PHYSICS 9A LECTURES WORK AND ENERGY I Learning Goals for Chapter Looking forward at … • what it means for a force to do work on a body, and how to calculate the amount of work done. • the definition of the kinetic energy (energy of motion) of a body, and how the total work done on a body changes the body’s kinetic energy. • how to use the relationship between total work and change in kinetic energy when the forces are not constant, the body follows a curved path, or both. • how to solve problems involving power (the rate of doing work). © 2016 Pearson Education Inc. Introduction • A baseball pitcher does work with his throwing arm to give the ball a property called kinetic energy. • In this lecture, the introduction of the new concepts of work, energy, and the conservation of energy will allow us to deal with problems in which Newton’s laws alone aren’t enough. © 2016 Pearson Education Inc. Work © 2016 Pearson Education Inc. Work • A force on a body does work if the body undergoes a displacement. These people are doing work as they push on the car because they exert a force on the car as it moves. © 2016 Pearson Education Inc. Units of work • The SI unit of work is the joule (named in honor of the 19th-century English physicist James Prescott Joule). • Since W = Fs, the unit of work is the unit of force multiplied by the unit of distance. • In SI units: 1 joule = (1 newton) (1 meter) or 1 J = 1 N ∙ m • If you lift an object with a weight of 1 N a distance of 1 m at a constant speed, you do 1 J of work on it. © 2016 Pearson Education Inc. Work done by a constant force • The work done by a constant force acting at an angle to the displacement is: (the projected effect along a direction) • This can be written more compactly as: © 2016 Pearson Education Inc. Positive work • When the force has a component in the direction of the displacement, work is positive. © 2016 Pearson Education Inc. Negative work • When the force has a component opposite to the direction of the displacement, work is negative. © 2016 Pearson Education Inc. Zero work • When the force is perpendicular to the direction of the displacement, the force does no work on the object. © 2016 Pearson Education Inc. Sign of work (recap) © 2016 Pearson Education Inc. Checkpoint • The work done by a dissipative force like friction is ... A. always 0. B. always > 0. C. always ≤ 0. © 2016 Pearson Education Inc. Checkpoint • The work done by a dissipative force like friction is ... A. always 0. B. always > 0. C. always ≤ 0. fs exists in two cases: 1. There is no displacement at all (W = f s d cos θ = 0 ) 2. The displacement is perpenducular to fs , i.e. fs ⟂ v (W = 0 again) fk acts opposite to the object's displacement (W<0). Never W > 0 © 2016 Pearson Education Inc. Clicker Q • A car on a level road turns a quarter circle ccw. Here the static friction causes the centripetal acceleration. The work done by static friction is _____. A. positive B. negative C. zero © 2016 Pearson Education Inc. Clicker Q • A car on a level road turns a quarter circle ccw. Here the static friction causes the centripetal acceleration. The work done by static friction is _____. A. positive B. negative C. zero © 2016 Pearson Education Inc. Zero work • A weightlifter does no work on a barbell as long as he holds it stationary. No work?! Do you even lift bro? © 2016 Pearson Education Inc. Lowering the barbell to the floor: Slide 1 © 2016 Pearson Education Inc. Lowering the barbell to the floor: Slide 2 © 2016 Pearson Education Inc. Lowering the barbell to the floor: Slide 3 © 2016 Pearson Education Inc. Total work • The work done by the net force on a particle as it moves is called the total work Wtot. • The particle speeds up if Wtot > 0, slows down if Wtot < 0, and maintains the same speed if Wtot = 0. © 2016 Pearson Education Inc. Kinetic energy • The energy of motion of a particle is called kinetic energy: • We then extend this definition to any system of particles by adding up the kinetic energies of all the constituent particles: • Like work, the kinetic energy of a particle is a scalar quantity; it depends on only the particle’s mass and speed, not its direction of motion. • Kinetic energy can never be negative, and it is zero only when the particle is at rest. • The SI unit of kinetic energy is the joule. © 2016 Pearson Education Inc. Kinetic energy • Kinetic energy does not depend on the direction of motion. © 2016 Pearson Education Inc. Kinetic energy • Kinetic energy increases linearly with the mass of the object. © 2016 Pearson Education Inc. Kinetic energy • Kinetic energy increases with the square of the speed of the object. © 2016 Pearson Education Inc. The work-energy theorem • The work-energy theorem: The work done by the net force on a particle equals the change in the particle’s kinetic energy. © 2016 Pearson Education Inc. The work-energy theorem (proof) © 2016 Pearson Education Inc. The work-energy theorem: from Kinematics © 2016 Pearson Education Inc. The work-energy theorem: individual forces ≠ ΔKE1 + ΔKE2 + ... = ΔKE (there is only one change in KE of each object in the sysyem) Read LibreText 3.1 to see an example © 2016 Pearson Education Inc. Work and kinetic energy in composite systems • Vertival net force is zero. • The boy's palms (where the force F is applied) do not move while the force acts. • So, the work done by the external forces acting on the skater is zero. • But the skater’s kinetic energy changes nonetheless! • The explanation is that it’s not adequate to represent the boy as a single point mass. © 2016 Pearson Education Inc. Work and energy with varying forces • Many forces are not constant. • Suppose a particle moves along the x-axis from x1 to x2. © 2016 Pearson Education Inc. Work and energy with varying forces • We calculate the approximate work done by the force over many segments of the path. • We do this for each segment and then add the results for all the segments. © 2016 Pearson Education Inc. Work and energy with varying forces • The work done by the force in the total displacement from x1 to x2 is the integral of Fx from x1 to x2: • On a graph of force as a function of position, the total work done by the force is represented by the area under the curve between the initial and final positions. © 2016 Pearson Education Inc. Work done by a constant force © 2016 Pearson Education Inc. Springs and elastic force • The spring exerts no force at its equilibrium position (Unstretched). The spring exerts a force in the opposite direction to an extension or stretch, and a compression. ▪ The minus sign is the mathematical indication of a restoring force. ▪ The constant k is called the spring constant of the spring. © 2016 Pearson Education Inc. Stretching a spring • The force required to stretch a spring a distance x is proportional to x: Fon spring, x = kx. • The area under the graph represents the work done on the spring to stretch it a distance X: Won spring = 1/2 kX 2. © 2016 Pearson Education Inc. Stretching a spring • The force applied on the hand when the spring is streched to a distance x: Fon hand, x = -kx. • The work done on the hand (by the spring) when the spring is streched to a distance X: Won hand = -1/2 kX 2. +x Fspring on hand, x = -kx © 2016 Pearson Education Inc. Spring's work (general case) ▪ Shown is a spring acting on an object as it moves from si to sf. ▪ The spring’s work can be computed by integrating: ▪ The solution to this integral is © 2016 Pearson Education Inc. Work–energy theorem for motion along a curve (LibreText 3.1: Line integral) • A particle moves along a curved path from point P1 to P2, acted on by a force that varies in magnitude and direction. • The work can be found using a line integral: © 2016 Pearson Education Inc. Power • Power is the rate at which work is done. • Average power is: • Instantaneous power is: • The SI unit of power is the watt (1 W = 1 J/s), but another familiar unit is the horsepower (1 hp = 746 W). © 2016 Pearson Education Inc. Power: Lifting a box slowly © 2016 Pearson Education Inc. Power: Lifting a box quickly © 2016 Pearson Education Inc. Power • In mechanics we can also express power in terms of force and velocity: (for a constant force we can write the following) • See LibreText example 3.3.1 © 2016 Pearson Education Inc. Power • Here is a one-horsepower (746-W) propulsion system. © 2016 Pearson Education Inc. Example 1 © 2016 Pearson Education Inc. Example 1 (continued) © 2016 Pearson Education Inc. Example 2 If the toy car makes it to the apex of the loop with an appropriate speed such that it is still (even barely) in contact with the track (N>0), it already overcame the biggest challenge not to fall off the track. So it is good to go for the rest of the revolution and can successfully complete the loop. The top of the loop is the hardest place to keep it up and the first and most likely place that the car's lack of kinetic energy may get it into trouble. © 2016 Pearson Education Inc. Example 2 (continued) © 2016 Pearson Education Inc. Example 3 © 2016 Pearson Education Inc. Example 3 (continued) © 2016 Pearson Education Inc.
