1. Potential Energy and Conservation of Energy
Chapter 8
Potential Energy and Conservation of Energy

2. Units of Chapter 8 Conservative and Nonconservative Forces
Potential Energy and the Work Done by Conservative Forces
Conservation of Mechanical Energy
Work Done by Nonconservative Forces
Potential Energy Curves and Equipotentials

3. 8-1 Conservative and Nonconservative Forces
Conservative force: the work it does is stored in the form of energy that can be released at a later time
Example of a conservative force: gravity
Example of a nonconservative force: friction
Also: the work done by a conservative force moving an object around a closed path is zero; this is not true for a nonconservative force

4. 8-1 Conservative and Nonconservative Forces
Work done by gravity on a closed path is zero:

5. 8-1 Conservative and Nonconservative Forces
Work done by friction on a closed path is not zero:

6. 8-1 Conservative and Nonconservative Forces
The work done by a conservative force is zero on any closed path:

7. Example
Calculate the work one by gravity as a 3.2 kg object is moved from point A to point B on the figure along paths 1, 2, and 3

8. 8-2 The Work Done by Conservative Forces
(8-1)

9. 8-2 The Work Done by Conservative Forces
Gravitational potential energy:

10. 8-2 The Work Done by Conservative Forces
Springs:
(8-4)

11. ConcepTest 8.4 Elastic Potential Energy
How does the work required to stretch a spring 2 cm compare with the work required to stretch it 1 cm?
1) same amount of work
2) twice the work
3) 4 times the work
4) 8 times the work

12. ConcepTest 8.4 Elastic Potential Energy
How does the work required to stretch a spring 2 cm compare with the work required to stretch it 1 cm?
1) same amount of work
2) twice the work
3) 4 times the work
4) 8 times the work
The elastic potential energy is 1/2 kx2. So in the second case, the elastic PE is 4 times greater than in the first case. Thus, the work required to stretch the spring is also 4 times greater.

13. Pre- and Post- Tests

14. 8-3 Conservation of Mechanical Energy
Definition of mechanical energy:
(8-6)
Using this definition and considering only conservative forces, we find:
Or equivalently:

15. 8-3 Conservation of Mechanical Energy
Energy conservation can make kinematics problems much easier to solve:

16. ConcepTest KE and PE
You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on?
1) only B
2) only C
3) A, B and C
4) only A and C
5) only B and C
A) skier’s PE B) skier’s change in PE C) skier’s final KE

17. ConcepTest KE and PE
You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on?
1) only B
2) only C
3) A, B and C
4) only A and C
5) only B and C
A) skier’s PE B) skier’s change in PE C) skier’s final KE
The gravitational PE depends upon the reference level, but the difference DPE does not! The work done by gravity must be the same in the two solutions, so DPE and DKE should be the same.
Follow-up: Does anything change physically by the choice of y = 0?

18. ConcepTest 8.6 Down the Hill
Three balls of equal mass start from rest and roll down different ramps. All ramps have the same height. Which ball has the greater speed at the bottom of its ramp?
4) same speed
for all balls 1 2 3

19. ConcepTest 8.6 Down the Hill
Three balls of equal mass start from rest and roll down different ramps. All ramps have the same height. Which ball has the greater speed at the bottom of its ramp?
4) same speed
for all balls 1 2 3
All of the balls have the same initial gravitational PE, since they are all at the same height (PE = mgh). Thus, when they get to the bottom, they all have the same final KE, and hence the same speed (KE = 1/2 mv2).
Follow-up: Which ball takes longer to get down the ramp?

20. ConcepTest 8.8a Water Slide I
Paul and Kathleen start from rest at the same time on frictionless water slides with different shapes. At the bottom, whose velocity is greater?
1) Paul
2) Kathleen
3) both the same

21. ConcepTest 8.8a Water Slide I
Paul and Kathleen start from rest at the same time on frictionless water slides with different shapes. At the bottom, whose velocity is greater?
1) Paul
2) Kathleen
3) both the same
Conservation of Energy:
Ei = mgH = Ef = 1/2 mv2
therefore: gH = 1/2 v2
Since they both start from the same height, they have the same velocity at the bottom.

22. Example
In the figure below, the water slide ends at a height of 1.50 m above the pool. If the person starts from rest at A and lands in the water at B, what is the height h of the slide?

23. 8-4 Work Done by Nonconservative Forces
In the presence of nonconservative forces, the total mechanical energy is not conserved:
Solving,
(8-9)

24. 8-4 Work Done by Nonconservative Forces
In this example, the nonconservative force is water resistance:

25. ConcepTest 8.10a Falling Leaves
You see a leaf falling to the ground with constant speed. When you first notice it, the leaf has initial total energy PEi + KEi. You watch the leaf until just before it hits the ground, at which point it has final total energy PEf + KEf. How do these total energies compare?
1) PEi + KEi > PEf + KEf
2) PEi + KEi = PEf + KEf
3) PEi + KEi < PEf + KEf
4) impossible to tell from
the information provided

26. ConcepTest 8.10a Falling Leaves
You see a leaf falling to the ground with constant speed. When you first notice it, the leaf has initial total energy PEi + KEi. You watch the leaf until just before it hits the ground, at which point it has final total energy PEf + KEf. How do these total energies compare?
1) PEi + KEi > PEf + KEf
2) PEi + KEi = PEf + KEf
3) PEi + KEi < PEf + KEf
4) impossible to tell from
the information provided
As the leaf falls, air resistance exerts a force on it opposite to its direction of motion. This force does negative work, which prevents the leaf from accelerating. This frictional force is a non-conservative force, so the leaf loses energy as it falls, and its final total energy is less than its initial total energy.
Follow-up: What happens to leaf’s KE as it falls? What net work is done?

27. 8-5 Potential Energy Curves and Equipotentials
The curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy:

28. 8-5 Potential Energy Curves and Equipotentials
The potential energy curve for a spring:

29. Summary of Chapter 8 Conservative forces conserve mechanical energy
Nonconservative forces convert mechanical energy into other forms
Conservative force does zero work on any closed path
Work done by a conservative force is independent of path
Conservative forces: gravity, spring

30. Summary of Chapter 8
Work done by nonconservative force on closed path is not zero, and depends on the path
Nonconservative forces: friction, air resistance, tension
Energy in the form of potential energy can be converted to kinetic or other forms
Work done by a conservative force is the negative of the change in the potential energy
Gravity: U = mgy
Spring: U = ½ kx2

31. Summary of Chapter 8
Mechanical energy is the sum of the kinetic and potential energies; it is conserved only in systems with purely conservative forces
Nonconservative forces change a system’s mechanical energy
Work done by nonconservative forces equals change in a system’s mechanical energy
Potential energy curve: U vs. position