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Unformatted text preview: CHAPTER 8 Potential energy and conservation of energy 1. Work and conservative forces A conservative force is one that does work independently of the path taken to bring an object from point A to point B . A B F F Fig. 1. Path independence of work done by a conservative force. An example of conservative force is the weight: the work done by weight in going from A to B can be calculated as: W weight = integraldisplay path F d r = integraldisplay y f y (- mg ) dy =- mg ( y f- y ) (1) since the force is just along the y axis, and the magnitude of the force doesnt change with position. The fact that the work is path-independent is obtained by the fact that, in the above equation, there are only the initial and final points of the trajectory. Another example is the elastic force, for which we showed that W elastic = integraldisplay path F d r = integraldisplay x f x o- kxdx = 1 2 kx 2 o- 1 2 kx 2 f (2) A corollary of the path-independence of work done by conservative forces is that the work done by a conservative force along a closed path is null: integraldisplay closed path F d r = 0 (closed path) (3) 2 2. Potential energy For conservative forces one can define a potential energy U as W ( A B ) = U A- U B =- U (4) where W ( A B ) is the work done by the conservative force in going from A to B . Notice the negative sign between U and W , which is part of the definition. Equation 4 can be applied to any force which has the characteristic of being conservative, i.e., its work does not depend on the specific path. 2.1. Gravitational potential energy We have shown in the example above that the gravitational force near the surface of the Earth (weight) is a conservtive force. For a path from y A to y B the work done by the weight is W ( A B ) =- mg ( y B- y A ). Accordingly, the potential energy is defined as U G = mg y (5) It is important to realize that only changes in potential energy are meaningful, not the actual value of U . Usually, one fixes the value of U at a reference point, say U = 0 on the ground ( y = 0), and then considers the potential energy values with respect to the reference point. The reference point is arbitrary. If we chose U = 0 on the ground, then we can give an absolute value for the gravitational potential energy: braceleftBigg U G = mgh assuming U=0 at h =0 (6) The physical meaning of potential energy is that of an object possessing energy by virtue of its position . At a position h above the ground, the gravitational force can do a positive work in going back to the ground, W ( h 0) = mg h > 0. This is why the negative sign in Equation 4....
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