Potential Energy

Physics 7C lecture 07 • Potential Energy Thursday October 17, 8:00 AM – 9:20 AM Engineering Hall 1200

Goals for Chapter 7 • To use gravitational potential energy in vertical motion • To use elastic potential energy for a body attached to a spring • To solve problems involving conservative and nonconservative forces • To determine the properties of a conservative force from the corresponding potential-energy function • To use energy diagrams for conservative forces

Introduction • How do energy concepts apply to the descending duck? • We will see that we can think of energy as being stored and transformed from one form to another.

Gravitational potential energy • Energy associated with position is called potential energy. • Gravitational potential energy is Ugrav = mgy. • Figure 7.2 at the right shows how the change in gravitational potential energy is related to the work done by gravity.

Q7.1 A piece of fruit falls straight down. As it falls, A. the gravitational force does positive work on it and the gravitational potential energy increases. B. the gravitational force does positive work on it and the gravitational potential energy decreases. C. the gravitational force does negative work on it and the gravitational potential energy increases. D. the gravitational force does negative work on it and the gravitational potential energy decreases.

A7.1 A piece of fruit falls straight down. As it falls, A. the gravitational force does positive work on it and the gravitational potential energy increases. B. the gravitational force does positive work on it and the gravitational potential energy decreases. C. the gravitational force does negative work on it and the gravitational potential energy increases. D. the gravitational force does negative work on it and the gravitational potential energy decreases.

The conservation of mechanical energy • The total mechanical energy of a system is the sum of its kinetic energy and potential energy. • A quantity that always has the same value is called a conserved quantity. • When only the force of gravity does work on a system, the total mechanical energy of that system is conserved. This is an example of the conservation of mechanical energy. Figure 7.3 below illustrates this principle.

An example using energy conservation • Refer to Figure 7.4 below as you follow Example 7.1. • Notice that the result does not depend on our choice for the origin.

Q7.2 You toss a 0.150-kg baseball straight upward so that it leaves your hand moving at 20.0 m/s. The ball reaches a maximum height y2. What is the speed of the ball when it is at a height of y2/2? Ignore air resistance. v2 = 0 y2 v1 = 20.0 m/s m = 0.150 kg y1 = 0 A. 10.0 m/s B. less than 10.0 m/s but greater than zero C. greater than 10.0 m/s D. not enough information given to decide

A7.2 You toss a 0.150-kg baseball straight upward so that it leaves your hand moving at 20.0 m/s. The ball reaches a maximum height y2. What is the speed of the ball when it is at a height of y2/2? Ignore air resistance. v2 = 0 y2 v1 = 20.0 m/s m = 0.150 kg y1 = 0 A. 10.0 m/s B. less than 10.0 m/s but greater than zero C. greater than 10.0 m/s D. not enough information given to decide

Q7.3 As a rock slides from A to B along the inside of this frictionless hemispherical bowl, mechanical energy is conserved. Why? (Ignore air resistance.) A. The bowl is hemispherical. B. The normal force is balanced by centrifugal force. C. The normal force is balanced by centripetal force. D. The normal force acts perpendicular to the bowl’s surface. E. The rock’s acceleration is perpendicular to the bowl’s surface.

A7.3 As a rock slides from A to B along the inside of this frictionless hemispherical bowl, mechanical energy is conserved. Why? (Ignore air resistance.) A. The bowl is hemispherical. B. The normal force is balanced by centrifugal force. C. The normal force is balanced by centripetal force. D. The normal force acts perpendicular to the bowl’s surface. E. The rock’s acceleration is perpendicular to the bowl’s surface.

Q7.4 The two ramps shown are both frictionless. The heights y1 and y2 are the same for each ramp. A block of mass m is released from rest at the left-hand end of each ramp. Which block arrives at the right-hand end with the greater speed? A. the block on the curved track B. the block on the straight track C. Both blocks arrive at the right-hand end with the same speed. D. The answer depends on the shape of the curved track.

A7.4 The two ramps shown are both frictionless. The heights y1 and y2 are the same for each ramp. A block of mass m is released from rest at the left-hand end of each ramp. Which block arrives at the right-hand end with the greater speed? A. the block on the curved track B. the block on the straight track C. Both blocks arrive at the right-hand end with the same speed. D. The answer depends on the shape of the curved track.

When forces other than gravity do work • Refer to Problem-Solving Strategy 7.1. • Follow the solution of Example 7.2.

Work and energy along a curved path • We can use the same expression for gravitational potential energy whether the body’s path is curved or straight.

Energy in projectile motion • Two identical balls leave from the same height with the same speed but at different angles. • Follow Conceptual Example 7.3 using Figure 7.8.

Motion in a vertical circle with no friction • Follow Example 7.4 using Figure 7.9.

Motion in a vertical circle with friction • Revisit the same ramp as in the previous example, but this time with friction. • Follow Example 7.5 using Figure 7.10.

Moving a crate on an inclined plane with friction • Follow Example 7.6 using Figure 7.11 to the right. • Notice that mechanical energy was lost due to friction.

Work done by a spring • Figure 7.13 below shows how a spring does work on a block as it is stretched and compressed.

Elastic potential energy • A body is elastic if it returns to its original shape after being deformed. • Elastic potential energy is the energy stored in an elastic body, such as a spring. • The elastic potential energy stored in an ideal spring is Uel = 1/2 kx2. • Figure 7.14 at the right shows a graph of the elastic potential energy for an ideal spring.

Situations with both gravitational and elastic forces • When a situation involves both gravitational and elastic forces, the total potential energy is the sum of the gravitational potential energy and the elastic potential energy: U = Ugrav + Uel. • Figure 7.15 below illustrates such a situation. • Follow Problem-Solving Strategy 7.2.

Motion with elastic potential energy • Follow Example 7.7 using Figure 7.16 below. • Follow Example 7.8.

Q7.5 A block is released from rest on a frictionless incline as shown. When the moving block is in contact with the spring and compressing it, what is happening to the gravitational potential energy Ugrav and the elastic potential energy Uel? A. Ugrav and Uel are both increasing. B. Ugrav and Uel are both decreasing. C. Ugrav is increasing; Uel is decreasing. D. Ugrav is decreasing; Uel is increasing. E. The answer depends on how the block’s speed is changing.

A7.5 A block is released from rest on a frictionless incline as shown. When the moving block is in contact with the spring and compressing it, what is happening to the gravitational potential energy Ugrav and the elastic potential energy Uel? A. Ugrav and Uel are both increasing. B. Ugrav and Uel are both decreasing. C. Ugrav is increasing; Uel is decreasing. D. Ugrav is decreasing; Uel is increasing. E. The answer depends on how the block’s speed is changing.

A system having two potential energies and friction • In Example 7.9 gravity, a spring, and friction all act on the elevator. • Follow Example 7.9 using Figure 7.17 at the right.

Conservative and nonconservative forces • A conservative force allows conversion between kinetic and potential energy. Gravity and the spring force are conservative. • The work done between two points by any conservative force • a) can be expressed in terms of a potential energy function. • b) is reversible. • c) is independent of the path between the two points. • d) is zero if the starting and ending points are the same. • A force (such as friction) that is not conservative is called a nonconservative force, or a dissipativeforce.

Frictional work depends on the path • Follow Example 7.10, which shows that the work done by friction depends on the path taken.

Q7.12 You push a block up an inclined ramp at a constant speed. There is friction between the block and the ramp. The rate at which the internal energy of the block and ramp increases is A. greater than the rate at which you do work on the block. B. the same as the rate at which you do work on the block. C. less than the rate at which you do work on the block. D. not enough information given to decide

A7.12 You push a block up an inclined ramp at a constant speed. There is friction between the block and the ramp. The rate at which the internal energy of the block and ramp increases is A. greater than the rate at which you do work on the block. B. the same as the rate at which you do work on the block. C. less than the rate at which you do work on the block. D. not enough information given to decide

Conservative or nonconservative force? • Follow Example 7.11, which shows how to determine if a force is conservative or nonconservative.

Conservation of energy • Nonconservative forces do not store potential energy, but they do change the internal energy of a system. • The law of the conservation of energy means that energy is never created or destroyed; it only changes form. • This law can be expressed as K + U + Uint = 0. • Follow Conceptual Example 7.12.

Q7.9 Fx The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? x 0 a A. U = 0 at x = a B. U is a maximum at x = a. C. U is a minimum at x = a. D. U is neither a minimum or a maximum at x = a, and its value at x = a need not be zero.

A7.9 Fx The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? x 0 a A. U = 0 at x = a B. U is a maximum at x = a. C. U is a minimum at x = a. D. U is neither a minimum or a maximum at x = a, and its value at x = a need not be zero.

Q7.10 Fx The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? x 0 a A. U = 0 at x = a B. U is a maximum at x = a. C. U is a minimum at x = a. D. U is neither a minimum or a maximum at x = a, and its value at x = a need not be zero.

A7.10 Fx The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? x 0 a A. U = 0 at x = a B. U is a maximum at x = a. C. U is a minimum at x = a. D. U is neither a minimum or a maximum at x = a, and its value at x = a need not be zero.

Q7.11 Fx The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx > 0 and dFx/dx < 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? x 0 a A. dU/dx > 0 at x = a B. dU/dx < 0 at x = a C. dU/dx = 0 at x = a D. Any of the above could be correct.

A7.11 Fx The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx > 0 and dFx/dx < 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? x 0 a A. dU/dx > 0 at x = a B. dU/dx < 0 at x = a C. dU/dx = 0 at x = a D. Any of the above could be correct.

Force and potential energy in one dimension • In one dimension, a conservative force can be obtained from its potential energy function using • Fx(x) = –dU(x)/dx • Figure 7.22 at the right illustrates this point for spring and gravitational forces. • Follow Example 7.13 for an electric force.

Force and potential energy in two dimensions • In two dimension, the components of a conservative force can be obtained from its potential energy function using • Fx = –U/dx and Fy = –U/dy • Follow Example 7.14 for a puck on a frictionless table.

Energy diagrams • An energy diagram is a graph that shows both the potential-energy function U(x) and the total mechanical energy E. • Figure 7.23 illustrates the energy diagram for a glider attached to a spring on an air track.

Force and a graph of its potential-energy function • Figure 7.24 below helps relate a force to a graph of its corresponding potential-energy function.

Q7.6 The graph shows the potential energy U for a particle that moves along the x-axis. The particle is initially at x = d and moves in the negative x-direction. At which of the labeled x-coordinates does the particle have the greatest speed? A. at x = a B. at x = b C. at x = c D. at x = d E. more than one of the above

A7.6 The graph shows the potential energy U for a particle that moves along the x-axis. The particle is initially at x = d and moves in the negative x-direction. At which of the labeled x-coordinates does the particle have the greatest speed? A. at x = a B. at x = b C. at x = c D. at x = d E. more than one of the above

Q7.7 The graph shows the potential energy U for a particle that moves along the x-axis. The particle is initially at x = d and moves in the negative x-direction. At which of the labeled x-coordinates is the particle slowing down? A. at x = a B. at x = b C. at x = c D. at x = d E. more than one of the above

A7.7 The graph shows the potential energy U for a particle that moves along the x-axis. The particle is initially at x = d and moves in the negative x-direction. At which of the labeled x-coordinates is the particle slowing down? A. at x = a B. at x = b C. at x = c D. at x = d E. more than one of the above

Q7.8 The graph shows the potential energy U for a particle that moves along the x-axis. At which of the labeled x-coordinates is there zero force on the particle? A. at x = a and x = c B. at x = b only C. at x = d only D. at x = b and d E. misleading question—there is a force at all values of x

A7.8 The graph shows the potential energy U for a particle that moves along the x-axis. At which of the labeled x-coordinates is there zero force on the particle? A. at x = a and x = c B. at x = b only C. at x = d only D. at x = b and d E. misleading question—there is a force at all values of x

Potential Energy