3/31/20111Physics 1CLecture 12B"Imagine if we taught baseball the way we teach science. Undergraduates might be allowed to reproduce famous historic baseball plays. But only in graduate school would they, at last, actually get to play a game."--Alison GopnikSHM: Mathematical ModelEquations of motion for SHM:Remember, simple harmonic motion is notuniformly accelerated motionSHM: Mathematical ModelThe maximum values of velocity and acceleration for an object in SHM:The velocity is 90oout of phase with the displacement and the acceleration is 180oout of phase with the displacementSHM –Example 1Initial conditions at t = 0are: x (0)= A andv (0) = 0This means f= 0The acceleration reaches extremes of w2AThe velocity reaches extremes of wASHM –Example 2Initial conditions at t = 0are: x (0)= 0andv (0) =ViThis means f= -p /2The graph is shifted one-quarter cycle to the right compared to the graph of x (0) = AMass-Spring EnergyLet’s assume a spring-mass system is moving on a frictionless surfaceThis is an isolated system, therefore the total energy of the system is constantThe kinetic energy is: KE= 1/2 mv 2= 1/2 mw2A2sin2(wt+ f)•The elastic potential energy is: PE= 1/2 kx 2= 1/2 kA2cos2(wt+ f)•The total energy is: KE+ PE= 1/2 kA 2
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