Chapter 8 Potential Energy and Conservation of Energy

1. Chapter 8 Potential Energy and Conservation of Energy

2. Reading and Review

3. Force and Work a) one force b) two forces c) three forces d) four forces e) no forces are doing work A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box?

4. displacement N T f mg Force and Work a) one force b) two forces c) three forces d) four forces e) no forces are doing work A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box? Any force not perpendicularto the motion will do work: N doesno work T doespositivework f does negative work mg does negative work

5. Free Fall I Two stones, one twice the mass of the other, are dropped from a cliff. Just before hitting the ground, what is the kinetic energy of the heavy stone compared to the light one? a) quarter as much b) half as much c) the same d) twice as much e) four times as much

6. Free Fall I Two stones, one twice the mass of the other, are dropped from a cliff. Just before hitting the ground, what is the kinetic energy of the heavy stone compared to the light one? a) quarter as much b) half as much c) the same d) twice as much e) four times as much Consider the work done by gravity to make the stone fall distance d: DKE = Wnet = F d cosq DKE = mg d Thus, the stone with thegreater masshas thegreater KE, which istwiceas big for the heavy stone. Follow-up: How do the initial values of gravitational PE compare?

7. Free Fall II a) quarter as much b) half as much c) the same d) twice as much e) four times as much In the previous question, just before hitting the ground, what is the final speed of the heavy stone compared to the light one?

8. Free Fall II a) quarter as much b) half as much c) the same d) twice as much e) four times as much In the previous question, just before hitting the ground, what is the final speed of the heavy stone compared to the light one? All freely falling objects fall at the same rate, which is g. Because the acceleration is the same for both, and the distance is the same, then the final speeds will be the same for both stones.

9. Topics 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

10. 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

11. 8-1 Conservative and Nonconservative Forces Work done by gravity on a closed path is zero:

12. 8-1 Conservative and Nonconservative Forces Work done by friction on a closed path is not zero:

13. 8-1 Conservative and Nonconservative Forces The work done by a conservative force is zero on any closed path:

14. 8-2 The Work Done by Conservative Forces If we pick up a ball and put it on the shelf, we have done work on the ball. We can get that energy back if the ball falls back off the shelf; in the meantime, we say the energy is stored as potential energy. (8-1)

15. 8-2 The Work Done by Conservative Forces Gravitational potential energy:

16. Sign of the Energy II Is it possible for the gravitational potential energy of an object to be negative? a) yes b) no

17. Sign of the Energy II Is it possible for the gravitational potential energy of an object to be negative? a) yes b) no Gravitational PE is mgh, where height h is measured relative to some arbitrary reference level where PE = 0. For example, a book on a table has positive PE if the zero reference level is chosen to be the floor. However, if the ceiling is the zero level, then the book has negative PE on the table. Only differences (or changes) in PE have any physical meaning.

18. Question 8.2KE 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? a) only B b) only C c) A, B, and C d) only A and C e) only B and C A) skier’s PE B) skier’s change in PE C) skier’s final KE

19. Question 8.2KE 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? a) only B b) only C c) A, B, and C d) only A and C e) 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 differenceDPE 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?

20. 8-2 The Work Done by Conservative Forces (8-4) Springs:

21. 8-3 Conservation of Mechanical Energy Definition of mechanical energy: (8-6) Using this definition and considering only conservative forces, we find: Or equivalently:

22. 8-3 Conservation of Mechanical Energy Energy conservation can make kinematics problems much easier to solve:

23. Question 8.2KE 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? a) only B b) only C c) A, B, and C d) only A and C e) only B and C A) skier’s PE B) skier’s change in PE C) skier’s final KE

24. 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? a) only B b) only C c) A, B, and C d) only A and C e) 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 differenceDPE 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?

25. Example: Two water slides are shaped differently, but start at the same height h and are of equal length. Two rides, Paul and Kathy, start from rest at the same time on different slides. • Which is travelling faster at the bottom? • Which makes it to the bottom first?

26. 8-4 Work Done by Nonconservative Forces In the presence of nonconservative forces, the total mechanical energy is not conserved: Solving, (8-9)

27. 8-4 Work Done by Nonconservative Forces In this example, the nonconservative force is water resistance:

28. 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:

29. 8-5 Potential Energy Curves and Equipotentials The potential energy curve for a spring:

30. 8-5 Potential Energy Curves and Equipotentials Contour maps are also a form of potential energy curve:

31. 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

32. 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

33. 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