Lecture27 - Summary Simple Harmonic Motion TYPICAL EQUATION...

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Summary Simple Harmonic Motion d 2 x + c x =0 dt 2 TYPICAL EQUATION for SHM Solution x(t) = A cos ( ω t + ϕ ) ω = c Τ = 2 π / ω x (t=0) = x 0 v (t=0) = v 0 a (t=0) = a 0 Initial Condition – to determine A, ϕ t x A T ϕ Velocity and acceleration Note : v and a always opposite to x a(t) = - A ω 2 cos ( ω t + ϕ ) v(t) = - A ω sin ( ω t + ϕ ) = + ω A 2 – x 2 Being x(0) = A cos ϕ = x 0 ϕ = arccos (x 0 /A)
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Example L eq L 1 m 2 m 1 System in equilibrium. Break the spring and m 1 starts to oscillate. Find x(t) for m 1 . Neglect friction
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“Vertical” spring as a simple harmonic oscillator New equilibrium position Oscillation around new position Σ F = - ky + mg - ky + mg = 0 (at equilib.) From Newton’s second law: d 2 y dt 2 = - k m + g 1 st step : Reduce it to simple differential equation 1) Add gravity to equation of motion Only change is in the equilibrium position! y
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Lecture27 - Summary Simple Harmonic Motion TYPICAL EQUATION...

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