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**Unformatted text preview: **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 μ 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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