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L01_a - Oscillations Review(Chapter 12 Oscillations motions...

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Oscillations: Review (Chapter 12) Oscillations: motions that are periodic in time (i.e. repetitive) o Swinging object (pendulum) o Vibrating object (spring, guitar string, etc.) o Part of medium (i.e. string, water) as wave passes by Oscillation requires o Restoring force (pushes “system” back toward equilibrium) o Inertia (keeps “system” going past equilibrium)
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Oscillator example: Mass on a Spring: o (Restoring) force from spring varies with “extension” o Hooke’s Law: kx F x = (linear restoring force) If mass m is displaced by x from equilibrium and released will oscillate If force is linear and restoring (and thus conservative) o Motion is simple harmonic motion (SHM) o System is a simple harmonic oscillator (SHO)
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Acceleration in SHM: not constant depends on x Newton’s 2 nd law says = x x ma F o For SHM, implies x ma kx = so that x m k a x = At max. positive displacement from equil. ( ) A x = A m k a x = and 0 = x v At equilibrium ( x=0 ) 0 = x a and max v v x = At max. negative displacement from equil. ( ) A x = A m k a x + = and 0 = x v
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How do we find an expression for x ( t ) in SHM? First: find an equation for which x ( t ) is the solution. Remember: dt dx v x = and 2 2 dt x d dt dv a x x = = But for mass on spring: x m k a x = So for mass on spring: x m k dt x d = 2 2 This is a differential equation.
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