light on these nerve endin gs in the retina which
causes nervous impulses to be tra smitted from the nerve endings
through the nerves to the brain
—
N ear the front of the eye ball embedded in the transparent
medium is a convex lens shaped tra sparent body which has
a slightly greater refractive index than the rest of the eye Th e
light has to pass through this lens and is made to con v erge
sli ghtly more in doing so Th e curvature of th e surfaces of this
lens can be varied by the contraction of an annular muscle which
surrounds it In this way the eye can change its focal length so
that sharp images of external obj ects at di fferent distances can
'
.
.
,
.
n
n
S
.
.
a
n
r
.
a
.
.
h
e
ye
.
.
o
a
,
c
.
n
.
-
,
n
,
.
.
n
.
,
-
,
n
,
.
.
.
L I G HT
3 58
I
v
P
AR
T
[
be for m ed on the retina Just i f o t of th e le s there is an
opaque diaphragm with a circular aperture in the middle called
the iris This diaphragm expands and contracts so that the
diameter of the aperture varies In this way the amount of light
entering the eye is regulated In bright li ght the aperture
becomes very small and in faint light it gets much larger
Owing to optical defects eyes are often unable to p oduce
distinct or sharply focused images of external
t l
S
obj ects on the retina S uch defects can ofte be
remedied by using spectacles which consist S imply of a pair of
thin lenses which are supported one in fro t of each eye Th e
curvatures of the surfaces of the lenses S hould be designed so that
they counteract the defects of the eye and enabl e it to form
a sharply focused image on the retina F example it often
happens that people can see obj e cts very near to their eyes
distinctly but obj ects at a distance appear blurred and indistinct
This shows that the focal length of the eye or rather its range of
possible focal lengths is too short so that the image of a distant
obj ect is formed in front of the retina i stead of on it Th e focal
length can be increased by using concave lenses as spectacles
Another common case is that of people who can see di sta t
obj ects distinctly but not near obj ects This shows that the
focal lengths of their eyes are too long so that the image of near
obj ects lies behind the retina This defect is remedied by means
of convex lenses There are many other possible optical defects of
the eye which can be remedied by properly designed spectacles
It often happens that the lens required for one eye is di fferent
from that required for the other S pectacles should always be
prescribed by a specialist because much harm to the eyes may be
done by using unsuitable lenses
P eople can usually see obj ects mo st distinctly at a distance
about ten inches from the eye This is called the normal
of most distinct vision Fi g 42 shows how the rays of li
an Obj ect form an image on the retina of the eye Th e
takes place almost entirely at the front surface CD of the
A B is the image of AB
Th e bundle of rays from any point
A on the obj ect enters the pupil as a nearly parallel beam
diameter of the aperture in the iris is small compared with
'
'
n
.
r
n
n
.
.
.
.
r
p ec
ac
es
.
n
.
,
n
.
or
.
’
.
,
,
,
,
n
.
.
n
.
.
.
.
.
,
.
.
.
.
.
’
’
.
.
L I G HT
3 60
P
A
R
T
I
v
[
away from the eye to be seen distinct ly Th e obj ect d le s
S hould be placed so that the image seen appears as distinct
possible Th e distance of the virtual i m g e f m the eye is then
an
.
n
as
a
.
F i g 43
.
ro
.
equal to the distance of most distinct vision which is about
If the focal length of the
1 0 inches or 25 c m s for a normal eye
magnifying glass is f we have
,
‘
.
.
1
1
f
1
25
u
’
where is the distance of the obj ect from the lens
supposed to be close to the eye If f 3 e m s we get
u
.
.
1
3
Th e
lens is
.
1
1
25
77
’
27 cms
which gives
Th e formation of the virtual image seen
by the eye in this case is shown in Fi g 44 Three rays are shown
u
.
.
.
F i g 44
.
.
coming from the top of the obj ect A These are refracted by the
lens so that they meet at the top of the virtual image A when
.
'
CH
.
V II ]
I
I N S TRU M E NTS
O PT CAL
3 61
produced backwards They are refracted by the eye so that they
meet on the retina at A LB is equal to about 25 cms Th e
magni fying power of the lens is the ratio of the height of the
vi rtual image A B C to the height of the obj ect ABC
Th is ratio is equal to LB /LB or since LB is only slightly
less than the focal length of the lens the magnifying power is
appr ximately equal t 25/f Fo example a lens of foc l length
A good magni fying glass
25 cms magnifies about 1 0 times
consists of a properly designed achromatic combination of two or
more lenses which act lik e a S ingle lens
A magnifying glass with a focal length of less than about
1 cm is inconve n i ent so that when it is desired
T
Mi
to obtain a higher magn i fying power than about
25 a simple magn i fyi ng glass is not suitable
Th e microscope is
the instrume t used when high magni fying powers are required
It consists essentially of two lenses or systems of lenses called the
obj ective and the eye piece
Th
obj ective is a convex lens of short focal length which
forms a real inverted image of the Obj ect Th i s real image is
viewed through the eye piece which acts as a S imple magnifying
glass and for m s a virtual image of the real image at the distance
from the eye of most distinct vision Th e obj ective is placed at
a distance from the obj ect only slightly g reater than its fo al
length so that the real image is larger tha the obj ect
Le t f be the focal length of the obj ective
the distance of the
obj ect from it and the distance of th e real image from the
bj ective We have
.
'
”
.
.
’
’
'
.
’
,
r
o
o
a
,
.
.
.
.
h
e
c ro s c o
pe
.
.
n
.
-
.
e
,
.
-
,
.
c
n
,
.
u
7)
O
.
Fo r
1
1
1
1
1
l
f
u
7)
exam ple s ppose f 1 cm and 7)
1 5 cms
This gives
1
1
l
o
em s
In
th
i s case the ob ect s
e m s from
j
15
lZ
Il
,
u
:
.
.
'
B
—
r
u
—
i
.
.
the obj ective and the real image is 1 5 cms from it on the opposite
side Th e height of the real image is to the height of the bj ect
the rat i o
or i n the part cular case cons dered 1
so that
.
O
.
7)
In
77
i
i
0
ii
the r al image in this cas e is 1 4 times as high as the obj ect
Le t f denote the focal length of the eye piece
Its magnifyin g
e
.
-
.
3 62
I G HT
L
A
R
I
v
P
T
[
power is approximately equal to 25/f so that the total magnifying
power of the microscope is equal to
’
25
7)
u
If 7)
1 5 cms which is about the value usually adopted and
if f is small then u and f are nearly equal so that the magnifying
power is nearly equal to
.
,
,
,
15
x
25
3 75
example if f 05 cm and f 3 e m s the magnifying p ower
is about 25 0 Fi g 45 shows how the real and virtual images are
’
Fo r
.
.
.
,
.
F i g 45
.
.
formed in a microscope A B C is the small obj ect Tw rays
from C are shown that are refracte d by the obj ective 0 and
form a real image of C at C These rays then fall o n the
eye piece E and are refracted so that when produced backwards
they meet at the virtual image C which is 25 cms from the eye
Th e two rays are refracted at th e surface of the eye and form
a real image of C on the retina at C as shown
.
.
o
’
.
-
.
’
”
.
.
L I G HT
3 64
P
I
V
A
R
T
[
equal to f —z f To get a large magnifyi n g power it is therefore
necessary to use a mirror of lo g focal length n d an eye piece of
short focal length
A refractor is shown in Fi g 47
It works in the same way as
a reflector and its magnifying power is equal to f /f where f is the
focal length of the obj ect glass d f that of the ey e piece
’
“
-
.
n
a
-
.
.
.
’
,
an
-
Fi g
.
.
47
glass of a telescope is always an achromatic lens and
the curvatures of its surface S hould be designed so that it may
produce an u distorted distinct image of a distant obj ect Th e
design of a good obj ect glass is a complicated problem which will
not be discussed i n this b ook Th e eye piece usually consists of
a combination of two or more lenses designed so as to be achromatic
and not to distort the image
magic la tern or proj ector is the instrument used for
Th
proj
ecti
g
enlarged
image
of
an
obj
ect
on
to
M
T
m
a screen so that t can be seen by a large number
of people Th e obj ect is strongly illuminated d a convex lens
is used to produce the image of it A magic lantern is S hown in
Fi g 48
S is a powerful source of white light such as an arc
Th e
'
o
b j ect
’
-
,
n
.
-
,
-
.
.
n
e
h
e
La
a
e
n
g ic
an
m
i
an
.
.
.
.
F i g 43
.
.
lamp C and C are two large convex lenses which focus the rays
from S on to a convex lens 0 Th e rays drawn pass through the
lens 0 at its centre so that they are not deviated by this lens
f
.
.
,
.
CH
.
V II ]
P I L I N S TRU M E NTS
3 65
O T CA
light passi g through the lens 0 illuminates the white
creen A B If a partly tran parent obj ect like a lantern slide
is put up at L in front of the lens C the positio of the lens 0
can be adj usted so that it produces a S har ply focused image of
the obj ect on the screen Fi g 49 shows how this image is formed
Th e
s
n
s
.
’
.
.
.
F i g 49
.
Three
n
.
rays are shown starti g from different points on the source
S which meet at P on the obj ect and are focused by the lens 0
on the screen at P Thus P is a real image of P Th e middle
ray of the three passes through 0 at its centre so it is like one of
the rays drawn in the previous figure Th e le ses C and C form
a rea l image of the source S at near to the lens 0 Th e lens O
is usually replaced by an achromatic combination of two or more
lenses designed so as to produce
undistorted image sharply
focused all over the screen
n
’
'
.
.
,
’
n
.
or
.
an
.
F
N
RE ERE CE
Th e Th e o r y
f
o
Op ti c a l I n s tr u m en
ts ,
E T
.
.
Whi tta k e r
.
C HAP T ER
VEL OCI TY
THE
V III
OF L I G HT
I LE O attempted to determine the velocity of l ight Tw
observers
situated
some
miles
apart
on
a
clear
night
V
l it
m
were pr vided with lanterns which could be covered
when desired by scree s O observer A uncovered h i s lantern
and the other B uncovered his lantern as soon as he saw the light
from A It was found that A saw th light fro m B as soon as the
lantern at A was uncovered so that the time taken by light to go
f om A to B d back was too small to be measured in this way
If the distance f o m A to B and back was 1 0 miles and the smallest
time interval which could have been detected was say 0 1 second
such an experimen t would how that the velocity of light was not
less than 1 00 miles per second
In 1 6 7 6 Roemer a Danish astronomer working in the Paris
Th
Observatory made the first estimate of the velocity of light
planet J upiter has several satellites or moons which revolve round
it in nearly circular orbits with nearly uniform velocity Th
satellite nearest to J upiter goes behi d J upiter as seen from the
earth once in each revolution Roemer observed the times at
which this satellite disappeared behind J upiter during more than
Th average interval between successive disappear
wh l year
ane cs is 42 hours 28 minutes 3 6 secs Thus w may regard the
satellite and J upiter as sending out a series of sig als at equal
intervals of time Roemer fou d that the observed time intervals
between successive disappearances were not equal but varied
throughout the year i a regular way While the ear th was
moving away from J upiter the intervals were longer than the
average and while it was moving towards J upi ter they were horter
than the average Roemer attributed this to the finite velocity of
light d calculated this velocity from his results
G AL
Th
e
e
oc
.
y
o
o
f
g“
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n e
.
e
.
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an
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,
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;
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.
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.
a
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.
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n
.
n
.
S
.
an
.
L I G HT
3 68
P
I
V
A
R
T
[
months it is reversed so that then the direction of the light is
changed in the opposite direction
Th angular diameter of the small circle described by the star
is therefore equal to twice the angle BA C so that we have
,
.
e
F i g 50
.
BAC
.
If C denotes the velocity of light and 7) that of the
earth in its orbit then
205
,
7)
ta n BA C
tan
C
1
205
1 0000
.
velocity of the earth in its orbit round the sun is equal to
the circumference 2
x 1 0 cms divided by the number of
seconds in a year which is nearly 3 1 6 x 1 0 This gives
so that C = 1 0 x 3 x 1 0 3 x 1 0 cms
e
3 x 1 0 cms per
7)
per sec Thus Bradley s method gives practically the same result
as Roemer s
Th e velocity of light in a vacuum 3 x 1 0 cms per s e or
miles per second is so great that it is di fficult to measure it on the
earth Two methods however have been sed successfully Th e
first method was used by Fizeau in 1 8 49 Fi g 5 1 is a diagram
of his apparatus S is a source of light the rays from which pass
through a convex lens L and are reflected from a pla e mirror M
so that they form an image of S at F They then pass through
a lens L which makes them parallel Th e parallel beam passes
through a lens L which causes it to converge to a focus on the
surface of a plane mirror C which reflects the rays back through
’
L
d L so that they form a second image of the source at F
Th e distance between L and L in Fi e u s experiment was about
This long distance is omitted in the figure
Th e
ms
rays then fall again on the mirror M This mirror is a glass plate
the surface of which is coated with a layer of silver so thin that it
o ly reflects abou t half the light which falls on it d allows the
other half to pass through About half the light returning from
Th e
71
13
x
.
7
.
,
6
s
.
e
6
4
1°
.
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.
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.
10
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.
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.
C H vm
.
]
THE V ELO C I TY o r
L I G HT
3 69
thro gh L therefore passes through 11 13 an eye piece L through
which it pas es i t the observer s eye E where it forms an image
of S on the retina at D Th e eye E is draw on a much larger
scale than the rest of the apparatus At F there is a wheel W
which can be made to rotate at a known speed about the axis AA
Th e circumference of this wheel is cut into a number of teeth or
cog s with spaces between the cogs equal in width to th e teeth
Th
wheel is placed so that the image at F lies close t the
circumfe ence and as the wheel rotates the light f om S is
stopped by the cogs but passes through the spaces between the
cogs
If the wheel is turned slowly the eye E sees an intermitte t
image of S for each cog stops the light If the wheel is turned
0
’
u
0
1
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e
o
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r
,
.
n
,
.
,
Fi g 5 1
.
.
more quickly the light which has passed through between the two
cogs is stopped on its way back to E by a cog which has mo v ed to
the positio of the image at F while the light travelled from F to
C and back
If the speed of the wheel is the doubled the ligh t
gets through to E again because whe it gets back to F the cog
passed d the ext space between two cogs is there If the
h
speed is made three times the first speed the light is stopped
again and so
It is possible t measure a se ies of speeds of
the wheel proportio al to the numbers 1 2 3 4 5 6 etc at which
the light is stopped when the number is odd and gets through
whe it is eve
,
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.
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.
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[
In Fi
s experiment the wheel had 7 20 teeth and it was
f und that increasing th e speed by 1 2 6 revolutions per second
caused the image of the source seen by the observer to change
from ze o inte sity to its maximum intensity or i creasi g th e
speed 25 2 revolutio s per second caused the brightness of the image
to cha ge from e zero of intensity to the next zero This shows
’
that the time taken by the light to go from F to C and back was
3 70
LI G HT
’
ze a u
o
n
r
n
n
,
n
on
n
.
1
equal to
x
was
second
7 20
.
d stance fro m
Th e
i
F
to
and back
’
C
cms so that the velocity of the light worked out as
.
1 7 3 26 00
X
x
7 20
x
31
10
1°
cms per s e e
.
.
In 1 8 7 4 Cornu repeated Fi e u s experiment with improved
apparatus d found the velocity of light to be almost exactly
3
1 0 cms per second
Another m e th d o f measuring the velocity of light was invented
by Foucault in 1 8 5 0 Th e principle of this method is as follows
In Fi g 5 2 S is a source of l i ght which passes through a lens L
and is then reflected from a plane mirror M on to a concave
mirror M at the surface of which the lens L forms an image of S
Th e mirror M re fl ects the light back so that it returns alo g its
h
M
e mirror M is made to
ath
to
T
rotate
rapidly
in
the
p
direction shown by the arrow about an axis at M perpendicular to
the plane of the paper
While the light goes from M to M and back the mirror M
turns through an angle so that the returning beam is not reflected
from M back i to L but i a direction like MS Th returning
d forms
light is recei v ed in a telescope L L
image of S at
"
S which is observed by the eye E through the eye piece L
Th e
mirror JV
I is placed so that its ce t e of curvature lies on the axis
of rotatio of the rotati g mir or M
that as the image of S
sweeps across M it is all the time eflec ted back to M If the
rotating mirror makes revolutions per seco d d the a gle
the time take by the light to go from M to M d
S MS
x
back is
so that if MM d the velocity of light is
’
a
z
an
x
10
.
.
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o
.
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.
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,
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on
77
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9
l 440 7i d
7 20 77
H
an
C HAP T ER
IX
I N TERF EREN CE AN D D I
FF RACTI O N
fact that rays of light i a u iform medium like air or
a vacuum are straight was explained by N ewton on the theory
t hat light consists of material particles shot out from hot bodies
in all directio s with very great velocities Th e theory that light
is a wave motion i a medium that fills all space was opposed by
N ewton on the ground that th i s theory could o t explain the
rectilinear propagation of light He argued that a wave m tion
would travel rou d
bstacle so that sharp shado ws ought not to
be formed and poi ted out that sou d which is a longitudinal wave
motio in air travels around corners and does not give shadows
like light
Th e wave theory was first put forward in 1 6 7 8 by Huyge s
who showed how it co ld explain e fl e i
d refraction
Huygens however was not able to explain the formation of
shadows on his theory ; so that the corpuscular theory supported
by N ewton was ge erally accepted u til the beginning of the
i eteenth century Th corpuscular theory was verthrow and
the wave theory firmly established early in the nineteenth ce tury
chiefly by the investigatio s of Young in E gland d Fresnel in
You g discove ed the principle of interference between
Fra ce
two trai s of waves d howed that certai optical phen me a
could be explai ed by it which cou ld t be explai ed satisfactorily
the corpuscular theo y Fi g 5 3 shows
experime t due to
You g S is a arrow slit the le gth of which is perpe dicular to
the paper in an opaque scree which is stro gly illumi ated by
mea s of a so rce of white light L and a co vex lens Th light
f om the slit S falls on a seco d scree co taining two narrow slits
A and B parallel to the slit S
Th e slits A and B are only a short
THE
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CH
.
Ix
I NTERFE RENC E
]
D I FF RACTI O N
AN D
3 73
distance say two or three millimetres apart Th e light from
Th e distances S A and A C
A and B falls on a white screen CD
between the screens may each be one or two metres According
to the corpuscular theory we should expect to get on the screen
CD two bright lines of light in the positions got by j oining S A
,
,
.
.
.
Fi g 5 3
.
.
and S B by straight lines and producing these lines to meet CD
You g found that the light from each of the slits A and B was
di ffused over a considerable area of the screen CD and that where
the illumination due to A was superposed on that due to B a series
Th e bands
o f p rallel bright and dark coloured bands appeared
were parallel to the lengths of the slits Th e central band at the
bisector of the angle A S B was white and on each side of it was
a dark ba d and then nearly white bands Outside these came
dark bands and then a series of coloured bands getting rapidly
more confused and indisti ct a s the distance from the central white
band in creased If instead of white
light coloured light is used to illu
m i nate the slit S then bright bands
are obtained of the same colour as
the light used with dark bands
between them Th e bands are
f rther apart with red light than
with blue light Fi g 5 4 shows the
appearance of the h ds They
are called interference bands
F ° 54
A more con v enient way of
(R p d d b y p m m f m
Op t i
S h t
obtaining them is by means of an
)
instrument invented by Fresnel called Fresnel s b i prism This
is shown in Fi g 5 5 S is a slit in an opaque screen which is
illuminated from the left hand side Th e b i prism A ED has three
plane sides AD AE and E1) Th e angles at A and I ) are equal
and very small so that the angle at E is nearly
.
n
a
.
.
,
n
.
,
n
.
,
,
.
u
.
.
an
.
.
1
e
ro
er
uce
c
°
°
’
u s er s
i ss
cs
n
ro
.
’
-
.
.
.
-
-
.
,
,
.
L I G HT
3 74
I
v
T
AR
P
[
Th e light from S after passi n g through the b i prism falls
a white screen CC S E should be about 25 cms and E0 about
light that passes between A d E i
Th
200 cms or more
deviated downwards by the prism d illuminates the screen
on
-
’
.
.
s
an
e
.
.
an
Fi g 5 5
.
.
between 0 and F Th e light that passes between E and D i
"
deviated upwards and illumi ates the screen between C and F
Thus between F and F the screen is illumi ated by light that
has passed through AE d also by light t h at has passed through
ED
Between F and F a series of interference bands is forme d
the screen like those obtained in Young s experiment Much
o
brighter bands are ob tai ed with the b i —prism than with two slits
because the prism allows much more light to pass
Th e formation of such interfere ce bands is easily explained
by the wa v e theory of light and the wave length can be calculated
from the dista ce between the bands In Fi g 5 6 let A d B b e
the two slits in Young s experi m ent and P Q the screen on which
L e t P be a point such that
the i terference bands are see
AP
At P there is a bright band
BP
If the light from S i s
monochromatic light the according to the wave theory it consist s
of trains of waves These waves pass through the slits A d B
and then diverge from A and B so that the screen is illumi ated
by two series of trains of waves At P the two paths from S S AP
and S EP are of equal length so that waves reaching P from A will
agree in phase with those from B At P therefore the two sets of
waves reinforce each other so that there is a bright band at P
N w consider a point Q
Th e two series of waves which arrive at Q
s
.
’
n
.
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n
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’
.
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.
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.
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.
.
.
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,
an
.
n
.
,
,
,
.
,
o
.
,
,
.
L I GH T
light is used The it will be found that
b ight ba ds occupy about 06 cm Hence
v
P
R
I
A
T
[
3 76
r
n
spaces be twee
te n
n
.
n
.
a
s
3 00
or
6
x
This
°m
s
.
is about the wave length f yellow light
T explain the la ws f
fl
i n and refractio of light by the
wave theory Huyge s used a principle called after
H
P i
l
him Accordi g t this principle when the ether
at any point is disturbed by the passage of a light wave over the
poi t then the point becomes a ce tre of disturba ce i th e ether
In Fi g 5 7 let S be
d a spherical light wa v e dive ges from it
a source f light and AB C a spherical wave which started f om
At every poi t
the surface of the sphere ABC the ether is
o
o
u
yg
r n c
O
re
en s
ip
e
n
ex o
n
,
.
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.
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.
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i
.
.
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O
n
on
Fi g 5 7
.
Fi g 58
.
.
.
disturbed so that every point on AB C must be supposed to se d
out a spherical wave After a time 75 these waves will all have
radii equal to c t where c denotes the v elocity of light Fi g 5 8
shows a number of circles f equal radii drawn with their ce tres
on the circle ABC between the points A d B All these circles
touch a circle DEF of radius + c t and a circle CH of radius
n
,
.
.
O
.
n
an
r
.
CH 1 x
.
I NT ER F ERE N C E
]
AN D
D I FF RACTI ON
3 77
is the radius of the circle AB C Th e wave from S
is i the position AB at the time 0 and at the time t each element
of it must have mo v ed a distance c t from its position on AB and
must therefore lie on the surface of the sphere of radius c t w i th
centre at the position of the eleme t at time 0 Huygens supposed
that a surface drawn so as to touch all the spheres of radii c t would
coi cide with the light wa v e after the time t F example in
Fi g 5 8 the wa v e between A and B after the time t would be i
the position D EF Th rays of light Huygens supposed coincide
with the radi i of the sphe rical waves which end on the surface
which they all touch We might expect accordi g to Huygens
principle that the wave AB would also produce a wave CH moving
inwards which does n t happen in fact Also it is not clear why
the spherical wa v es produce no effect except at the surface of the
sphere DEF which they all touch Huygens principle alo e
the efore does not help us v ery much but Fresnel showed that by
combining Young s principle of i terference with Huygens principle
the nearly rectil i near propagation f light could be explained
Before discussing the eleme ts of F e s e l s theory we may consider
how Huyge s explained the laws of fl i and refraction
Le t A CB i Fi g 5 9 be a plane light wave and AB the surface
Here
ct
r
.
‘
7
.
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,
n
.
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or
.
.
,
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.
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.
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.
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.
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,
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.
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n
re
ex on
.
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.
Fi g 59
.
.
"
of a plane mirror Le t the wave at B move along BB B and C
along CC so tha t CC and BB B are rays of light and are pe pe n
di c u l to the wave AB
Where the wave meets the mirror at A
’
.
’
ar
’
.
”
r
L I GH T
3 78
P
A
R
I
V
T
[
we suppose following Huyge s that a spherical wave starts out
f om the mirror the radius f which after a time t is t By the
time the wave gets to the mir or at B the radius of the wave
starting from A will be equal to BB Describe a circle with centre
Also with ce tre C and radius equal
A and radius equal to BB
This circle represe ts the spherical
t B B describe another circle
wave starti g from C when the wave has got to B If we draw
a pla e B O A touching the circles it will be i the position of
the reflected wave at the instant when B has got to the mirror at
"
B
S i ce the a gles at B and A are right angles and AA
BB
it follows that AB d B A are equally incli ed to the mirror i
"
C C is the reflected ray
accorda ce with the laws of e fl e i o
cor esponding to CC for it j oins the centre of the sphe rical wave
starting from C to the point C where it touches the wave su face
n
,
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c
.
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.
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a similar way Huygens explained the laws of refraction
supposed that the ratio of the velocity of light in the first
In
He
.
Fi g 6 0
.
.
medium to that in the seco d was equal to the refractive index of
the second with respect to the fi st medium If the first medium is
air in which the velocity of light is c a d the seco d glass or any
other tra sparent substance i which the velocity Of light is c the
where
is
the
refractive
i
dex
of
glass
Huyge s supposed
c c
/
with respect to air
n
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.
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,
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L I GH T
3 80
A
R
T
P
IV
[
could be satisfactorily explai ed on the wave theory He also
showed that the observed deviations fro m exact rectilinear pro
were
acco
da
ce
with
the
wave
theory
In
this
book
t
i
i
p g
we shall co sider o ly e simple case i an eleme tary way
I Fi g 6 1 let S be a small source of light a d A B an opaque
screen with a straight edge at A perpendicular to the plane of the
paper which throws a shadow
a pla e white screen CD Th e
scree is illuminated by S from C to C but from C to D the paque
n
a
a
on
.
n
n
on
n
n
n
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n
,
.
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.
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on
,
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.
O
,
Fi g 6 1
.
.
screen AB casts a shado w If S is a arrow slit illuminated with
mo ochromatic light from the left side with its length parallel to
the straight edge t A d the dista ces S A and A O are both o
or two metres then on examining the edge of the shadow at C it
is fou d that j ust above C there are narrow bright d dark bands
getting rapidly arrower d less distinct as the distance above 0
is i creased while below 0 the illumination of the screen rapidly
but gradually gets fainter so that a short distance below 0 it is not
appreciable Th e appearance of the edge of the shadow is shown
in Fi g 6 2 CC is the level of the geometrical shadow that is
where the straigh t line S A C meets the screen Bright bands are
marked BB and dark bands DD According to the corpuscular
theory we should have expected the screen to be uniformly bright
above CC d u niformly dark below it Th e slight bending Of
the light into the region below CC is called di ffraction d the
bands seen above CC are called diff action bands Th e problem
is to explain the bands on the wave theory d to show why the
n
.
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,
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a
an
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.
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.
[
I NT ER FERE N C E
X]
381
A ND D i FF RAC TI O N
illumination extends o ly suc h a little way below C C Huyge s
could t solve this problem and the wave theory was not accepted
until this w do e by Fres el
'
n
n
.
n o
as
n
n
.
Fi g
.
62
.
in Fi g 6 3 represe t a pla e area d suppose that
a train of plane waves of wave le gth 7x is movi g perpendicular
Le t AB CD
n
.
n
n
an
n
the area so that CP which is ormal to
i a ray
Th wave surface are the p lanes parallel to the area
to
n
e
s
n
s
of
light
A B CD
.
.
L I GH T
3 82
A
P
R
T
I
v
[
disturba ce in the ether at P may be regarded as produced by
the disturbances which diverge as spherical waves from every poi t
on the area A B CD Th resulta t e ffect at P at any i sta t can
be got by findi g th resultant f all the e ffects at P at that
i stant coming from all the di fferent parts of AB CD Th e
amplitude of the vibration at P due to the train of spherical
AB CD may be take to be
waves comi g from a small area
proportional to the small area and to dimi ish as its dista ce from
C i creases
Describe a series of concentric ircles o the plane AB CD with
centres at C and radii equal to N/wx V2w7x V355 and so on Here
= CP
S uch a series of concentric Circles is shown in Fi g 6 4
Th e
n
n
n
n
n
e
.
n
o
e
n
.
on
n
n
n
n
n
.
C
n
,
x
»
,
.
.
.
.
Fi g 6 4
.
.
If the wave length of the light is 6 x
cm and is equal to
1
x
0
or
cms then the rad Of the circles are
7;
1 cm V2 cms V3 cms etc respectively
Th area of the first
ci cle is m k that of the second 2 m) and so
Thus the a eas
between successive circles are all equal to
Th e circles there
fore divide the area ABCD into equal parts These equal areas
.
1
5
.
.
r
.
,
ar
,
,
ii
,
.
,
e
.
7
on
c
.
.
r
L I GH T
3 84
P
A
R
T
I
v
[
Hence
each of these series is equal to 5S since the sum of the two
is equal to S
esultant amplitude of the vibration at P is the ef e
Th
equal to e half that due to the disturbance comi g from the firs t
circle described rou d C
]
,
e
,
.
r
or
r
on
n
-
n
.
If we describe a circle with centre
and radi us
0
that is
of half the area of the first Fres el zone the we may regard
the disturbance at P as coming fro m this small area at C d the
disturbances from all the rest of the area AB CD can be regarded
as destroying each other by interfe e ce at P and so hav i g
e ffect there
In Fi g 6 5 let RP be a ray f light and A CB surface
perpe dicular to it at C It follows f om the above that the light
n
on e
n
,
an
r
n
n
n o
.
’
o
.
n
a
r
.
Fi g
.
65
.
at any point on th e ray say P may be regarded as comi g from
a very small area o A OB surrounding C N o w 0 can be taken
anywhe e along the ray RP so that it follows tha t the light tra v els
along the ray and the light at P will t be cut Off by opaque
screen u less it crosses the ray RP comes so ear to R P that it
c ts ff pa t of the small area arou d the ray from which the light
at P comes If CP
the radius of the small area at C is
,
n
,
n
.
r
an
n o
n
u
n
r
O
as
.
ask
example if
Fo r
2
n
or
w = 1 00
cms
.
an
d A
6
cm the
x
.
radius is V3 x
about { 55 cm
In this way Fresnel explai ed the for m ation of shadows on the
wave theory It is lear that the approximately rectili ear
propagation of light depends on the fact that the wave length is
1 00 m
so smal l F example i f we take A 1 0 cms d
or
.
n
.
.
or
n
C
.
.
an
7
. ;
o
s
.
C H. I x
I NT ER FER EN C E
]
AN D
D I FF RAC TIO N
3 85
we get
224 cms so that the illumination at P would come
from an area of radius 22 4 cms at C and so could not be regarded
as travelling along close to the line CP
Th e series S
S
S
may be written
S
.
,
.
.
2
l
3
4
if g
s
{( ,
s,)
-
terms ( S S ) ( S S ) etc are all v ery small for there is
no reason to suppose that the e ffect due to a small area dimi ishes
at all rapidly with its dis tance from C
C nsequently the successive rings r u d 0 may be supposed
to nearly destroy each other s effects ; the utside half of each
ring destroys the e ffect due to the i side half of the next ri g
Th e t tal e ffect is therefore that due to the inside half of the
first circle at 0
L t us now consider more i
detail the e ffect of an opaque
s ree on the illuminati n at P Le t the paque screen cover the
wh le f the area AB CD ( Fi g 6 4) except for a circular area with
its ce tre at C If the circular hole in the scree has a radius
equal to V37? it will let thr ugh the first Fres el zone only so that
the amplitude at P will be equal to S and so will be double that
when the scree is removed Th e intensity f the illumi ation t
P is proportional to the square of the amplitude of V ibration so the
inte si ty will be f ur times that when the screen is removed If
the hole in the scree has a radius equal to V2“ it will let through
the first two zones d the amplitude at P will be S S which is
v e y small Thu i creasi g the radius f the hole from V to
V23 1 diminishes the illumination at P from four times the value
of this value These
without the screen t a very small fi c ti o
at first sight surprisi g results of the the ry have been verified
experimentally S uch results clearly dispr ve the corpuscular
theory of light
If instead of a screen with a circular hole i it we use an opaque
ci cular disk we get equally remarkable results S upp se the
ce t e of the disk is at C and its radius is equal to that f one of
the Fres el zones the it cuts ff some of the first terms in the
erie
Th e
2
l
3
,
Q
.
,
,
n
.
o
o
n
’
o
n
n
.
o
.
e
c
n
n
o
o
o
.
o
.
n
n
.
o
3
n
1
n
,
,
o
.
n
a
,
n
o
.
n
an
r
s
.
l
n
n
2
o
3
o
a
n
n
.
o
o
.
.
n
r
n
r
o
n
s
o
.
,
n
o
s
Sl
w
.
P
.
S2
S3
S,
L I GH T
386
P
T
R
A
Iv
[
S uppose
the first four circles are covered by the disk then the
amplitude at P is equal to
,
which is equal to
;
L
and so is early equal to
its value when the
n
screen is removed According to this there should be a bright
spot in the centre of the shadow of a circular disk cast by a distant
small source of light Fresnel f und this to be the case
If the screen has a straight edge d covers half the area A BCD
so that the point C is on its edge the the sc een cuts ff the light
from half of each of the Fres el zo es so that the amplitude at P
.
o
.
.
an
n
,
n
S
I
o
r
n
and the illumination is one quarter of that when the screen
—
is removed By considering what parts of the zones are cut ff it
can be shown that the illuminatio ear the boundary of the
shadow of a screen with a strai ght edge should vary in exactly
the way which is bserved d which W s described earlier in this
chapter
O e of the best ways of finding experime tally the wave le gth
f
f
of
light
is
by
means
f
what
is
called
a
di
raction
it t
grating Di ffraction gratings are made by ruling
a series of equidistant parallel lines either on a plate of glass or on
a plane mir or Gratings ruled on glass are called tra smission
gratings a d we shall consider these o ly here
In making a tra smissio grating a plate of plane glass of u iform
thickness is taken d the lines are ruled o it with a diamond
poi t which scratches the glass Th li es are ruled by a machine
called a dividing engine which rules the lines straight and ve y
accurately equidistant Th e lines are ruled ve y close together
and consist of exceedingly a row scratches G ratings have bee
made with
or more lines to the centimetre A useful grating
has an area about 3 cms square ruled with about
lines or
5 000 li es per cm
In Fi g 6 6 let P Q be a small part of a grating draw on a
greatly enlarged scale P Q represents a section of the glass plate
d AB CD EF represe t cross sections of the lines or scratches
the le gths of which are supposed to be perpendicular to the plane
of the paper Th transparent spaces between the lines are
o
.
n
n
a
an
O
.
n
n
n
O
D ffr a c i o n
G ra i n g s
'
.
r
n
.
n
n
n
.
n
n
n
an
n
.
e
n
”
r
,
r
.
n
r
n
.
.
.
n
.
n
.
.
an
,
n
,
n
.
e
L I GH T
388
P
A
R
T
I
v
[
travelli g along BR has gone a distance BL f om B d the train
from D has gone a distance DM N o w B L — DM = B
Also
B
BD sin BD
B t BD
d so that B
1 s i 0 where 9
is the a gle betwee the parallel li es like BR d the ormal to
the surface of the grati g In the same way the waves at M have
t ve lle d
distance equal to d s i 79 furt h er than the waves at N
Thus if we consider pa allel rays from the top of each slit at a plane
perpendicular to the rays the waves from the top of each slit have
travelled a distance greater than those from the next lower slit by
d sin 9
In the same way we may consider rays from the middle
f each slit or from
corresponding
points
i
each
slit
T
h
e
y
ays fr m y slit have to go d sin 9 further than the correspo ding
rays from the next slit to reach the pla e represe t ed by J L
S uppose now that a convex lens is placed i
fro t of the grati g
parallel to the plane J L All the rays from the grating parallel
to BR will be brought to a focus on the axis of this le s At the
focus then we shall have a series of wave trains which have travelled
dista ces di fferi g by multiples of d s i 9 If d s i His equal to
A or a whole number of times A say A then the trains of waves
from each slit will agree in phase with those from the corresponding
points in all the other slits so that there will be a bright point at
the focus of the lens If d s i 0 is t exactly equal to K then
the light from the di fferent slits will interfere and there will be no
light at the focus F example suppose d sin 0 1 0 01 A Then
the light from the first slit will be behind that f om the 5 01 s t slit
by a dista ce equal to 5 00 x 1 0 01 A
so that the light
from the slit number 5 01 will destroy that from the fi st slit In
the same way the light from the 5 02 d slit will destroy that from
the second slit and so o
Thus unless d s i His v ery accurately
equal to k there will be no light at the focus of the lens
Fi g 6 7 shows the apparatus used to measure the wave length
Th e
O f the light emitted by a source S with a di ffraction grati g
light fro m S is focused o the slit L f a collimator L O Th
c llimator is adj usted so that the dista ce L C is equal to the focal
length of the le s at C Th light from each poi t o the slit L
then forms a beam of parallel rays after passi g through the colli
mator Th e grating C C is put up with its pla e perpendicular
to the axis of the collimato and its li es parallel to the slit Th
n
r
an
a
.
a
a
u
.
n
n
(
a
,
n
n
n
.
,
an
n
.
‘
ra
a
n
.
r
,
.
an
O
an
o
r
n
.
7
n
n
n
n
.
n
n
.
n
,
.
,
n
n
n
,
77
n
.
,
,
n
.
.
n o
77
or
,
.
r
n
:
r
.
n
n
n
.
77
.
.
n
o
n
.
e
.
n
o
n
n
e
.
n
n
’
n
.
r
n
.
e
CH
.
Ix
I NT ER F ERE N C E
]
D I FFRACTI O N
AN D
light coming from the grating is examined with a telescope TE
focused for parallel rays Th e axis of this telescope lies in a plane
perpendicular to the grati g lines d c t i n i g th e axis of the
collimator Th e telesc pe can be rotated in this plane about an
axis c i cidi ng with the central line on the grating Th e angle 9
between the a is of the telescope and a normal to the grating can
be read on a g aduated c i rcle
If the source S emits light of wave length A and the angle 79
is made equal to one of the roots of the equation A d sin
an
image of the slit L is formed at I on the axis of the telescope
Fo each point on the slit gives a parallel beam and each of these
,
.
an
n
on
a
n
o
.
o n
.
x
r
.
77
.
r
Fi g 6 7
.
.
beams produces a bright point at the focus of the telescope when
A d sin
Th e image of the slit is observed through the eye
piece E Th e light o f wave leng th 7L gives a series of images of
the slit If
we get
and there is an image in this
position with light of any wave length This central image is
formed b y rays which have passed straight through the grating
without deviatio
If
1 we get A d sin 9 so that 9 is equal
77
.
.
.
n
77
.
1
1
7k
to i sin d There are therefore two images of the slit
each side of the ce tral image making angles of 79 and
“
1
n
it
.
If
77
2
we get
1
,
sin
92
—I
QX
—
d
,
on e
01
on
with
and there are two more images
one on each side of the central image and nearly t wice as far from
it as the first pair
In the same way there i s a pair of images correspond i ng to each
of the other values of 3 4 5 etc Th e wave length of the light
.
77 :
,
,
,
.
L I GH T
3 90
P
v
A
R
T
I
[
be determined by measuring the values of 9 and calculati g
the value of 7x by means of the equation
x
d sin 9
F example with the light from a B unsen flame containing vapour
of a thallium salt d a grati g for which 1
cm
green
images
00
of the slit are bserved with the following values of 9
Ca n
n
77
.
or
an
— 1
5 0
7
n
.
O
°
7\ i n m
If
77 g
(1
77 A
on
there is no possible value of
4
sin
illi th
fin
cannot be satisfied unless
n
—
X
d
s o f 3.
9
,
mm
.
for the equation
is less than unity
Th e
.
values of 9 in the above table give the values of 7 shown in the
third column
If the slit of the collimator is illuminated with white light then
we get a white central image d a series of spectra on each side
of it ; each of the spectra is like th e s pe c t u m f white light
produced by a prism Th e violet ends are nearer the ce tral image
because the wave length of violet light is shorter than that of red
light
A simple way of Observing the spectrum of a small source of
light is to look at it through a grating held close to the eye without
using a collimator or telescope We then see the source with
a series of images of it on each side similar to the im ages of the
slit seen in the telescope
7
»
.
,
an
’
r
O
n
.
.
.
.
L I GH T
3 92
P
R
T
I
v
A
[
angles of 1 01 55 each Th e remaining six corners at B C D E F
’
and H are contained by one obt se angle of 1 01 5 5 and two acute
angles of 7 8 5 each If the S par has been sp lit so that all the
sides of the rhombohedron are of equal length then the line A G
j oining the two obtuse corners is equally incli ed to the three faces
that meet at A d to the three that meet at G
e of the faces of the crystal
S ppose a ray of light RP falls on
at P in Fi g 6 9 If RP is perpendicular to the surface of the
'
°
,
.
,
,
0
u
°
,
'
.
,
n
an
.
on
u
.
.
Fi g 69
.
.
O e the
crystal we get two refracted rays P N C and
ordina y y goes through the crystal without deviatio ; but the
other P MK is deviated at P and M and emerges parallel to RP
Th e ray P M K is called the extraordinary ray
Th e two rays are
of equal intensity If the incident ray RP is o t perpendicular to
the surface of the crystal the ordinary ray is ref acted accordi g
to the ordinary laws and the refractive index of the spar f it is
Th e extraordi ary ray is refracted according to quite
about
di ffere t laws which need t be considered here
If the rhombohedron of Iceland spar is laid over a black dot on
n
r
ra
,
n
,
.
.
n
.
n
r
or
,
n
n
n o
.
POL ARI Z ATI ON
]
OH x
.
EF RACTI ON
3 93
AN D D OU B LE R
a sheet of white paper then o lookin g at the dot through the spar
two images of it are seen If the spar is turned round one
of these images remains fixed and the other moves round with the
spar
Huygens discovered that both the ordi nary and extraordinary
rays after emerging from the crystal possess remarkable pro
T
h
different
from
those
of
ordinary
light
two
rays
e
e ti e
p
are rather near together so that it is convenient to get rid of
one of them in order that the properties of the other may be
examined separately
A device which enables the ordinary ray to be stopped was
in vented by N icol in 1 8 28 a d is called Nicol s prism A crystal
of Iceland spar is split so that a rhombohedron at least three times
This crystal
a s long as it is wid e is obtained as shown in Fi g 7 0]
n
,
.
,
,
.
s
r
.
,
.
’
n
.
.
,
Fi g 7 0
.
.
is sawn in two at the plane AJ GK and the two surfaces are
polished and con ected together agai n with Canada balsam If
now a y of light enters the end AB CD parallel to the length of
the prism it is split up into ordinary and extraord i nary rays which
fall
the surface of the thin layer of Canada balsam Th
refractive index of the balsam is greater than that of the spar f
the ordinary ray and this ray is totally reflected from the balsam
surface but the extraordi nary ray passes through the balsam and
emerges from the prism through the face EFH G Th e long sides
of the prism are blackened so that the ordi nary ray is absorbed by
them
Fi g 7 1 shows the paths of the two rays RP C and RP X through
the prism Th e ordi ary ray B BC is absorbed at C after total
fl
i o at the plane GK A J
,
n
.
ra
,
on
.
e
or
,
.
.
.
.
re
ex
n
n
.
LI
3 94
GH T
A
R
T
I
v
P
[
shows an arrangement that may be used to exami e
the properties of the extraordinary ray from a N icol s p ism S is
a so rce like arc lamp the rays from which are focused on to
a small circular hole i a scree P Th e rays passing through thi s
hole pass through the N icol prism N and then through a convex
lens L which forms image of the hole i P o a white screen M
Fi g 7 2
n
.
’
r
.
an
u
n
n
.
,
n
n
an
Fi g 7 1
.
.
.
Fi g 7 2
.
.
If a rhombohedron of Iceland spar is put in the path of the rays
between L d M we get in general two images at M instead of
one Th e extraordinary ray from the N icol gives rise to an
ordinary and an extraordinary ray in the crystal Le t the crystal
of Icela d spar be turned so that its sides are parallel to the
corresponding sides of the N icol prism a d the let it be rotated
about an axis parallel to the rays of light between L and M through
an angle d
Th e following table gives the intensity of illumination or
brightness of the images on the screen for di fferent values of the
angle 1
di
E t
m
I
g
y
an
.
.
n
n
)
n
.
()
x raor
n ar
1
t
0
4
;
I
t
0
t
l
a
e
L I GH T
3 96
AR
T
P
Iv
[
When u polarized light enters a crystal like Iceland spar the
disturbances in the light are resolved parallel to tw perpendicular
directions d those parallel to one direction are refracted in
a di fferent way to those parallel to the other Thus the unpolarized
light is separated i to two plane polarized beams polarized in
perpendicular planes
Th e action of two N icol prisms on a ray of light may be clearly
illustrated by passing a stretched string through two nar ow slits
in which it can j ust move freely S uch an arra gement is shown
i Fi g 7 3
If the string at A is shaken about in all directions
n
o
an
.
n
.
r
n
.
n
.
.
Fi g 7 3
.
.
transverse to its length the transverse waves produced will be
stopped by the slit S except those in which the displaceme t is
parallel to the slit Th stri g between S and S will therefore
vibrate parallel to S If S is parallel to S the waves bet ween
S and S will get through to B but if S is at right angles to S they
will all be stopped In the same way when the two Nicol prisms
are parallel the pla e polarized light from the first gets through and
when they are crossed it is stopped entirely
Th fact that the vibratio s in light are tr nsverse that is to
say the directio of the disturbance lies in a plane perpendicular
to the ray was first discovered by Fresnel and You g
,
n
'
n
e
.
'
.
’
,
.
n
,
,
.
e
n
a
,
n
,
n
,
F
N
RE ERE CE
T/7e Th eo ry
f
o
Op ti c s , Dr u de
.
.
C HAP T ER
N
E ERG
Y
OF L I G HT
.
WH EN light falls on
I
XI
N VI S I BL E
RADI ATI O
NS
surface of a black body it is absorbed
and completely disappears It is f und that the black body gets
hotter so that the light has bee converted i to heat When light
is emitted by a very hot body like the filame t of an incandescent
electric lamp the hot body loses heat In this case heat is con v erted
i to light We have seen that heat is a form of energy so that the
s me is true of light
A trai of sound waves in air possesses energy
When a tuning
fork is vibrating it alternately compresses d
fie s the air and
sets it i moti n Th e fork does work on the air and so gives it
e ergy This ene gy travels through the i i the sound waves
In the s me way at the surface of a very hot body the vibrating
atoms produce disturbances in the ether which travel away as light
waves Th vibrating atoms do work on the ether and give it
energy which takes the f rm of light
If the light from a powerful source like arc lamp is focused
by a large con v ex lens
to a small vessel of water s me of the
light is absorbed d the water soon begi s to boil
T study the amount of energy in light by measuring the heat
produced when it is abs rbed by a black body it is con v enient
to have some mea s of measuring very small changes of tempera
ture because the heati g e ffect is small unless the light is very
i te se
A convenient i trument for this purp se is the thermopile of
which a modern f rm is show i F i g 7 4 S S S S S S etc
f silver each about 1 cm lo g
B B B B etc
e bar
bars
P P P etc
e small square plates of blacke ed
f bi muth
silver which are each s ldered on to the middles of a silver d
a bismuth bar
th e
o
.
n
n
,
.
n
.
,
n
.
,
a
.
n
.
an
n
n
o
.
a r
r
.
rare
n
.
a
e
.
o
.
an
on
o
an
n
.
o
o
n
n
,
n
n
.
o
n s
n
o
s o
a r
o
s
n
.
2,
.
3
o
.
,
.
ar
.
n
.
,
1
.
1
]
,
1
2
2
,
”
2,
3
.
3
.
,
are
n
an
L I GH T
3 98
P
A
R
T
I
v
[
bismuth bar at the lower edge of each plate is soldered at
its e ds to the ends of the silver bar at the upper edge of the next
plate Th top silver bar S S is co ected by copper wires
S C and S C to a wire leadi g to a galvanometer G and the bottom
bismuth bar is con ected i a similar way to the other wire
leading to the galvanometer Th e thermopile is contained in
a box i one of the walls of which there is a slit which is pposite
to the row of blacke ed silver plates so that light entering the slit
falls on these plates but t on the other parts of the pile Th e
Th e
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Fi g 74
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light heats the plates and this causes an electric current to fl w
round the circuit contai i g the pile and galva ometer This
curre t can be measured by the galva ometer With a se sitive
galva ometer a very small amount of energy given to the pile
produces a measurable c rre t Th e current is approximately
proportional to the energy received by the pile per second If the
light from a candle or an electric lamp is allowed to fall on such
a thermopile it is found that the cu rent p oduced varies inversely
as the square of the dista ce between the pile and the lamp This
shows that the current is proportio al to the energy received by
the pile in unit time for we know that the light waves from a small
o
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the spectrum
d the distribution
hanges As the temperature of a body rises it first appea s red
the yell wish d finally white or e v e bluish
In Fi g 7 5 the distributio of the energy i the spectrum Of
the light emitted by a black body at several temperatures is show
graphically Th wave lengths are plotted horizo tally and
relative energies vertically
,
C
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i thousandths of a millimetre and
wave lengths
e ergy is in arbit ary units Th limits f the visible spectrum
are marked A and B on the scale of wave l e gths Th e upper of
the three curves shows the distribution of the e ergy in the
spectrum of the light from a black body at a temperature f 1 1 04 C
At this temperature a body is bright red hot but ot white hot
Th greatest am u t of energy in this case is around the wave
le gth 00 021 mm At 8 1 4 C the maximum is near 00 026 mm
Th e maximu m e ergy occurs at a
d at 56 4 C ear 00 03 5 mm
Th e
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CH
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XI
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L I G HT I N V I S I BLE
OF
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D I ATI ONS
401
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wa v e len g th 00 006 mm which is that of yellow light when the
temperat re is about 45 00 C
Th e so u rces of artificial white light which are now available all
ha v e temperatures very much below 4500 C so that they emit
far more energy in th form of invisible in fra red light than in the
f rm of visible light There is much more energy beyond the red
in the spectrum of the light from an arc lamp than there is in the
v i ible part of the spectrum
Th e energy f the ultra violet light from an a e light is com
paratively small I t can nly be detected with a very sensitive
thermopile d galvanometer There are however other methods
of detecting ultra violet light
It is fo u nd that certain substa ces when exposed to ultra violet
light emi t visible light This phenomenon is called fluorescence F
example barium platinocya i de which is a pale yellow substance
when exposed to ultra violet light becomes luminous emitting
gree light If a piece of paper coated with a thin layer of barium
platinocyanide is put up beyond the violet end f the spectrum of
an
lamp it emits green light thus showin g the presence of the
ultra violet light U ltra violet light is strongly absorbed by fli nt
glass so t h at if a flint glass lens or prism is used to produce the
spectrum very little ultra violet light will be f u d in it Q uartz
is more t ra sparent to ultra violet light than glass so that to
ob t i a spectrum rich in ultra v iolet light it is best to use lenses
and prisms made of quartz
.
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F ll f Em m
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ii + 94
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