Lecture 15 Potential Energy and Force Fundamental theorem of calculus In One dimension the potential energy difference is de nite as This is just the interrogation of Newton s Second Law in the x Direction Putting the math and physics together yields The x component of the force is the negative derivative of the potential energy Worked Example Potential Energy and Force Consider the potential energy associated wit a spring U x kx where is the displacement of the spring from its equilibrium The negative slope of the plot of U x vs x is the x component of the force and is given by Group Problem Potential Energy and Force Consider the gravitational potential energy function associated with the inverse square Universal Law of Gravitation a Plot U r vs r b Determine the radial component F r of the gravitational force c Make a new plot of F r vs r next to your plot of U r vs r Energy Diagram Zero Non conservative Work Choose zero point for potential energy U x 0 0 Potential energy function U x kx U x 0 0 Mechanical energy is represented by a horizontal line since it is constant E K x U x mv kx Kinetic energy is difference between mechanical energy and potential energy K x E U x Discussion Energy Diagram The gure above shows a graph of potential energy U x verses position for a particle executing one dimensional motion along the x axis The total mechanical energy of the system is indicated by the dashed line At t 0 the particle is somewhere between points A and G For later times answer the following questions a At which point s will the magnitude of the force be a maximum b At which point s will the kinetic energy be a maximum At points A and G At point C c At which point s will the velocity be zero At points A and G d At which point s will the force be zero At points C D and E
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