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oscillation_summary - Oscillations Summary Free...

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Oscillations Summary Free Oscillations: Differential Equation: d 2 x dt 2 + ω 2 x = 0 Solution: x ( t ) = A cos( ωt + δ ) Common Oscillators: Mass on Spring: ω 0 = k m Simple Pendulum: ω 0 = g L Physical Pendulum: ω 0 = mgd I Uncommon Oscillators: Use Forces or Work/Energy to obtain ω 0 Damped Oscillations: Solution: x ( t ) = A 0 e - bt 2 m cos( ωt + δ ) The amplitude is time-dependent ! A ( t ) = A 0 e - bt 2 m Frequency Shift ! System oscillates at ω = ω 2 0 - ( b 2 m ) 2 Classification: if ω is real, the system is underdamped. If it is imagi- nary, the system is overdamped. If it is zero, the system is critically
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