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12B x,v,a eqns for SHM

# 12B x,v,a eqns for SHM - SHM Mathematical Model Physics 1C...

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3/31/2011 1 Physics 1C Lecture 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 Gopnik SHM: Mathematical Model Equations of motion for SHM: Remember, simple harmonic motion is not uniformly accelerated motion SHM: Mathematical Model The maximum values of velocity and acceleration for an object in SHM: The velocity is 90 o out of phase with the displacement and the acceleration is 180 o out of phase with the displacement SHM Example 1 Initial conditions at t = 0 are: x (0)= A and v (0) = 0 This means f = 0 The acceleration reaches extremes of w 2 A The velocity reaches extremes of w A SHM Example 2 Initial conditions at t = 0 are: x (0)= 0 and v (0) = V i This means f = - p /2 The graph is shifted one- quarter cycle to the right compared to the graph of x (0) = A Mass-Spring Energy Let’s assume a spring -mass system is moving on a frictionless surface This is an isolated system, therefore the total energy of the system is constant The kinetic energy is: KE = 1/2 mv 2 = 1/2 m w 2 A 2 sin 2 ( w t + f ) The elastic potential energy is: PE = 1/2 kx 2 = 1/2 kA 2 cos 2 ( w t + f ) The total energy is: KE + PE = 1/2 kA 2

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