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**Unformatted text preview: **Chapter 25 Potential Energy, Conservation of Energy, Power 171 25 Potential Energy, Conservation of Energy, Power The work done on a particle by a force acting on it as that particle moves from point A to point B under the influence of that force, for some forces, does not depend on the path followed by the particle. For such a force there is an easy way to calculate the work done on the particle as it moves from point A to point B . One simply has to assign a value of potential energy (of the particle 1 ) to point A (call that value U A ) and a value of potential energy to point B (call that value U B ). One chooses the values such that the work done by the force in question is just the negative of the difference between the two values. ) ( A B U U W − − = U W Δ − = (25-1) A B U U U − = Δ is the change in the potential energy experienced by the particle as it moves from point A to point B . The minus sign in equation 25-1 ensures that an increase in potential energy corresponds to negative work done by the corresponding force. For instance for the case of near- earth’s-surface gravitational potential energy, the associated force is the gravitational force, a.k.a. the gravitational force. If we lift an object upward near the surface of the earth, the gravitational force does negative work on the object since the (downward) force is in the opposite direction to the (upward) displacement. At the same, time, we are increasing the capacity of the particle to do work so we are increasing the potential energy. Thus, we need the “ − “ sign in U Δ − = W to ensure that the change in potential energy method of calculating the work gives the same algebraic sign for the value of the work that the force-along-the path times the length of the path gives. Note that in order for this method of calculating the work to be useful in any case that might arise, one must assign a value of potential energy to every point in space where the force can act on a particle so that the method can be used to calculate the work done on a particle as the particle moves from any point A to any point B . In general, this means we need a value for each of an infinite set of points in space. This assignment of a value of potential energy to each of an infinite set of points in space might seem daunting until you realize that it can be done by means of a simple algebraic expression. For instance, we have already written the assignment for a particle of mass m 2 for the case of the universal gravitational force due to a particle of mass m 1 . It was equation 17-5: r 2 1 m m G U − = 1 The potential energy is actually the potential energy of the system consisting of the particle, whatever the particle is interacting with, and the relevant field. For instance, if we are talking about a particle in the gravitational field of the earth, the potential energy under discussion is the potential energy of the earth plus particle and gravitational field of the earth plus particle. field of the earth plus particle....

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