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Chapter 8 Potential Energy and Conservation of Energy In this chapter we will introduce the following concepts: Potential Energy Conservative and non-conservative forces Mechanical Energy Conservation of Mechanical Energy The conservation of energy theorem will be used to solve a variety of problems As was done in Chapter 7 we use scalars such as work, kinetic energy, and mechanical energy rather than vectors. Therefore the approach is mathematically simpler. (8-1) A B g v o h v o Work and Potential Energy: A tomato of mass m is taken together with the earth as the system we study. The tomato is thrown upwards with initial speed v o at point A. Under the gravitational force, it slows down and stops at point B. Then it falls back to point A and reaches the original speed v o . During the trip from A to B the gravitational force F g does work U W = - (8-2) A A B B k m k F S F S v o v o m Consider our system with the mass m attached to a spring of spring constant k . The mass has an initial speed v o at point A. Under the spring force F S m slows down and stops at point B (with a spring compression x ). Then the mass reverses the direction of its motion and reaches point A. Its speed reaches the original value v o . During the trip from A to B the spring force F s does work U W = - (8-3) m m A B v o f f x d Conservative and non-conservative forces. The gravitational force and the spring force are called conservative because they can transfer energy between kinetic energy and potential energy with no loss. Frictional and drag forces are called non-conservative for reasons that are explained below. Consider a block of mass m on the horizontal floor . The block starts to move with initial speed v o at point A. The coefficient of kinetic friction between the floor and the block is k . The block will slow down and stop at point B after traveling a distance d . During the trip from A to B the frictional force does work (8-4) Closed-loop test : A force is conservative if the net work done on a particle during a round trip is always equal to zero (see fig.b). In the examples of the tomato-earth and mass-spring system W net = W a1b + W b2a = 0 (because ) So gravitational force and spring force are conservative forces! Next, we prove the path independence of work done by a conservative force Path Independence of Conservative Forces How to decide whether a force is conservative or non-conservative? net W = (8-5) Thus, the work done by a conservative force does not depend on the path taken but only on the initial and final points choose a simple path 1 2 , = - =- =- a b a b b a b a U W W U U W U U 1 2 = =- a b a b a b W W U U Path-independence of work done by conservative forces Work done by conservative forces ( W = - U ) is path-independent, it results in changes in the object's potential energy. ...
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