Chapter 8Potential Energy and Conservation of EnergyIn this chapter we will introduce the following concepts:Potential EnergyConservative and non-conservative forcesMechanical EnergyConservation of Mechanical EnergyThe conservation of energy theoremwill be used to solve a variety of problemsAs 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)
ABgvohvoWork and Potential Energy:A tomato of mass mis taken together with the earth as the system we study. The tomato is thrown upwards with initial speed voat 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 vo.During the trip from A to B the gravitational force Fgdoes work UW∆= -(8-2)
AABBkmkFSFSvovomConsider our system with the mass mattached to a spring of spring constant k. The mass has an initial speed voat point A. Under the spring force FS mslows 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 vo.During the trip from A to B the spring force Fsdoes work UW∆= -(8-3)
mmABvoffxdConservative 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 vo at point A. The coefficient of kinetic friction between the floor and the block is μk. 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 Wnet= Wa1b+ Wb2a= 0 (because )So gravitational force and spring force are conservative forces!Next, we prove the path independenceof work done by a conservative force Path Independence of Conservative ForcesHow to decide whether a force is conservative or non-conservative?0netW=(8-5)Thus, the work done by a conservative force does notdepend on the path taken but only on the initial and final points choose a simple path12,∆= -→=-=-a babb abaUWWUU WUU12==-a ba babWWUU
Path-independence of work done by conservative forcesWork done by conservative forces(W = - ∆U) is path-independent, it results in changes in the object's potential energy.
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