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PH 201 Dr. Cecilia Vogel Lecture 20

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Potential Energy Graph It is very often useful to look at a graph of potential energy as a function of position. Why? Can determine forces and subsequent motion of an object subjected to this potential energy.

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U(x) Graph and Energy If there are no external forces acting on the system, and no energy is dissipated, then energy is conserved. K+U = E = constant Where U is small, K = E-U is large object moves fast. Where U is large, but less than E, K is small object moves slowly.

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U(x) Graph and Energy If energy is conserved, K+U = E = constant Where U = E, K = 0 object stops (momentarily) turning point Where U > E, K = E-U object cannot go there with that amount of energy! “forbidden region” (note — quantum) K can’t be negative!

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Pendulum Example The potential energy of a pendulum can be plotted as a function of angle. U = mgh h = L-Lcos U = mgL(1-cos ) h L L cos

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Pendulum U( ) U = mgL(1-cos ) If it starts from rest at 10 o, How much energy does it have? Will it go right or left? It can’t go right. U can’t be larger than 30J. As it goes left, what happens to the speed? AS U gets smaller, K gets bigger — moves faster. 10 o E = U i +K i = 30J 5o5o 0 10 20 30 40 -5 o -10 o U (J)

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Pendulum U( ) U = mgL(1-cos ) Starts from rest at 10 o, E=30J As it goes left, moves faster. Until… =0, it goes fastest, K=30J continues left (inertia) As U gets bigger, K gets smaller, slows down. Until… 10 o 5o5o 0 10 20 30 40 -5 o -10 o U (J)

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Pendulum U( ) U = mgL(1-cos ) Starts from rest at 10 o, with E=30J As it goes left, moves faster. Until =0, where it continues, but now slowing down, until it stops at = -10 o, where U=30J, K=0 This is turning point. Then starts back right… back and forth… 10 o 5o5o 0 10 20 30 40 -5 o -10 o U (J)

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Graph Ordinate Graph of U vs. x is not a picture of the motion. The y-axis is potential energy, not height. For example, for a mass on a spring sliding along a surface, U= ½kx 2. The graph looks like this, even though the mass goes along a line, never up or down.

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U(x) Graph and Forces Consider the graphs of potential energy and force for gravity and a spring GRAVITYSPRING U= ½kx 2 x F= - kx x U= mgy y F= - mg y NOTE: F = opposite the slope of U graph.

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Pendulum Forces When the pendulum is at -5 o, and moving left (-) since the slope is (-), the force is (+), so it is slowing down. When the pendulum is at +5 o, and moving left (-) since the slope is (+), the force is (-), so it is speeding up. 10 o 5o5o 0 10 20 30 40 -5 o -10 o U (J)

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Stable and Unstable Equilibrium Suppose the force a potential F = -d U /dx Zero force implies Unstable equilibrium means if you displace it just a bit it will move

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Stable and Unstable Equilibrium If you displace it just a bit what happens?

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