Copyright © 2010 Pearson Education, Inc. Lecture Outline Chapter 8 Physics, 4 th Edition James S. Walker

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

Copyright © 2010 Pearson Education, Inc. 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

Copyright © 2010 Pearson Education, Inc. 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

Copyright © 2010 Pearson Education, Inc. 8-1 Conservative and Nonconservative Forces Work done by gravity on a closed path is zero: Work done by gravity on a closed path is zero Gravity does no work on the two horizontal segments of the path. On the two vertical segments of the amounts of work done are equal in magnitude but opposite in sign. There fore, the total work done by gravity on this- or any – closed path is zero. W total = 0 + (-mgh) mgh = 0

Copyright © 2010 Pearson Education, Inc. 8-1 Conservative and Nonconservative Forces Work done by friction on a closed path is not zero: Work done by friction on a closed path is nonzero The work done by friction when an object moves through a distance d is -µ k mgd. Thus, the total work done by friction on a closed path is nonzero. In this case, it is equal to - 4µ k mgd.

Copyright © 2010 Pearson Education, Inc. 8-1 Conservative and Nonconservative Forces The work done by a conservative force is zero on any closed path: The work done by a conservative force is independent of path Considering paths 1 and 2, we see that W 1 + W 2 = 0, or W 2 = W 1. From paths 1 and 3, however, we see that W 1 + W 3 = 0, or W 3 = -W 1. It follows, then, that W 3 = W 2, since they are both equal to –W 1 ; hence the work done in going from A to B in independent of the path.

Copyright © 2010 Pearson Education, Inc. 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) Potential Energy, U When a conservative force does an amount of work W c, the corresponding potential energy U is changed according to the following definition: The work done by a conservative force is equal to the negative of the change in potential energy. For example, when an object falls, gravity does positive work on it and its potential energy decreases. Similarly, when an object is lifted, gravity does negative work and the potential energy increases.

Copyright © 2010 Pearson Education, Inc. 8-2 The Work Done by Conservative Forces Gravitational potential energy: Gravitational Potential energy A person drops from a diving board into a swimming pool. The driving board is at the height y, and the surface of the water is at y = 0. We choose the gravitational potential energy to be zero at y = 0; hence, the potential energy is mgy at the diving board.

Copyright © 2010 Pearson Education, Inc. 8-2 The Work Done by Conservative Forces Gravitational potential energy: Gravitational Potential energy A candy bar called the Mountain Bar has a calorie content of 212 Cal = 212 Kcal, which is equivalent to an energy of 8.87 x 10 5 J. If an 81.0-kg mountain climber eats a Mountain Bar and marginally converts it all to potential energy, what gain of altitude would be possible? U = mgh h = U/mg = 8.87 x 10 5 J/ (81.0 kg) (9.81 m/s 2 ) = 1120 m h = 1120 m

Copyright © 2010 Pearson Education, Inc. 8-2 The Work Done by Conservative Forces Springs:(8-4) Consider a spring that is stretched from its equilibrium position a distance x. The work required to cause this stretch is W = 1/2kx 2. If the spring is released- and allowed to move from the stretched position back to the equilibrium position- it will do the same work, 1/2kx 2. From our definition of potential energy, then, we see that Note that in this case U f is the potential energy when the spring is at x = 0 and U i is the potential energy when the spring is stretched by the amount x.

Copyright © 2010 Pearson Education, Inc. 8-3 Conservation of Mechanical Energy Definition of mechanical energy: (8-6) Using this definition and considering only conservative forces, we find: Or equivalently: In systems with conservative forces only, the mechanical energy E is conserved; that is, E = U + K = constant.

Copyright © 2010 Pearson Education, Inc. 8-3 Conservation of Mechanical Energy Energy conservation can make kinematics problems much easier to solve: 9a). A set of keys fall to the floor. Ignoring frictional forces, we know that the mechanical energy at points I and f must be equal; E i = E f. Using this condition, we can find the speed of the keys just before they land. (b). The same physical situation as in part (a), except this time we have chosen y = 0 to be at the point where the keys are dropped. As before, we set E i = E f to find the speed of the keys just before they land.

Copyright © 2010 Pearson Education, Inc. 8-4 Work Done by Nonconservative Forces In the presence of nonconservative forces, the total mechanical energy is not conserved: Solving, (8-9)

Copyright © 2010 Pearson Education, Inc. 8-4 Work Done by Nonconservative Forces In this example, the nonconservative force is water resistance:

Copyright © 2010 Pearson Education, Inc. 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:

Copyright © 2010 Pearson Education, Inc. 8-5 Potential Energy Curves and Equipotentials The potential energy curve for a spring: A mass on a spring (a) A spring is stretched by an amount A, giving it a potential energy of U = 1/2kA 2. (b) The potential energy curve, U = 1/2kx 2, for the spring in (a). Because the mass starts at rest, its initial mechanical energy is E o = 1/2kA 2. The mass oscillates between x = A and x = - A

Copyright © 2010 Pearson Education, Inc. 8-5 Potential Energy Curves and Equipotentials Contour maps are also a form of potential energy curve: A contour map A small mountain (top, in side view) is very steep on the left, more gently sloping on the right. A contour map of this mountain (bottom) shows a series of equal altitude contour lines from 50 ft. to 450 ft. Notice that the contour lines are packed close together where the terrain is steep, but are widely spaces where it is more level.

Copyright © 2010 Pearson Education, Inc. 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

Copyright © 2010 Pearson Education, Inc. 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 kx 2

Copyright © 2010 Pearson Education, Inc. 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