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ƌŽŶŽůŽŐşĂƚĞĐŶŽůſŐŝĐĂ ϭϲϱϬ WƌŽƚĂŐŽŶŝƐƚĂƐ /ŶƐƚƌƵŵĞŶƚŽƐŵĞĐĄŶŝĐŽƐ͕ ŚŝĚƌĄƵůŝĐŽƐ͕ĂŐĂƐ͘͘͘ ƐŝŐŶĂƚƵƌĂƐ ƌƋşŵĞĚĞƐ͕<ĞƉůĞƌ͕'ĂůŝůĞŽ͕,ŽŽŬĞ͕ EĞǁƚŽŶ͕ƵůĞƌ͕,ƵLJŐĞŶƐ͕Ě͛ůĞŵďĞƌƚ DĞĐĄŶŝĐĂ ƉƌŝŵĞƌŽƐƐŝƐƚĞŵĂƐĚĞǀĂƉŽƌ ϭϳϱϬ ϭϳϴϬͲϭϴϯϬ͗ϭǐƌĞǀŽůƵĐŝſŶ ŝŶĚƵƐƚƌŝĂů ;ŵĄƋƵŝŶĂƐĚĞǀĂƉŽƌͿ ϭϴϱϬ ϭϵϱϬ ĂƌŶŽƚ͕:ŽƵůĞ͕<ĞůǀŝŶ͕ůĂƵƐŝƵƐ͕ ŽůƚnjŵĂŶŶ͕,ĞůŵŚŽůƚnj͕'ŝďďƐ dĞƌŵŽĚŝŶĄŵŝĐĂ >ĂǀŽŝƐŝĞƌ͕'ĂLJͲ>ƵƐƐĂĐ͕ǀŽŐĂĚƌŽ͕ĂůƚŽŶ͕ ƌƌŚĞŶŝƵƐ͕DĞŶĚĞůĞĞǀ͕<ĞŬƵůĠ͕>ĞǁŝƐ YƵşŵŝĐĂ &ƌĂŶŬůŝŶ sŽůƚĂ KŚŵ <ŝƌĐŚŽĨĨ &ƌĂŶŬůŝŶ͕sŽůƚĂ͕KŚŵ͕<ŝƌĐŚŽĨĨ͕ ůĞĐƚƌŽŵĂŐŶĞƚŝƐŵŽ ů ƚ ƚŝ ϭϴϳϬͲϭϵϬϬ͗ϮǐƌĞǀŽůƵĐŝſŶ ϭϴϳϬ ϭϵϬϬ Ϯǐ ů ŝſ ŵƉğƌĞ͕&ĂƌĂĚĂLJ͕DĂdžǁĞůů͕,Ğƌƚnj ŝŶĚƵƐƚƌŝĂů ;ĞůĞĐƚƌŝĐŝĚĂĚн ƋƵşŵŝĐĂŝŶĚƵƐƚƌŝĂů;ƉĞƚƌſůĞŽ͕ĨĄƌŵĂĐŽƐ͘͘͘ͿͿ DĂƚĞƌŝĂůĞƐ ϭϵϲϬͲϭϵϵϬ͗ϯǐƌĞǀŽůƵĐŝſŶ ŝŶĚƵƐƚƌŝĂů ;ĞůĞĐƚƌſŶŝĐĂ;ƐŝůŝĐŝŽͿ͕ ŝŶĨŽƌŵĄƚŝĐĂ͕ŽƉƚŽĞůĞĐƚƌſŶŝĐĂͿ MƉƚŝĐĂĂƉůŝĐĂĚĂ ,ŽLJLJŵĂŹĂŶĂ͗ŐĞŶĠƚŝĐĂ͕ŶĂŶŽƚĞĐŶŽůŽŐşĂ͕ĞŶĞƌŐşĂƐůŝŵƉŝĂƐ dŝĞŵƉŽ ;ĂŹŽĚ͘͘Ϳ ƉůŝĐĂĐŝŽŶĞƐĚĞůĂŵĞĐĄŶŝĐĂ͗ŵĄƋƵŝŶĂƐƐŝŵƉůĞƐ;ĂŵƉůŝĨŝĐĂŶůĂĨƵĞƌnjĂŚƵŵĂŶĂͿ͗ ƉůĂŶŽŝŶĐůŝŶĂĚŽ͕ŵĂƌƚŝůůŽ͕ƉŽůĞĂ͕ƉŽůŝƉĂƐƚŽ͕ƉĂůĂŶĐĂ;ŶŽƌŵĂůĞŚŝĚƌĄƵůŝĐĂͿ͕ƚŽƌŶŽ LJĂĚĞŵĄƐ͗ƌƵĞĚĂƐ͕ƐƵƐƉĞŶƐŝŽŶĞƐ͕ĂŵŽƌƚŝŐƵĂĚŽƌĞƐ͕ƌĞůŽũĞƐĚĞƉĠŶĚƵůŽ͙ DĞĐăŶŝĐĂ&ŽŶĂŵĞŶƚĂů ŚƚƚƉ͗ͬͬĂƚĞŶĞĂ͘ƵƉĐ͘ĞĚƵ ƌŽďĞƌƚŽ͘ŵĂĐŽǀĞnjΛƵƉĐ͘ĞĚƵ ;ĚĞƐƉĂĐŚŽϭϭ͘ϰϱ͕ƉůĂŶƚĂϭϭͿ ŚƚƚƉ͗ͬͬŐĐŵ͘ƵƉĐ͘ĞĚƵͬŵĞŵďĞƌƐͬƌŽďĞƌƚŽͲŵĂĐŽǀĞnj DĞĐĄŶŝĐĂ͗ ^dh/K> DKs/D/EdK z ^h^ h^^ z^h^h^^ ^ŝƐƚĞŵĂƐƌşŐŝĚŽƐ ;ĐƵĞƌƉŽƐƐſůŝĚŽƐͿ ^ŝƐƚĞŵĂƐďůĂŶĚŽƐ ŽĞůĄƐƚŝĐŽƐ;ůşƋƵŝĚŽƐ͕ Ͳ ŝŶĞŵĄƚŝĐĂ Ͳ ƐƚĄƚŝĐĂLJŝŶĄŵŝĐĂ Ͳ &ůƵŝĚŽĞƐƚĄƚŝĐĂ Ͳ KƐĐŝůĂĐŝŽŶĞƐLJKŶĚĂƐ & dp& F= dt ŐĂƐĞƐ͕ŵƵĞůůĞƐͿ NOTA = 0.6 EXfinal + 0.1MQ + 0.1EvC1 + 0.1EvC2 + 0.1LAB ϰƉƌƵĞďĂƐĞƐĐƌŝƚĂƐ ϯƐĞƐŝŽŶĞƐ ϭǐƐĞƐŝſŶĚĞ>͗ŵŝƚĂĚĚĞƐĞƉƚŝĞŵďƌĞ͊>͗ƉůĂŶƚĂϲ ŶƚƌĞŐĂƌƉŽƌƉĂƌĞũĂƐĞůƉƌŽďůĞŵĂϭ͘ϱ͘ϭ ;ĞƐĐƌŝƚŽĂŵĂŶŽͿ >ĞĞƌƉĄƌƌĂĨŽƐϭ͘ϯLJϭ͘ϱĚĞůĂƐŶŽƚĂƐĚĞĐůĂƐĞ DKs/D/EdKĚĞƵŶĐƵĞƌƉŽсdZE^>/MEнZKd/MEн&KZD/ME Αϭ͘ϳŝŶĞŵăƚŝĐĂĚĞůĂƉĂƌƚşĐƵůĂ hŶĂƉĂƌƚşĐƵůĂ;Ž͞ƉƵŶƚŽŵĂƚĞƌŝĂů͟ͿĞƐƵŶŽďũĞƚŽĚĞĚŝŵĞŶƐŝŽŶĞƐĚĞƐƉƌĞĐŝĂďůĞƐƌĞƐƉĞĐƚŽĂůĂƐ ĚŝŵĞŶƐŝŽŶĞƐĚĞƐƵƚƌĂLJĞĐƚŽƌŝĂ;ĚŝƐƚĂŶĐŝĂƐLJƌĂĚŝŽĚĞĐƵƌǀĂƚƵƌĂͿ͕LJƋƵĞŶŽŐŝƌĂƐŽďƌĞƐŝŵŝƐŵĂ͘ ůǀĞĐƚŽƌƉŽƐŝĐŝſŶ ĚĞƵŶĂƉĂƌƚşĐƵůĂ;Ž͞ƉƵŶƚŽŵĂƚĞƌŝĂů͟ͿĞƐĞůǀĞĐƚŽƌ͗ k̂ & & 2 2 2 ˆ ˆ ˆ r = ( x, y , z ) = xi + yj + zk , de módulo r = r = x + y + z & r = ( x, y, z) ŶĞůĞƐƉĂĐŝŽ͕ůĂƉĂƌƚşĐƵůĂƚŝĞŶĞƚƌĞƐ ŐƌĂĚŽƐĚĞůŝďĞƌƚĂĚ͕LJĂƋƵĞƐĞ ĵ ŶĞĐĞƐŝƚĂŶϯŶƷŵĞƌŽƐƉĂƌĂĞƐƉĞĐŝĨŝĐĂƌƐƵƉŽƐŝĐŝſŶ;ĞŶĞůƉůĂŶŽ͕ƐſůŽϮͿ iˆ >ĂƚƌĂLJĞĐƚŽƌŝĂƚĞŵƉŽƌĂůĚĞůĂƉĂƌƚşĐƵůĂĞƐůĂĐƵƌǀĂ & r (t ) = ( x (t ), y (t ), z (t ) ) ĚĞƐĐƌŝƚĂƉŽƌůĂƉĂƌƚşĐƵůĂ͘ůĚĞƐƉůĂnjĂŵŝĞŶƚŽ ĞŶƚƌĞĚŽƐ & & & ƉƵŶƚŽƐĚĞůĂƚƌĂLJĞĐƚŽƌŝĂĞƐĞůǀĞĐƚŽƌ Δr = r2 − r1 = ( x2 − x1 , y2 − y1 , z2 − z1 ) & ^ŝůŽƐĚŽƐƉƵŶƚŽƐĞƐƚĄŶŝŶĨŝŶŝƚĂŵĞŶƚĞƉƌſdžŝŵŽƐ͗ dr = ( dx, dy , dz ) & & & & Δr d r & § dx dy dz · ůǀĞĐƚŽƌǀĞůŽĐŝĚĂĚŝŶƐƚĂŶƚĄŶĞĂĚĞůĂƉĂƌƚşĐƵůĂ v (t ) = =r =¨ , , v = lim Δ t → 0 ¸ Δt dt ĞƐůĂĚĞƌŝǀĂĚĂǀĞĐƚŽƌŝĂůĚĞůĂƉŽƐŝĐŝſŶ͗ © dt dt dt ¹ & v (t ) ůǀĞĐƚŽƌǀĞůŽĐŝĚĂĚĞƐƚĂŶŐĞŶƚĞĂůĂƚƌĂLJĞĐƚŽƌŝĂĞŶĐĂĚĂƉƵŶƚŽ & ;ƐĞǀĞŐƌĄĨŝĐĂŵĞŶƚĞƚŽŵĂŶĚŽĞůůşŵŝƚĞͿ r (t ) & & & & dv d 2 r & § d 2 x d 2 y d 2 z · = =r =¨ 2 , 2 , 2 ¸ >ĂĂĐĞůĞƌĂĐŝſŶĚĞůĂƉĂƌƚşĐƵůĂĞƐ͗ a = v = dt dt 2 © dt dt dt ¹ ƐƚĂƐĚĞĨŝŶŝĐŝŽŶĞƐƐĞƉƵĞĚĞŶŝŶǀĞƌƚŝƌƉĂƌĂƐĂĐĂƌ;ƉŽƌĞũĞŵƉůŽͿůĂƉŽƐŝĐŝſŶĚĞůĂǀĞůŽĐŝĚĂĚ͗ & & & r (t ) = ³ v (t )dt + C = ³ vx (t )dt + C x , ³ v y (t )dt + C y , ³ v z (t )dt + C z & & WͲϭ͘ϯ͘ϭ;ƐŝŶƉƵŶƚŽ;ĚͿͿ͕WͲϭ͘ϳ͘Ϯ a (t ) = (0,0,− g ) v (t = 0) = (2,0,1) WͲϭ͘ϳ͘ϯнůŽŵŝƐŵŽĐŽŶLJ ( ) & v (t ) = v(t )vˆ(t ) = vvˆ ¯vˆ = vector tangente unitario ­ & ůǀĞĐƚŽƌǀĞůŽĐŝĚĂĚĞƐƐŝĞŵƉƌĞƚĂŶŐĞŶƚĞĂůĂƚƌĂLJĞĐƚŽƌŝĂ v (t ) ® & & dv d (vˆ v ) dv dvˆ ˆ = = vˆ + v >ĂĂĐĞůĞƌĂĐŝſŶĞƐ a (t ) = ͘zĂƋƵĞ dv ⊥ vˆ dt dt dt dt dt aT vˆ & a 2 d (vˆ ⋅ vˆ) dvˆ d ( vˆ ) n̂ ;ĞƐƚŽƐŝŐƵĞĚĞͿ͕ůůĂŵĂŶĚŽĞůǀĞĐƚŽƌ = 2vˆ ⋅ = =0 dt dt dt & & & dvˆ dv a (t ) = vˆ + v nˆ = aT + a N ĚĞŵſĚƵůŽϭŽƌƚŽŐŽŶĂůĂ v̂ dt dt & & r (t ) aN nˆ a ƚŝĞŶĞƵŶĂĐŽŵƉŽŶĞŶƚĞ ǀĂƌŝĂĐŝſŶĚĞů ǀĂƌŝĂĐŝſŶĚĞ & & ƚĂŶŐĞŶƚĞĂůĂƚƌĂLJĞĐƚŽƌŝĂLJŽƚƌĂŶŽƌŵĂů ŵſĚƵůŽĚĞ v ĚŝƌĞĐĐŝſŶĚĞ v vˆ(t ) ĐĞůĞƌĂĐŝſŶŶŽƌŵĂůLJƌĂĚŝŽĚĞĐƵƌǀĂƚƵƌĂ͗ & & dθ WĂƌĂƵŶĚĞƐƉůĂnjĂŵŝĞŶƚŽŝŶĨŝŶŝƚĠƐŝŵŽ͕ůĂĐƵƌǀĂƐĞĐŽŶĨƵŶĚĞ dr & r (t ) d r ĐŽŶĞůƐĞŐŵĞŶƚŽƌĞĐƚŝůşŶĞŽLJĐŽŶĞůĂƌĐŽĚĞĐşƌĐƵůŽƚĂŶŐĞŶƚĞ͗ vˆ(t + dt ) & d θ dr = d" = RCURV dθ ͕ĚŽŶĚĞĞƐĞů͞ƌĂĚŝŽĚĞĐƵƌǀĂƚƵƌĂ͟ RCURV & RCURV dr dvˆ vˆ = 1 ^ĞǀĞŐƌĄĨŝĐĂŵĞŶƚĞƋƵĞ͘ĂĚŽƋƵĞ͕ = dθ = RCURV & vˆ dvˆ dθ 1 dr v dvˆ & v2 = = = RCURV = v ͘WŽƌůŽƚĂŶƚŽLJ a N = nˆ dt dt RCURV dt RCURV dt RC & & & & & v v a = aT + a N ZĞƐƵŵĞŶĨſƌŵƵůĂƐ͗ vˆ(t ) = = & v v ;ƚŽĚĂƐƐĞĐĂůĐƵůĂŶ & dvˆ v2 & dv dvˆ dvˆ & ˆ ˆ = = a n v n ˆ a v = N T ˆ n(t ) = v (t ) ĂƉĂƌƚŝƌĚĞͿ dt RC dt dt dt dvˆ dvˆ = vˆ dθ = dθ RC = v dvˆ dt ĂƐŽƐŝŵƉŽƌƚĂŶƚĞƐ͗ŵŽǀŝŵŝĞŶƚŽƌĞĐƚŝůşŶĞŽƵŶŝĨŽƌŵĞ͕ƵŶŝĨŽƌŵĞŵĞŶƚĞĂĐĞůĞƌĂĚŽ ŽƉĞƌŝſĚŝĐŽ͕ ŵŽǀŝŵŝĞŶƚŽĐŝƌĐƵůĂƌƵŶŝĨŽƌŵĞ WͲϭ͘ϳ͘ϰ;ƐſůŽĐŝƌĐƵůĂƌƵŶŝĨŽƌŵĞͿ͕WͲϭ͘ϳ͘ϱ dĞŵĂϮ͗DĞĐĄŶŝĐĂĚĞƵŶĂƉĂƌƚşĐƵůĂ >ĂƐůĞLJĞƐĚĞĞƐƚĞƚĞŵĂƐŽŶǀĄůŝĚĂƐƉĂƌĂƉĂƌƚşĐƵůĂƐ͘DĄƐ ĂĚĞůĂŶƚĞ;§ϯ͘ϮͿǀĞƌĞŵŽƐƋƵĞƚĂŵďŝĠŶǀĂůĞŶƉĂƌĂĚĞƐĐƌŝďŝƌĞůŵŽǀŝŵŝĞŶƚŽĚĞƚƌĂŶƐůĂĐŝſŶ ĚĞƵŶ ĐƵĞƌƉŽƌşŐŝĚŽƐŽŵĞƚŝĚŽĂĨƵĞƌnjĂƐĞdžƚĞƌŝŽƌĞƐ͘ŶĐŽŶĐƌĞƚŽ͕ůĂĂĐĞůĞƌĂĐŝſŶĚĞůĐĞŶƚƌŽĚĞŵĂƐĂƐĚĞů ĐƵĞƌƉŽĞƐĚĂĚĂƉŽƌůĂϮǐ>ĞLJĚĞEĞǁƚŽŶ;ƚŽŵĂŶĚŽƐſůŽůĂƐĨƵĞƌnjĂƐĞdžƚĞƌŝŽƌĞƐͿ͘ ΑϮ͘ϭLJϮ͘Ϯ>ĞLJĞƐĚĞEĞǁƚŽŶ͕ĨƵĞƌnjĂ͕ŵĂƐĂ͕ŵŽŵĞŶƚŽůŝŶĞĂů ϭǐ>ĞLJĚĞEĞǁƚŽŶ͗ ƵŶĐƵĞƌƉŽƐŽďƌĞƋƵĞŶŽĂĐƚƷĂŶŝŶŐƵŶĂ͞ĐĂƵƐĂ͟;ĨƵĞƌnjĂͿ͕ƐĞŵƵĞǀĞĐŽŶ & ǀĞůŽĐŝĚĂĚĐŽŶƐƚĂŶƚĞ;ŵŽǀŝŵŝĞŶƚŽƌĞĐƚŝůşŶĞŽƵŶŝĨŽƌŵĞͿ;^dd/Æ F = 0 Ϳ͘ Ϯǐ>ĞLJĚĞEĞǁƚŽŶ͗ůĂĨƵĞƌnjĂƋƵĞĂĐƚƷĂƐŽďƌĞƵŶĂƉĂƌƚşĐƵůĂĞƐƉƌŽƉŽƌĐŝŽŶĂůĂůĂ & ĂĐĞůĞƌĂĐŝſŶĚĞůĂŵŝƐŵĂ͖ůĂĐŽŶƐƚĂŶƚĞĚĞƉƌŽƉŽƌĐŝŽŶĂůŝĚĂĚĞƐůĂŵĂƐĂĚĞ & &. F = ma = p ůĂƉĂƌƚşĐƵůĂ͕ŽƐĞĂůĂĐĂŶƚŝĚĂĚĚĞŵĂƚĞƌŝĂ;ÅÆƌĞůĂĐŝŽŶĂĚĂĐŽŶƐƵƉĞƐŽͿ͘ & & ůǀĞĐƚŽƌƐĞůůĂŵĂŵŽŵĞŶƚŽůŝŶĞĂůŽĐĂŶƚŝĚĂĚĚĞŵŽǀŝŵŝĞŶƚŽ p = mv >ĂϮǐ>ĞLJŝŵƉůŝĐĂůĂϭǐ͘ŵďĂƐƐŽŶǀĄůŝĚĂƐƐſůŽĞŶƐŝƐƚĞŵĂƐĚĞƌĞĨĞƌĞŶĐŝĂŝŶĞƌĐŝĂůĞƐ & ;ĞƐĚĞĐŝƌ͕ĨŝũŽƐŽĞŶŵŽǀŝŵŝĞŶƚŽƌĞĐƚŝůşŶĞŽƵŶŝĨŽƌŵĞͿ & F1 F ŽŵŽůĂƐĂĐĞůĞƌĂĐŝŽŶĞƐ͕ƋƵĞƐŽŶǀĞĐƚŽƌĞƐ͕ ůĂƐĨƵĞƌnjĂƐƚĂŵďŝĠŶƐĞƐƵŵĂŶĐŽŵŽǀĞĐƚŽƌĞƐ͗ & m >ĂƵŶŝĚĂĚĚĞĨƵĞƌnjĂĞŶĞů^/ĞƐĞůŶĞǁƚŽŶ;EͿ͗ 1N = 1kg 2 F2 s ĞĨŝŶĞůĂĨƵĞƌnjĂ͗ ůĂĨƵĞƌnjĂĞƐůĂĐĂƵƐĂĚĞůĂĂĐĞůĞƌĂĐŝſŶĚĞůĂƉĂƌƚşĐƵůĂ͖ƉƵĞĚĞƐĞƌ & & ĨƵŶĐŝſŶĚĞůĂƉŽƐŝĐŝſŶLJĚĞůĂǀĞůŽĐŝĚĂĚ;LJĚĞůƚŝĞŵƉŽͿ͕ƉĞƌŽŶŽ ĚĞůĂĂĐĞůĞƌĂĐŝſŶ͘ & & F = ma ĞĨŝŶĞůĂŵĂƐĂ͗ ƐŝƐŽďƌĞůĂŵŝƐŵĂƉĂƌƚşĐƵůĂĂĐƚƷĂŶĨƵĞƌnjĂƐĚŝĨĞƌĞŶƚĞƐ͕ƋƵĞ F 1 , F2 ... & & ĐĂƵƐĂŶĂĐĐĞůĞƌĂĐŝŽŶĞƐ͕ĞůĐŽĐŝĞŶƚĞĚĞůŽƐŵſĚƵůŽƐĞƐĐŽŶƐƚĂŶƚĞ͕ŝŐƵĂůĂůĂ a1 , a2 ... F1 F2 ŵĂƐĂ;ŝŶĞƌĐŝĂůͿ ĚĞůĂƉĂƌƚşĐƵůĂ͗ m= a1 = a2 = ... & & & t2 & F ^ĞĚĞĨŝŶĞĞůŝŵƉƵůƐŽƐƵŵŝŶŝƐƚƌĂĚŽƉŽƌƵŶĂĨƵĞƌnjĂĞŶĞůŝŶƚĞƌǀĂůŽĐŽŵŽ I t1 t2 I = Fdt & & t1 ĞůĂϮǐ>ĞLJĚĞEĞǁƚŽŶƐŝŐƵĞŝŶŵĞĚŝĂƚĂŵĞŶƚĞĞů͗ dĞŽƌĞŵĂĚĞůŵŽŵĞŶƚŽůŝŶĞĂů͗ I =Δp ³ ΑϮ͘ϱƉůŝĐĂĐŝſŶĚŝƌĞĐƚĂĚĞůĂϮǐ>ĞLJĚĞEĞǁƚŽŶ WͲϮ͘ϭ͘Ϯ͕WͲϮ͘ϭ͘ϯ͕WͲϮ͘Ϯ͘ϭ͕WͲϮ͘ϱ͘ϱ WͲϮ͘Ϯ͘Ϯ͕ WͲϮ͘Ϯ͘ϯ͕WͲϰ͘ϭ͘ϭ >ĂϮĂůĞLJĚĞEĞǁƚŽŶŶŽƐĚŝĐĞĐŽŵŽƐĞŵƵĞǀĞůĂƉĂƌƚşĐƵůĂ͗ ĞƐƵŶĂĞĐƵĂĐŝſŶĚŝĨĞƌĞŶĐŝĂůƋƵĞĞŶ & ĂůŐƵŶŽƐĐĂƐŽƐƉƵĞĚĞƐĞƌŝŶƚĞŐƌĂĚĂ;ĚŽƐǀĞĐĞƐͿƉĂƌĂĞŶĐŽŶƚƌĂƌůĂƚƌĂLJĞĐƚŽƌŝĂƚĞŵƉŽƌĂů͘ r (t ) & & C1 C2 ŶĐĂĚĂŝŶƚĞŐƌĂĐŝſŶƐĞŝŶƚƌŽĚƵĐĞƵŶĂĐŽŶƐƚĂŶƚĞ;ǀĞĐƚŽƌŝĂůͿ͗LJ͘^ŝĐŽŶŽĐĞŵŽƐƉŽƐŝĐŝſŶLJ & & & & & & & & r0 = r (t0 ) v0 = v (t0 ) ǀĞůŽĐŝĚĂĚĞŶƵŶŝŶƐƚĂŶƚĞŝŶŝĐŝĂůƚĞŶĚƌĞŵŽƐ͗ ­ r0 = f ( C 1 , C 2 , t 0 ) t0 & & & & ° & ®& ∂f & & (C1 , C 2 , t0 ) ° v0 = ∂t ¯ & & & & r0 , v0 ^ŝĐŽŶŽĐĞŵŽƐůĂĨƵŶĐŝſŶĨ͕ƐĞĐĂůĐƵůĂŶĂƉĂƌƚŝƌĚĞ͕ƉĂƌĂĞŶĐŽŶƚƌĂƌ͗ ů Ĩ ſ Ĩ C1 , C2 ů ů& r (t ) = r (r0 , v0 , t ) & ũ͗͘ĨƵĞƌnjĂƋƵĞƐſůŽĚĞƉĞŶĚĞĚĞůƚŝĞŵƉŽ F = F (t ) & § · d 2r & 1 ¨ & & ¸ dt m F ( t ) r ( t ) F ( t ) dt = = Æ /ŶƚĞŐƌĂĐŝſŶĚŝƌĞĐƚĂÆ t t ¸ dt 2 m ¨ & © ¹ & & & F m a = F a = = ct ĂƐŽƉĂƌƚŝĐƵůĂƌ͗ĨƵĞƌnjĂĐŽŶƐƚĂŶƚĞ͘ŶĞƐƚĞĐĂƐŽůĂϮĂ>ĞLJĚĞEĞǁƚŽŶĞƐ͗ & m /ŶƚĞŐƌĂŶĚŽϮǀĞĐĞƐ͗ r& (t ) = C& + C& t + 1 F t 2 1 2 2m & & & & r (t0 ) = r0 , r (t0 ) = v0 ŽŶůĂƐĐŽŶĚŝĐŝŽŶĞƐŝŶŝĐŝĂůĞƐ͕ƐĞŚĂƉƵĞƐ͗ ³³ & & 1 F 2½ ­ & & F & & & & r (t0 ) = r0 = C1 + C2 t0 + t0 ° °C2 = v0 − t0 m 2m ° ° & & ¾® & F & & & 1F 2 & & ° ° r (t0 ) = v0 = C2 + t0 C = r −v t + t °¿ °¯ 1 0 0 0 2 m 0 m >ĂĞĐƵĂĐŝſŶĚĞůĂƚƌĂLJĞĐƚŽƌŝĂĞƐƉƵĞƐ͗ & & & & 1F r (t ) = r0 + v0 (t − t0 ) + (t − t0 ) 2 2m ĂƐŽƐƉĂƌƚŝĐƵůĂƌĞƐĚĞĨƵĞƌnjĂƐĐŽŶƐƚĂŶƚĞƐ͗ ƉĞƐŽLJ ĨƌŝĐĐŝſŶĚŝŶĄŵŝĐĂ & & & & & &ƵĞƌnjĂƉĞƐŽ͗ m a = F = mg a = y F & = g = ct m m y ^ŝƚŽŵĂŵŽƐƵŶƐŝƐƚĞŵĂĚĞƌĞĨĞƌĞŶĐŝĂĐŽŶĞůĞũĞƉĂƌĂůĞůŽĂůĂĨƵĞƌnjĂLJ ƚĂůƋƵĞůĂǀĞůŽĐŝĚĂĚŝŶŝĐŝĂůĞƐƚĠĞŶĞůƉůĂŶŽ ( x, y ) ͕ůĂƚƌĂLJĞĐƚŽƌŝĂ & & & & 1F r (t ) = r0 + v0 (t − t0 ) + (t − t0 ) 2 ĞƐƵŶĂƉĂƌĄďŽůĂĞŶƚĂůƉůĂŶŽ͘ 2m & & v & F & ZŽnjĂŵŝĞŶƚŽĚŝŶĄŵŝĐŽ͗ FRD = μ D N , FRD = − μ D Nvˆ v0 & & F v̂ FRD ^ŝсĐŽŶƐƚ͕сĐŽŶƐƚ͘^ŝĂĚĞŵĄƐсĐŽŶƐƚ͕сĐŽŶƐƚ N RD x v(t ) >ĂůĞLJŚŽƌĂƌŝĂ;ĚĞĐĞůĞƌĂĐŝſŶƵŶŝĨŽƌŵĞͿǀĂůĞƐſůŽŚĂƐƚĂƋƵĞůĂƉĂƌƚşĐƵůĂ ƐĞƉĂƌĂ͕ĞƐĚĞĐŝƌŚĂƐƚĂƋƵĞǀ сϬ͕ƉŽƌƋƵĞĞŶƚŽŶĐĞƐůĂĨƌŝĐĐŝſŶĚĞƐĂƉĂƌĞĐĞ t ϭͿĂůĐƵůĂƌĂƋƵĞ ĄŶŐƵůŽƌĞƐƉĞĐƚŽĚĞůƐƵĞůŽŚĂLJƋƵĞůĂŶnjĂƌƵŶŽďũĞƚŽƉĂƌĂƋƵĞůůĞŐƵĞůŽŵĄƐůĞũŽƐƉŽƐŝďůĞ͕LJĐĂůĐƵůĂƌůĂǀĞůŽĐŝĚĂĚĚĞůŽďũĞƚŽ v0 = 3 ms −1 ũƵƐƚŽĂŶƚĞƐƋƵĞƚŽƋƵĞĞůƐƵĞůŽ͖ϮͿ,ĂůůĂƌůĂĚŝŶĄŵŝĐĂĚĞƵŶĂŵĂƐĂƐƵũĞƚĂĂĨƌŝĐĐŝſŶĚŝŶĄŵŝĐĂ;ͿƋƵĞƐĂůĞĚĞůŽƌŝŐĞŶĐŽŶ μ D = 0.1 KƚƌŽĐĂƐŽĚĞĨƵĞƌnjĂŝŶƚĞŐƌĂďůĞ͗ ĨƵĞƌnjĂĚĞƵŶŵƵĞůůĞ;ŵŽǀŝŵŝĞŶƚŽĂƌŵſŶŝĐŽͿ x<0 &ƵĞƌnjĂĚĞƵŶŵƵĞůůĞ;ŽĚĞ,ŽŽŬĞͿ͗ FH = − k (l − l N ) x = l − lN ĞĨŝŶŝĞŶĚŽƵŶŶƵĞǀŽƐŝƐƚĞŵĂĚĞƌĞĨĞƌĞŶĐŝĂĞŶƋƵĞ͕ƐĞŚĂ͗ k x = − x F = ml = mx = −kx ͕ŽƐĞĂ͕͗ĐƵLJĂƐŽůƵĐŝſŶŐĞŶĞƌĂůĞƐ͗ m m FH = 0 x(t ) = A sin(ω (t − t0 ) + ϕ0 ) ͕ĐŽŶ͗ ω= k m x(t0 ) v(t0 ) 2 2E 2 sin ϕ = = + = A x ( t ) 0 ͕ ͕ 0 2 A ω k T = 2π ω WƌĄĐƚŝĐĂϭĚĞůĂďŽƌĂƚŽƌŝŽ FH > 0 x=0 m x>0 FH < 0 WͲϮ͘ϱ͘ϯ m KƚƌĂƐĞƐƚƌĂƚĞŐŝĂƐƉĂƌĂĚĞƚĞƌŵŝŶĂƌĞůŵŽǀŝŵŝĞŶƚŽ͗ŵŽŵĞŶƚŽĂŶŐƵůĂƌLJĞŶĞƌպà ΑϮ͘ϯDŽŵĞŶƚŽĚĞƵŶĂĨƵĞƌnjĂLJŵŽŵĞŶƚŽĂŶŐƵůĂƌ & & ^ĞĚĞĨŝŶĞĞůŵŽŵĞŶƚŽĚĞƵŶĂĨƵĞƌnjĂƌĞƐƉĞĐƚŽ M ( P) F & & ĂƵŶƉƵŶƚŽ W ĐŽŵŽ͗ & & M (P) M ( P ) = r( P ) × F & & & P r( P ) = r − rP ĞƐĞůǀĞĐƚŽƌƉŽƐŝĐŝſŶŵĞĚŝĚŽĚĞƐĚĞĞůƉƵŶƚŽĚĞ ĂƉůŝĐĂĐŝſŶW ĚĞůĂĨƵĞƌnjĂ͘ & P ^ĞĚĞĨŝŶĞĞůŵŽŵĞŶƚŽĂŶŐƵůĂƌĚĞƵŶĂƉĂƌƚşĐƵůĂƌĞƐƉĞĐƚŽĂ L( P ) & & & ĐŽŵŽ͗ L( P ) = r( P ) × p & dp& ƉĂƌƚŝƌĚĞůĂ>ĞLJĚĞEĞǁƚŽŶ͕ŵƵůƚŝƉůŝĐĂŵŽƐĂŵďŽƐ & F = dt ƚĠƌŵŝŶŽƐ Ă ůĂ ŝnjƋƵŝĞƌĚĂ ƉŽƌ r( P ) × ƚĠƌŵŝŶŽƐĂůĂŝnjƋƵŝĞƌĚĂƉŽƌ & & ^ŝWĞƐƵŶƉƵŶƚŽĨŝũŽĚĞůƐŝƐƚĞŵĂĚĞƌĞĨĞƌĞŶĐŝĂ;ŝŶĞƌĐŝĂůͿ͕ r( P ) = v & F & F & r( P ) & L( P) P & p & & p r( P) & & & dp d & & & & & = (r( P ) × p ) ͕LJĂƋƵĞ v × p = v × mv = 0 ͘ĞĞƐƚŽƐĞŽďƚŝĞŶĞƋƵĞ͗ ŶƚŽŶĐĞƐ r( P ) × dt dt & WͲϮ͘ϯ͘ϭ ^ŝĞŶƉĂƌƚŝĐƵůĂƌĞůŵŽŵĞŶƚŽĚĞůĂĨƵĞƌnjĂƋƵĞĂĐƚƷĂƐŽďƌĞůĂ & dL( P ) = M ( P ) ƉĂƌƚşĐƵůĂĞƐĐĞƌŽ͕ĞŶƚŽŶĐĞƐĞůŵŽŵĞŶƚŽĂŶŐƵůĂƌƐĞŵĂŶƚŝĞŶĞ WͲϮ͘ϯ͘ϰ dt ĐŽŶƐƚĂŶƚĞ;ĐƵŝĚĂĚŽ͗ƌĞƐƉĞĐƚŽĂůŵŝƐŵŽƉƵŶƚŽW͊͊Ϳ WƌŽĚƵĐƚŽ ǀĞĐƚŽƌŝĂů͗ iˆ & & A × B ≡ Ax Bx ˆj Ay By kˆ Az = ( Ay Bz − Az By , Az Bx − Ax Bz , Ax By − Ay Bx ) Bz & & Y(P) ĞĨŝŶŝĞŶĚŽĞůŝŵƉƵůƐŽĂŶŐƵůĂƌƐƵŵŝŶŝƐƚƌĂĚŽƉŽƌĞůŵŽŵĞŶƚŽĚĞƵŶĂĨƵĞƌnjĂĞŶĞů M (P) t1 t2 ŝŶƚĞƌǀĂůŽĐŽŵŽ͗ t2 & & & & Y( P ) = M ( P ) dt ͕ƚĞŶĞŵŽƐĞů dĞŽƌĞŵĂĚĞůŵŽŵĞŶƚŽĂŶŐƵůĂƌ͗ Y(P) =ΔL(P) ³ t1 ůŵŽŵĞŶƚŽĂŶŐƵůĂƌƐĞŵĂŶƚŝĞŶĞĐŽŶƐƚĂŶƚĞĞŶƉĂƌƚŝĐƵůĂƌĞŶƚƌĞƐĐĂƐŽƐŝŵƉŽƌƚĂŶƚĞƐ͗ & & ϭͿ ^ŝ;ŵŽǀŝŵŝĞŶƚŽƌĞĐƚŝůŝŶĞŽƵŶŝĨŽƌŵĞͿ͕ L F =0 ( P ) = const ∀P & ϮͿ ŶƵŶŵŽǀŝŵŝĞŶƚŽĐŝƌĐƵůĂƌƵŶŝĨŽƌŵĞ͕͕ƐŝĞŶĚŽK ĞůĐĞŶƚƌŽĚĞůĐşƌĐƵůŽ L( O ) = const & L = const ϯͿ ŶĐĂƐŽĚĞĨƵĞƌnjĂĐĞŶƚƌĂůĚŝƌŝŐŝĚĂŚĂĐŝĂƵŶƉƵŶƚŽ͕ ( C ) &ƵĞƌnjĂƐĐĞŶƚƌĂůĞƐ hŶĂĨƵĞƌnjĂĐĞŶƚƌĂůĞƐƵŶĂĨƵĞƌnjĂĐƵLJĂƌĞĐƚĂĚĞĂĐĐŝſŶƐŝĞŵƉƌĞƉĂƐĂ ƉŽƌ ƵŶ ŵŝƐŵŽ ƉƵŶƚŽ ͘ ũĞŵƉůŽƐ͗ ůĂ ĨƵĞƌnjĂ ŐƌĂǀŝƚĂƚŽƌŝĂ ĚĞů ^Žů ƐŽďƌĞ ƉŽƌƵŶŵŝƐŵŽƉƵŶƚŽ͘ũĞŵƉůŽƐ͗ůĂĨƵĞƌnjĂŐƌĂǀŝƚĂƚŽƌŝĂĚĞů^ŽůƐŽďƌĞ ƵŶƉůĂŶĞƚĂ͖ůĂĨƵĞƌnjĂĞůĞĐƚƌŽƐƚĄƚŝĐĂĚĞƵŶĂĐĂƌŐĂĨŝũĂƐŽďƌĞŽƚƌĂ ƉĂƌƚşĐƵůĂĐĂƌŐĂĚĂ͖ůĂƚĞŶƐŝſŶĚĞƵŶĂĐƵĞƌĚĂĨŝũĂĚĂĞŶƵŶĞdžƚƌĞŵŽ͘ & mm FGU = −G 1 2 2 rˆ r & FC C WͲϮ͘ϯ͘ϰ ŶĞůĐĂƐŽĚĞĨƵĞƌnjĂĐĞŶƚƌĂů͕ĞůŵŽŵĞŶƚŽĚĞůĂĨƵĞƌnjĂƌĞƐƉĞĐƚŽĂůƉƵŶƚŽ & ƐƐŝĞŵƉƌĞĐĞƌŽ͕LJƉŽƌƚĂŶƚŽ dL& & dL (C ) = M (C ) (C ) = 0 dt dt ^ĞƉƵĞĚĞĚĞŵŽŶƐƚƌĂƌƋƵĞƚŽĚĂĨƵĞƌnjĂĐĞŶƚƌĂůĐƵŵƉůĞůĂ͗ & Ϯǐ>ĞLJĚĞ<ĞƉůĞƌ;ĐŽŵŽĐŽŶƐĞĐƵĞŶĐŝĂĚĞůĂĐŽŶƐĞƌǀĂĐŝſŶĚĞͿ L(C ) Æ sĞƌĞϮ͘ϯ͘Ϯ & TB C & T WͲϮ͘ϯ͘ϯ WͲϮ͘ϯ͘ϱ & mm >ĞLJĞƐĚĞ<ĞƉůĞƌ ;ƐĞĚĞŵƵĞƐƚƌĂŶĂƉĂƌƚŝƌĚĞůĂϮǐ>ĞLJĚĞEĞǁƚŽŶĐŽŶͿ FGU = −G 1 2 2 rˆ r WƌŝŵĞƌĂ>ĞLJ b >ŽƐƉůĂŶĞƚĂƐĚĞƐĐƌŝďĞŶŽƌďŝƚĂƐĞůşƉƚŝĐĂƐĂůƌĞĚĞĚŽƌĚĞůƐŽů͕ ĐŽŶĞůƐŽůŽĐƵƉĂŶĚŽƵŶŽĚĞůŽƐĨŽĐŽƐĚĞůĂĞůŝƉƐĞ a ^ĞŐƵŶĚĂ>ĞLJ >ĂǀĞůŽĐŝĚĂĚĚĞƵŶƉůĂŶĞƚĂǀĂƌşĂĞŶĞůƚŝĞŵƉŽ͕ĚĞĨŽƌŵĂƋƵĞĞůǀĞĐƚŽƌƋƵĞƵŶĞĞůƐŽůĂůƉůĂŶĞƚĂ & ĐƵďƌĞĄƌĞĂƐŝŐƵĂůĞƐĞŶƚŝĞŵƉŽƐŝŐƵĂůĞƐ͘ƐƚĂůĞLJĞƐƵŶĂĐŽŶƐĞĐƵĞŶĐŝĂĚŝƌĞĐƚĂĚĞ L( C ) = const ĞͲϮ͘ϯ͘Ϯ dĞƌĐĞƌĂ>ĞLJ ůĐƵĂĚƌĂĚŽĚĞůƉĞƌŝŽĚŽĚĞ ƌĞǀŽůƵĐŝſŶĚĞƵŶƉůĂŶĞƚĂĞƐ ĚŝƌĞĐƚĂŵĞŶƚĞƉƌŽƉŽƌĐŝŽŶĂůĂ ůĂůŽŶŐŝƚƵĚĚĞůƐĞŵŝĞũĞŵĂLJŽƌ ĂůĐƵďŽ͗ T 2 ∝ a 3 KƚƌĂ͞ĂƉůŝĐĂĐŝſŶ͟ĞŶůĂƋƵĞƚĂŶ ƐſůŽƐĞĐŽŶƐĞƌǀĂůĂĚŝƌĞĐĐŝſŶ ĚĞ & L ͗ƉĠŶĚƵůŽĚĞ&ŽƵĐĂƵůƚÆ (C ) & Fi ΑϮ͘ϰdƌĂďĂũŽ͕ƉŽƚĞŶĐŝĂ͕ĞŶĞƌŐşĂƉŽƚĞŶĐŝĂů͕ĞŶĞƌŐşĂĐŝŶĞƚŝĐĂ & Wi ĚĞƵŶĂĨƵĞƌnjĂĂůŽůĂƌŐŽĚĞƵŶĂĐƵƌǀĂΓ F ůƚƌĂďĂũŽ ĚĞƵŶ i & & & r r1 r2 ƉƵŶƚŽĂŽƚƌŽĞƐĞůŝŶƚĞŐƌĂů͗ & & Wi = ³ Fi ⋅ dr & r1 2 & Γ: r1 & & r2 & dr Γ & ^ŝůĂĐƵƌǀĂΓ ĞƐůĂƚƌĂLJĞĐƚŽƌŝĂĚĞƵŶĂƉĂƌƚşĐƵůĂ͕͕LJĞůƚƌĂďĂũŽƉƵĞĚĞĞƐĐƌŝďŝƌƐĞĐŽŵŽ͗ dr = v dt Wi & & & F ℘i = Fi ⋅ v ;ŝŶƐƚĂŶƚĄŶĞĂͿĚĞƐĂƌƌŽůůĂĚĂƉŽƌůĂĨƵĞƌnjĂ i Wi = ³℘i dt ͕ƐŝĞŶĚŽůĂƉŽƚĞŶĐŝĂ t2 WͲϮ͘ϰ͘Ϯ >ĂƵŶŝĚĂĚĚĞƚƌĂďĂũŽĞŶĞů^/ĠƐĞůũƵůŝŽ͗ J = N m ͖ůĂƵŶŝĚĂĚĚĞƉŽƚĞŶĐŝĂĞƐĞůǀĂƚŝŽ;tͿ͗ 1W = 1 J ͎ĐŽŵŽƐĞĐĂůĐƵůĂĞůƚƌĂďĂũŽ͍ůĂƚƌĂLJĞĐƚŽƌŝĂĞƐƵŶĂĐƵƌǀĂϭLJƐĞƉƵĞĚĞĚĞƐĐƌŝďŝƌĂƚƌĂǀĠƐĚĞƵŶ s & ƷŶŝĐŽƉĂƌĄŵĞƚƌŽƌĞĂů;Ɖ͘Ğũ͕͘ƵŶĄŶŐƵůŽ͕ŽĞůƚŝĞŵƉŽƚͿ͘>ĂƌĞůĂĐŝſŶƐĞůůĂŵĂĞĐƵĂĐŝſŶ r (λ ) λ ƉĂƌĂŵĠƚƌŝĐĂĚĞůĂĐƵƌǀĂΓ ĚĞƐĐƌŝƚĂƉŽƌůĂƚƌĂLJĞĐƚŽƌŝĂ͘^ŝƐĞĐŽŶŽĐĞƵŶĂĞdžƉƌĞƐŝſŶƉĂƌĂŵĠƚƌŝĐĂĚĞ Γ͕ĞůŝŶƚĞŐƌĂůƋƵĞĚĞĨŝŶĞĞůƚƌĂďĂũŽƉƵĞĚĞĞƐĐƌŝďŝƌƐĞĞŶĨƵŶĐŝſŶĚĞůƉĂƌĄŵĞƚƌŽ λ & & & ^ŝůĂĞdžƉƌĞƐŝſŶƉĂƌĂŵĠƚƌŝĐĂĚĞůĂĐƵƌǀĂĞƐƉ͘Ğũ͘ r ( λ ) = ( x ( λ ), y ( λ ), z ( λ )), P = r ( λ ) , P = r (λ2 ) ͕ 1 1 2 & & r2 λ2 λ2 & & & & dr ĞŶƚŽŶĐĞƐ͗ dy dz · § dx Fi Wi = ³ Fi ⋅ dr = ³ Fi ⋅ dȜ = ³ ¨ Fx + Fy + Fz ¸ dȜ & & dȜ dȜ dȜ dȜ λ2 ¹ Γ: r1 λ1 λ1 © dr WͲϮ͘ϰ͘ϭƐŝŶ;ĐͿ WͲϮ͘ϰ͘ϲ & r (λ ) Γ λ1 ͎WĂƌĂƋƵĠƐŝƌǀĞĞůƚƌĂďĂũŽ͍^ŝĐŽŶƐŝĚĞƌĂŵŽƐĞůƚƌĂďĂũŽĚĞůĂĨƵĞƌnjĂƚŽƚĂů & & & & & & & & & & & F = FTOT = F1 + F2 + ... = ma ƚĞŶĞŵŽƐ͗ dWTOT = FTOT ⋅ dr = F1 ⋅ dr + F2 ⋅ dr + ... = dW1 + dW2 + ... t1 & & & & & dv & d 1 1 WŽƌŽƚƌŽůĂĚŽ FTOT ⋅ dr = ma ⋅ dr = m ⋅ v dt = §¨ mv 2 ·¸dt = d §¨ mv 2 ·¸ ͘/ŶƚĞŐƌĂŶĚŽ͗ dt dt © 2 ¹ ©2 ¹ WͲϮ͘ϭ͘ϭ 1 § 1 2· 2 W = Δ¨ mv ¸ = ΔEc ͘>ĂĐĂŶƚŝĚĂĚƐĞůůĂŵĂĞŶĞƌŐşĂĐŝŶĠƚŝĐĂ͘ƐşƉƵĞƐ͗ W1 + W2 + ... = ΔEc Ec = m v 2 ©2 ¹ && & & & § ∂U ∂U ∂U · ∃ U (r ) F(r ) ¸¸ hŶĂĨƵĞƌnjĂƐĞĚŝĐĞĐŽŶƐĞƌǀĂƚŝǀĂ ƐŝƚĂůƋƵĞ͗ , , F = −∇U = −¨¨ x y z ∂ ∂ ∂ © ¹ >ĂĨƵŶĐŝſŶh ƐĞůůĂŵĂĞŶĞƌŐŝĂƉŽƚĞŶĐŝĂů U ĂƐƐŽĐŝĂĚĂĂůĂĨƵĞƌnjĂ͘ůƚƌĂďĂũŽĚĞůĂĨƵĞƌnjĂǀĂůĞ͗ ' ' λ2 § ∂U ∂U · ∂U & dz ¸¸ = − ³ dU dy + dx + W = ³ F ⋅ dr = ³ ( Fx dx + Fy dy + Fz dz ) = − ³ ¨¨ dr ∂z ¹ ∂y © ∂x 2 & & λ1 r (λ ) & F ŶƚŽŶĐĞƐ͕ƉĂƌĂĐŽŶƐĞƌǀĂƚŝǀĂ͗ W1→2 = − ³ dU = −[U (2) − U (1)] = −ΔU F i 1 dƌĂďĂũŽĚĞƵŶĂĨƵĞƌnjĂĐŽŶƐĞƌǀĂƚŝǀĂсǀĂƌŝĂĐŝſŶĚĞĞŶĞƌŐşĂƉŽƚĞŶĐŝĂů && ⇔ F(r ) ĞƐĐŽŶƐĞƌǀĂƚŝǀĂ⇔ & & F ³ ⋅ dr = 0 ⇔ curva cerrada ³ B A W = −Δ U & & F ⋅ dr ĞƐŝŶĚĞƉĞŶĚŝĞŶƚĞĚĞůĂƚƌĂLJĞĐƚŽƌŝĂ & & >ĂĨƵŶĐŝſŶhƐĞĐĂůĐƵůĂĐŽŵŽ͗ U = − ³ F ⋅ dr + ct = − ³ ( Fx dx + Fy dy + Fz dz ) + ct Ͳ Ŷϭ͗ĐĂĚĂĨƵĞƌnjĂĞƐĐŽŶƐĞƌǀĂƚŝǀĂLJĂƋƵĞƐŝĞŵƉƌĞƐĞƉƵĞĚĞĚĞĨŝŶŝƌU ( x) = − F ( x) dx + ct ³ WͲϮ͘ϰ͘ϱ͕YͲϮ͘ϰ͘ϰнĚŝƐĐƵƚŝƌƉƵŶƚŽƐĚĞĞƋƵŝůŝďƌŝŽĞƐƚĂďůĞĞŝŶĞƐƚĂďůĞ & & Ͳ ŶϮ͗dU = −F ·dr = −Fx dx − Fy dy ⇔ Fx = − ∂U ∂x Fy = − ∂U ∂y ^ĞĚĞŵƵĞƐƚƌĂƋƵĞƵŶĂĨƵŶĐŝſŶĚŝĨĞƌĞŶĐŝĂďůĞĚĞĚŽƐǀĂƌŝĂďůĞƐƚŝĞŶĞůĂƉƌŽƉŝĞĚĂĚƋƵĞ͗ ∂ 2U ∂ 2U ƐƚŽŝŵƉůŝĐĂƋƵĞƵŶĂĨƵĞƌnjĂĞŶϮĞƐĐŽŶƐĞƌǀĂƚŝǀĂ ⇔ ∂Fx ∂Fy = WͲϮ͘ϰ͘ϯ = ∂y ∂x ∂x∂y ∂y∂x WͲϮ͘ϰ͘ϰ͕WͲϮ͘ϰ͘ϳ & & F Ͳ Ŷϯ͗^ŝĞƐƌĂĚŝĂůLJƐſůŽĚĞƉĞŶĚĞĚĞŶŽĚĞŽƚƌĂƐĐŽŵďŝŶĂĐŝŽŶĞƐĚĞdž͕LJ͕nj͕ŽƐĞĂ r =r & & & Ɛŝ͕ĞŶƚŽŶĐĞƐĞƐĐŽŶƐĞƌǀĂƚŝǀĂLJ F (r ) = F (r )rˆ U (r ) = − ³ F ( r )rˆ ⋅ dr + ct = − ³ F (r )dr + ct ( ) ĂƐŽƐŝŵƉŽƌƚĂŶƚĞƐ ;ΑϮ͘ϱLJΑϯ͘ϳͿ ϭͿhŶĂĨƵĞƌnjĂĐŽŶƐƚĂŶƚĞ;ĞŶŵŽĚƵůŽLJĚŝƌĞĐĐŝſŶͿĞƐĐŽŶƐĞƌǀĂƚŝǀĂ͗ & & & dU = − F ·dr °½ && & & Fg = mg U = m g y + ct ¾ U = − F ·r + ct ͘ũĞŵƉůŽ͗Æ F = cte °¿ & & k k mm F = − 2 rˆ U (r ) = − FGU = −G 1 2 2 rˆ ϮͿ͘ũĞŵƉůŽƐ͗>ĞLJĚĞŐƌĂǀŝƚĂĐŝſŶƵŶŝǀĞƌƐĂů r r r & 1 q1q2 ˆ r LJ>ĞLJĚĞŽƵůŽŵď FC = − 2 1 r πε 4 2 0 ϯͿ FH = −kx U(x) = −³ (−kx) dx = k x + ct 2 2 2 & & ˆ ˆ ϰͿ FRDS = − μ D Nv W1→2 = ³ (− μ D N )v ⋅ dr = − μ D N ³ d" = − μ D N" 1→2 1 Å 1 ^ſůŽǀĂůĞ ƐŝE сĐŽŶƐƚ ŶŵƵĐŚĂƐƐŝƚƵĂĐŝŽŶĞƐŚĂLJĨƵĞƌnjĂƐŽƌƚŽŐŽŶĂůĞƐĂůŵŽǀŝŵŝĞŶƚŽ;ŽƐĞĂ͕ĂůĂǀĞůŽĐŝĚĂĚͿ͖ŵƵLJĂ ŵĞŶƵĚŽĠƐƚĞĞƐĞůĐĂƐŽĚĞůĂƐůůĂŵĂĚĂƐ͞ƌĞĂĐĐŝŽŶĞƐ͕͟ĐŽŵŽůĂĨƵĞƌnjĂŶŽƌŵĂůĚĞďŝĚĂĂƵŶĂ ͕ & ƐƵƉĞƌĨŝĐŝĞŽůĂƚĞŶƐŝſŶĚĞƵŶĂĐƵĞƌĚĂĞŶƵŶŵŽǀŝŵŝĞŶƚŽĐŝƌĐƵůĂƌ͘/ŶĚŝĐĄŶĚŽůĂƐĐŽŶůĂůĞƚƌĂ͕ƐĞ & & & & & & R& & & & & & ŚĂ͗F = FTOT = ma = R1 + F2 + ... ͘DƵůƚŝƉůŝĐĂŵŽƐĞƐĐĂůĂƌŵĞŶƚĞƉŽƌ͗ & dr ma&⋅ dr &= R1 ⋅ dr + F2 ⋅ dr + ... & & & ůƐĞƌ͕ůĂĨƵĞƌnjĂĚĞƌĞĂĐĐŝſŶĞƐŽƌƚŽŐŽŶĂůĂ͕ŽƐĞĂ͘ƐƚŽŶŽƐĚĞũĂ dr R1 ⋅ dr = 0 R ⊥ v = dr dt & & & & ĐŽŶĞů͗ ƉƌŝŶĐŝƉŝŽĚĞĚ͛ůĞŵďĞƌƚ F2 + ... + Fn − ma ⋅ dr = 0 ( ) & & & & & & & & /ŶƚĞŐƌĂŶĚŽůĂĞĐ͘ĚĞĚ͛ůĞŵďĞƌƚ͕ ³ F ⋅ dr = ³ F1 ⋅ dr + ³ F2 ⋅ dr +... + ³ Fn ⋅ dr ͕ƐĞŽďƚŝĞŶĞůĂĞĐƵĂĐŝſŶ ĨƵŶĚĂŵĞŶƚĂůĚĞůĂĞŶĞƌŐşĂ͗ ΔEc = −ΔU Fcons + WF NO cons WF NO cons = Δ( Ec + U ) WͲϯ͘ϳ͘ϯ͕WͲϮ͘ϱ͘ϳ͕;WͲϯ͘ϭϬ͘ϰĐŽŶĂͿ WͲϮ͘ϱ͘ϭ͕WͲϮ͘ϱ͘ϲ͕YͲϮ͘ϭ͘ϭ͕WͲϯ͘ϳ͘ϰ͕WͲϯ͘ϭϬ͘ϭ͕WͲϯ͘ϭϬ͘ϵ͕YͲϯ͘ϭϬ͘ϭ͕YͲϯ͘ϭϬ͘Ϯ Ec + U = E = ct ^ŝŶŽŚĂLJĨƵĞƌnjĂƐĚŝƐŝƉĂƚŝǀĂƐ͕ĞŶƚŽŶĐĞƐ͕ŽƐĞĂƚĂŵďŝĠŶ͗ dE =0 ŶĂůŐƵŶŽƐĐĂƐŽƐ;ƉŽƌĞũĞŵƉůŽƐŝĞůƐŝƐƚĞŵĂƚŝĞŶĞƵŶƐŽůŽŐƌĂĚŽĚĞůŝďĞƌƚĂĚͿ͕ dt ĞƐƚĂĞĐƵĂĐŝſŶĞƐƐƵĨŝĐŝĞŶƚĞƉĂƌĂŚĂůůĂƌůĂƚƌĂLJĞĐƚŽƌŝĂƚĞŵƉŽƌĂůĚĞůƐŝƐƚĞŵĂ͘ WͲϯ͘ϭϬ͘ϯ dĞŵĂϯ͗^ŝƐƚĞŵĂƐĚĞN ƉĂƌƚşĐƵůĂƐ;ŝ сϭ͕Ϯ͕͘͘͘NͿ ϯǐ>ĞLJĚĞEĞǁƚŽŶŽ>ĞLJĚĞĂĐĐŝſŶLJƌĞĂĐĐŝſŶ͗ƐŝƵŶĐƵĞƌƉŽĂĐƚƷĂƐŽďƌĞƵŶƐĞŐƵŶĚŽĐŽŶƵŶĂ & & ĨƵĞƌnjĂ͕ĞŶƚŽŶĐĞƐĠƐƚĞƷůƚŝŵŽŐĞŶĞƌĂƵŶĂĨƵĞƌnjĂŝŐƵĂůLJŽƉƵĞƐƚĂƐŽďƌĞĞůƉƌŝŵĞƌŽ − F F ũĞŵƉůŽ͗ĨƵĞƌnjĂƐĚĞĐŽŶƚĂĐƚŽ WͲϯ͘ϭϬ͘Ϯ DƵĐŚĂƐǀĞĐĞƐĞƐƚĂƐĨƵĞƌnjĂƐĐƵŵƉůĞŶƚĂŵďŝĠŶŽƚƌĂĐŽŶĚŝĐŝſŶ͕ƋƵĞĞƐƋƵĞƐƵĚŝƌĞĐĐŝſŶĞƐ & & & & ƉĂƌĂůĞůĂĂůǀĞĐƚŽƌƋƵĞƵŶĞůĂƐƉŽƐŝĐŝŽŶĞƐĚĞůĂƐƉĂƌƚşĐƵůĂƐ͗ F // r (= r − r ) 1→ 2 1→ 2 2 1 >ĂƐĨƵĞƌnjĂƐĚĞŐƌĂǀŝƚĂĐŝſŶƵŶŝǀĞƌƐĂůLJŽƵůŽŵďĐƵŵƉůĞŶůĂϯǐ>ĞLJĚĞEĞǁƚŽŶĞŶĞƐƚĂĨŽƌŵĂ ͞ĨƵĞƌƚĞ͟;WͲϯ͘ϳ͘ϭͿ & (ext ) mi & F &ƵĞƌnjĂƐĞdžƚĞƌŶĂƐ ͗ƐŽŶĐĂƵƐĂĚĂƐƉŽƌĂŐĞŶƚĞƐ i (ext ) Fi ĞdžƚĞƌŝŽƌĞƐ͕ƋƵĞŶŽ ƉĞƌƚĞŶĞĐĞŶĂůƐŝƐƚĞŵĂĐŽŶƐŝĚĞƌĂĚŽ mi & Fji & Fij mj &ƵĞƌnjĂƐŝŶƚĞƌŶĂƐ͗ƐŽŶůĂƐĨƵĞƌnjĂƐĚĞůĂƐƉĂƌƚşĐƵůĂƐĚĞůƐŝƐƚĞŵĂĞŶƚƌĞĞůůĂƐ ƉŽƌůĂϯǐ>ĞLJĚĞEĞǁƚŽŶ͗ & & ext & int & int Fi → j = − F j →i & & &. + ¦ F jint→i = mi ai = p i & & ext & ext dP &. j &. = ¦ p i = P ͕ŽƐĞĂ͗ FTOT = F = dt i >ĂϮĂ>ĞLJĚĞEĞǁƚŽŶƉĂƌĂůĂƉĂƌƚşĐƵůĂŝ ĞƐ͗ Fi = Fi & & & ext & ext ¦i Fi =¦i Fi = FTOT & & Æ ^ŝ͕ĞŶƚŽŶĐĞƐ F ext = 0 P = ct ^ƵŵĂŶĚŽƐŽďƌĞŝ͗ FTOT = & & 1 RCM = mi ri M = ¦ mi ĞĨŝŶŝĞŶĚŽůĂĐŽŽƌĚĞŶĂĚĂĚĞůĞŶƚƌŽĚĞDĂƐĂƐ;DͿĐŽŵŽ͕͗ĐŽŶ͕ ¦ M i . & & & & & i & ĞŶƚŽŶĐĞƐ͗ P = ext LJ p = MV ¦ i i CM F = P = MACM ƉůŝĐĂĐŝſŶ͗ĨƵĞŐŽƐĂƌƚŝĨŝĐŝĂůĞƐ WͲϯ͘ϭϭ͘ϭ͕WͲϯ͘Ϯ͘ϴ YͲϯ͘Ϯ͘ϲ WͲϯ͘ϲ͘ϳ EŽƚĂŝŵƉŽƌƚĂŶƚĞƐŽďƌĞĞůĐĞŶƚƌŽĚĞŵĂƐĂƐ͘WĂƌĂĚŽƐƉĂƌƚşĐƵůĂƐ͗ & 1 RCM = (m1r&1 + m2 r&2 ) = 1 (( M − m2 )r&1 + m2 r&2 ) = r&1 + m2 (r&2 − r&1 ) M M M ƋƵşƐĞŚĂƵƚŝůŝnjĂĚŽ M = m1 + m2 m1 = M − m2 & & r2 − r1 × 1 & r1 & RCM 2 CM WŽƌůŽƚĂŶƚŽ͕ĞůĐĞŶƚƌŽĚĞŵĂƐĂƐĚĞƵŶĐŽŶũƵŶƚŽĚĞĚŽƐ ƉĂƌƚşĐƵůĂƐĞƐƚĄĂůŽůĂƌŐŽĚĞůĂƌĞĐƚĂƋƵĞůĂƐƵŶĞ͕ĞŶƵŶƉƵŶƚŽŝŶƚĞƌŵĞĚŝŽĞŶƚƌĞůĂƐĚŽƐ & & & & Αϯ͘ϰ ůŵŽŵĞŶƚŽĂŶŐƵůĂƌƚŽƚĂůĚĞƵŶƐŝƐƚĞŵĂĚĞƉĂƌƚşĐƵůĂƐĞƐ͗ L = Li (Q ) = ri (Q ) × mi vi (Q ) & & & & i i ĚŽŶĚĞ͘ĞƌŝǀĂŶĚŽƌĞƐƉĞĐƚŽĚĞůƚŝĞŵƉŽ͗ ri ( Q ) = ri − rQ WͲϯ͘ϰ͘ϭн RCM & сϬ & & dL(Q ) & & & & & & & & & & & = ¦ vi ( Q ) × mi vi + ¦ ri ( Q ) × mi ai = ¦ vi × mi vi − ¦ vQ × mi vi + ¦ ri ( Q ) × Fi = M (Q ) − vQ × ¦ mi vi dt i i i i i & int &i ^ŝůĂƐĨƵĞƌnjĂƐŝŶƚĞƌŶĂƐĐƵŵƉůĞŶůĂĐŽŶĚŝĐŝſŶ͕ĞŶƚŽŶĐĞƐƐƵŵŽŵĞŶƚŽƚŽƚĂůĞƐĐĞƌŽ͕LJ Fi → j // rij & & ext & & & M (Q ) = M ( Q )͘ŽŶĞƐƚŽ͕ůĂĞĐƵĂĐŝſŶĚĞůŵŽŵĞŶƚŽĂŶŐƵůĂƌƋƵĞĚĂ͗ dL(Q ) dt = M (extQ) − v&Q × MVCM ¦ ¦ ůϮǑƚĠƌŵŝŶŽĞƐĐĞƌŽƐŝ͗;ϭͿYĞƐƵŶƉƵŶƚŽĨŝũŽ͖Ž;ϮͿ͕YĐŽŝŶĐŝĚĞĐŽŶĞůĐĞŶƚƌŽĚĞŵĂƐĂƐ͘ƐşƉƵĞƐ͗ & & & ext & ext dL(Q ) L M = 0 ͕ĐŽŶYĨŝũŽŽYŁ D͘ƐƚŽŝŵƉůŝĐĂƋƵĞƐŝ ͕ĞŶƚŽŶĐĞƐ ( Q ) = ct (Q ) M (Q ) = dt ƉůŝĐĂĐŝŽŶĞƐ;YсDͿ͗ĨƷƚďŽůĂŵĞƌŝĐĂŶŽ͕ĂƌŵĂƐĚĞĨƵĞŐŽ͕ YƺĞƐƚŝŽŶƐ WĂƌĐŝĂůϮϬϭϬ YƺĞƐƚŝŽŶƐWĂƌĐŝĂůϮϬϭϬ WͲϯ͘Ϯ͘ϭ͕WͲϯ͘ϯ͘ϭ͕YͲϯ͘ϰ͘ϭ͕Y͘ϯ͘ϰ͘Ϯ͕YͲϯ͘ϰ͘ϯ ŐŝƌŽƐĐŽƉŝŽ ĞƐƚĂĐŝŽŶĞƐ ƚĞƌƌĞŵŽƚŽƐ ŐŝƌŽƐĐŽƉŝŽ͕ĞƐƚĂĐŝŽŶĞƐ͕ƚĞƌƌĞŵŽƚŽƐ Αϯ͘ϱ ŽŶƐŝĚĞƌĞŵŽƐĞůƚƌĂďĂũŽĚĞƵŶĂƉĂƌĞũĂĚĞĨƵĞƌnjĂƐŝŶƚĞƌŶĂƐ͗ & & & & & & & int int dW12 = F1int → 2 ⋅ dr2 + F2 →1 ⋅ dr1 = F1→ 2 ⋅ (dr2 − dr1 ) ;ƐĞŚĂƵƚŝůŝnjĂĚŽůĂϯǐ>ĞLJĚĞEĞǁƚŽŶͿ^ŝůĂĨƵĞƌnjĂƐſůŽĚĞƉĞŶĚĞĚĞůĂ & & & rrel = r2 − r1 ĐŽŽƌĚĞŶĂĚĂƌĞůĂƚŝǀĂLJĂĚĞŵĄƐĞƐĐŽŶƐĞƌǀĂƚŝǀĂ͕ĞŶƚŽŶĐĞƐ͗ & & & & & & dW12 = F1→2 (rrel ) ⋅ drrel = −∇U (rrel ) ⋅ drrel = − dU ƐƚŽƐŝŐŶŝĨŝĐĂƋƵĞĂĐĂĚĂƉĂƌĞũĂ;ŝ͕ũͿƐĞƉƵĞĚĞĂƐŽĐŝĂƌƵŶĂĞŶĞƌŐşĂƉŽƚĞŶĐŝĂůhƋƵĞƐſůŽ ĚĞƉĞŶĚĞĚĞůĂĐŽŽƌĚĞŶĂĚĂƌĞůĂƚŝǀĂ͘^ŝƚŽĚĂƐůĂƐĨƵĞƌnjĂƐŝŶƚĞƌŶĂƐLJĞdžƚĞƌŶĂƐƐŽŶĐŽŶƐĞƌǀĂƚŝǀĂƐ͗ U = U iext + U ijint E = EC + U = ct ͕ĐŽŶ͘^ŝŚĂLJĨƵĞƌnjĂƐĞdžƚĞƌŶĂƐŶŽĐŽŶƐĞƌǀĂƚŝǀĂƐ͗ ¦ i WͲϯ͘ϱ͘Ϯ͕WͲϯ͘ϳ͘Ϯ ¦ parejas WͲϯ͘ϳ͘ϰ͕WͲϯ͘ϱ͘ϯ͕YͲϯ͘ϯ͘ϭ͕YͲϯ͘ϱ͘ϭ͕YͲϯ͘ϱ͘Ϯ͕YͲϯ͘ϱ͘ϯ WFext NO CONS = Δ( EC + U ) Αϯ͘ϲŚŽƋƵĞƐ hŶĐŚŽƋƵĞĞƐƵŶĂĐŽůŝƐŝſŶĞŶƚƌĞĚŽƐĐƵĞƌƉŽƐ͕ŽƐĞĂƵŶĂŝŶƚĞƌĂĐĐŝſŶĚĞĐŽŶƚĂĐƚŽĚĞĚƵƌĂĚĂ ůŝŵŝƚĂĚĂ;ƵƐƵĂůŵĞŶƚĞƐĞǀĞƌŝĨŝĐĂƋƵĞȴƚ сϬ͘Ϭϭр Ϭ͘ϬϬϭƐĞĐͿ͘EŽĐŽŶƐŝĚĞƌĂŵŽƐůĂƌŽƚĂĐŝſŶ͘ YͲϮ͘Ϯ͘ϭ dŽŵĞŵŽƐĐŽŵŽƉƌŝŵĞƌĞũĞŵƉůŽĞůĚĞƵŶĂƉĞůŽƚĂĚĞŵĂƐĂϴϬŐƋƵĞƌĞďŽƚĂĐŽŶƚƌĂ ƵŶĂƉĂƌĞĚ͘ŽŶƐŝĚĞƌĞŵŽƐůŽƐŝŶƐƚĂŶƚĞƐũƵƐƚŽĂŶƚĞƐ;ͿLJũƵƐƚŽĚĞƐƉƵĠƐ;ͿĚĞůĂĐŽůŝƐŝſŶ͕LJ ŽůǀŝĚĠŵŽŶŽƐĚĞŵŽŵĞŶƚŽĚĞůĂƐĞǀĞŶƚƵĂůĞƐĨƵĞƌnjĂƐĞdžƚĞƌŝŽƌĞƐ͘^ŝĞůŵŽĚƵůŽĚĞůĂǀĞůŽĐŝĚĂĚ ĂŶƚĞƐLJĚĞƐƉƵĠƐĚĞůĐŚŽƋƵĞĞƐĚĞϯϬŵͬƐ͕ĞůŝŵƉƵůƐŽŐĞŶĞƌĂĚŽƉŽƌůĂĨƵĞƌnjĂĚĞďŝĚĂĂůĂ & & & & ƉĂƌĞĚĚƵƌĂŶƚĞĞůĐŚŽƋƵĞĞƐ I = Δp = m(vD − v A ), en modulo 4.8 N s >ĂĨƵĞƌnjĂŵĞĚŝĂƋƵĞƐĞŐĞŶĞƌĂĞŶĞůŝŵƉĂĐƚŽĞƐĞŶŽƌŵĞ ƉĂƌĂƵŶĂƉĞůŽƚĂĚĞϴϬŐ͕LJǀĂůĞ͕ĐŽŶȴƚ сϬ͘ϬϬϯƐĞĐ ͞WZ/E/W/KĚĞůDZd/>>K͟ Fmedia = & I Δt = 1000 N ŽŵƉĂƌĂĚŽĐŽŶĞƐƚŽ͕ůĂĨƵĞƌnjĂĚĞŐƌĂǀĞĚĂĚ;ƉŽƌĞũĞŵƉůŽͿƐŽďƌĞůĂƉĞůŽƚĂĞƐĚĞƐƉƌĞĐŝĂďůĞ͘ ŽŵƉĂƌĂĚŽ ĐŽŶ ĞƐƚŽ ůĂ ĨƵĞƌnjĂ ĚĞ ŐƌĂǀĞĚĂĚ ;ƉŽƌ ĞũĞŵƉůŽͿ ƐŽďƌĞ ůĂ ƉĞůŽƚĂ ĞƐ ĚĞƐƉƌĞĐŝĂďůĞ ŽŶƐŝĚĞƌĞŵŽƐĂŚŽƌĂĞůĐŚŽƋƵĞĞŶƚƌĞĚŽƐŽŵĄƐŵĂƐĂƐ͘^ĞƉƵĞĚĞƐƵƉŽŶĞƌƋƵĞŶŽŚĂLJĂ ĨƵĞƌnjĂƐĞdžƚĞƌŝŽƌĞƐ͕LJĂƋƵĞĞƐƚĂƐƐŽŶŵƵLJĚĠďŝůĞƐƉĂƌĂũƵŐĂƌƵŶƉĂƉĞůĚƵƌĂŶƚĞĞůĐŚŽƋƵĞ͘ WŽƌůŽƚĂŶƚŽ͕ůĂĐĂŶƚŝĚĂĚĚĞŵŽǀŝŵŝĞŶƚŽƚŽƚĂůĚĞůƐŝƐƚĞŵĂƐĞĐŽŶƐĞƌǀĂ͗ & & ext F ≈ 0 & P & = ct PD = PA WͲϯ͘ϲ͘ϴ;ĐŚŽƋƵĞĐŽŵƉů͘ŝŶĞůĄƐƚŝĐŽͿ & v1(ini) m1 m1 DŝĞŶƚƌĂƐƋƵĞĞůŵŽŵĞŶƚŽůŝŶĞĂůƐĞĐŽŶƐĞƌǀĂƐŝĞŵƉƌĞĞŶƵŶĐŚŽƋƵĞĞŶƚƌĞ & v2(ini ) ƉĂƌƚşĐƵůĂƐ͕ůĂĞŶĞƌŐşĂĞŶŐĞŶĞƌĂůŶŽ͘>ůĂŵĂŵŽƐĞůĄƐƚŝĐŽƵŶĐŚŽƋƵĞĞŶ m2 ƋƵĞůĂĞŶĞƌŐşĂĐŝŶĠƚŝĐĂƚŽƚĂů͕LJƉŽƌůŽƚĂŶƚŽůĂĞŶĞƌŐşĂƚŽƚĂů͕ƐĞĐŽŶƐĞƌǀĂ ;ƐŝŶĚĞƐƉůĂnjĂŵŝĞŶƚŽƐ͕ůĂĞŶĞƌŐşĂƉŽƚĞŶĐŝĂůĚĞ&ĞdžƚĞƌŶĂƐŶŽĐĂŵďŝĂͿ͘ WĂƌĂϮƉĂƌƚşĐƵůĂƐ͕ĐŚŽƋƵĞĞůĄƐƚŝĐŽ ⇔ m2 1 1 1 1 m1v12, D + m2 v22, D = m1v12, A + m2 v22, A 2 2 2 2 N m & vA , i = 1,2 & P = ct ĚĞŵĄƐĂƋƵşƚĞŶĚƌĞŵŽƐ͗ // i,D v m & vD n̂ /ŵƉŽƌƚĂŶƚĞ͗WĂƌĂƵŶĂĐŽůŝƐŝſŶĞŶƚƌĞƵŶĐƵĞƌƉŽLJƵŶĂƉĂƌĞĚĨŝũĂ͗ vD// = v A// EſƚĞƐĞƋƵĞĞŶĞƐƚĞĐĂƐŽŶŽ ƐĞĐŽŶƐĞƌǀĂĞůŵŽŵĞŶƚŽůŝŶĞĂůƚŽƚĂů /ŐƵĂůŵĞŶƚĞ͕ƉĂƌĂƵŶĐŚŽƋƵĞĞŶƚƌĞĚŽƐĐƵĞƌƉŽƐ͗ & v2( fi) WͲϯ͘ϲ͘ϯ ^ŝƐĞĐŽŶŽĐĞůĂŐĞŽŵĞƚƌşĂĚĞůĂĐŽůŝƐŝſŶLJŶŽŚĂLJƌŽnjĂŵŝĞŶƚŽ͗ ĞŶĞƐƚĞĐĂƐŽƉŽĚĞŵŽƐĐŽŶŽĐĞƌůĂĚŝƌĞĐĐŝſŶĚĞůĂƐĨƵĞƌnjĂƐŝŵƉƵůƐŝǀĂƐ ŐĞŶĞƌĂĚĂƐĞŶĞůŝŵƉĂĐƚŽ͕ƋƵĞĞƐŽƌƚŽŐŽŶĂůĂůƉůĂŶŽĚĞĐŚŽƋƵĞ͘ƐƚŽ ŝŵƉůŝĐĂƋƵĞŶŽŚĂLJŶŝŶŐƵŶĂĨƵĞƌnjĂƉĂƌĂůĞůĂĂĞƐƚĞƉůĂŶŽ͕LJƋƵĞ ĞŶƚŽŶĐĞƐůĂƐĐŽŵƉŽŶĞŶƚĞƐƉĂƌĂůĞůĂƐĚĞůĂƐǀĞůŽĐŝĚĂĚĞƐƐĞĐŽŶƐĞƌǀĂŶ͘ & v1( fi) & v1, D & v1, A 1 =v // i, A & v2 , A & v2 , D 2 n̂ WŽƌůŽƚĂŶƚŽůĂƐƷŶŝĐĂƐŝŶĐſŐŶŝƚĂƐƋƵĞƋƵĞĚĂŶƐŽŶůĂƐĐŽŵƉŽŶĞŶƚĞƐĚĞůĂǀĞůŽĐŝĚĂĚŽƌƚŽŐŽŶĂůĞƐ ĂůƉůĂŶŽĚĞĐŚŽƋƵĞ͘^ĞĚĞĨŝŶĞ ĞůĐŽĞĨŝĐŝĞŶƚĞĚĞƌĞƐƚŝƚƵĐŝſŶĞ ĐŽŵŽ͗ e= v1⊥( D ) − v2⊥( D ) v1⊥( A) − v2⊥( A) ^ŝĐŽŶŽĐĞŵŽƐĞůǀĂůŽƌĚĞĞ͕ĞůƉƌŽďůĞŵĂƐĞƌĞƐƵĞůǀĞƉŽŶŝĞŶĚŽĂƐŝƐƚĞŵĂůĂƐĞĐƵĂĐŝŽŶĞƐƋƵĞ ƌĞƐƵůƚĂŶĚĞůĂĐŽŶƐĞƌǀĂĐŝſŶĚĞůŵŽŵĞŶƚŽLJĚĞůĂĚĞĨŝŶŝĐŝſŶĚĞĞ͘ ^ĞƉƵĞĚĞĚĞŵŽŶƐƚƌĂƌƋƵĞƉĂƌĂƵŶĐŚŽƋƵĞĞůĄƐƚŝĐŽ͕Ğсϭ WͲϯ͘ϲ͘ϭ͕WͲϯ͘ϲ͘ϰ WͲϯ͘ϲ͘ϲ͕WͲϯ͘ϲ͘ϵ͕WͲϯ͘ϲ͘ϭϬ
