Chapter 8 Potential Energy and Conservation of Energy. Copyright 2010 Pearson Education, Inc.

Transcription

1 Chapter 8 Potential Energy and Conservation of Energy

2 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; Example: paint can in Problem 7.8. This is not true for a nonconservative force

3 8-1 Conservative and Nonconservative Forces Work done by gravity on a closed path is zero:

4 8-1 Conservative and Nonconservative Forces Work done by friction on a closed path is not zero:

5 8-1 Conservative and Nonconservative Forces The work done by a conservative force is zero on any closed path:

6 8- 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)

7 8- The Work Done by Conservative Forces Gravitational potential energy:

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

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10 Picture the Problem: The Hawkmoth wing behaves as an ideal spring. Strategy: Use Hooke s Law (equation 6-4) to find the force constant of the wing, then use equation 8-5 to find the energy stored. Combine both equations to find the force required to store twice the energy. Solution: 1. (a) Solve equation 6-4 for k: k F N 0.65 N/m 0.63 N/m x m. (b) Use equation 8-5 to find U: (c) Solve equation 8-5 for x, taking the negative root because the wing is depressed in the negative direction as in step 1: U kx N/m m J 7. J U U x x k k 4. Use Hooke s Law to find the force: F kx k U k Uk J 0.65 N/m N 4. mn 10

11 8-3 Conservation of Mechanical Energy Definition of mechanical energy: (8-6) Using this definition and considering only conservative forces, we find: Or equivalently:

12 8-3 Conservation of Mechanical Energy Energy conservation can make kinematics problems much easier to solve:

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14 Strategy: The work done by gravity equals the change in kinetic energy according to equation 7-7. The work done by gravity is always W = mgh as indicated in Example 7- and Conceptual Checkpoint 7-1. Solution: 1. (a) The work done by gravity on the pine cone equals the increase in its kinetic energy. Set the W K mgh mv energies equal and solve for v: v gh m/s 16 m 18 m/s. (b) Air resistance did negative work because the speed and therefore the kinetic energy of the pine cone when it landed was reduced. Air resistance removed energy from the pine cone. Insight: Kinetic friction always does negative work because the force is always opposite to the direction of motion. 14

15 8-4 Work Done by Nonconservative Forces In this example, the nonconservative force is water resistance:

16 8-4 Work Done by Nonconservative Forces In the presence of nonconservative forces, the total mechanical energy is not conserved: Solving, (8-9)

17 Calculate total work from A to B k = 480 N/m; m =.7 kg; µ = ) along path ) Directly from A to B 17

18 Picture the Problem: The physical situation is depicted at right. 1 Strategy: Use Wsp kx i xf for the work done by the spring. That way the work will always be negative if you start out at xi 0 because the spring force will always be in the opposite direction from the stretch or compression. The work done by kinetic friction is Wfr kmgd, where d is the distance the box is pushed irregardless of direction, because the friction force always acts in a direction opposite the motion. Solution: 1. (a) Sum the work done by the spring for each segment of path : W k x x x x W 1 sp sp 480 N/m m 0.00 m 0.00 m J 0 J J. Sum the work done by friction for each segment of path : 3. (b) Sum the work done by the spring for the direct path from A to B: 4. Sum the work done by friction for the direct path from A to B: W mg d d fr k kg 9.81 m/s m 0.5 J W k x x 1 sp A B 480 N/m m J 1 Wfr kmgd kg 9.81 m/s 0.00 m J 18

19 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:

20 8-5 Potential Energy Curves and Equipotentials The potential energy curve for a spring:

21 8-5 Potential Energy Curves and Equipotentials Contour maps are also a form of potential energy curve: