# lecture34 - Harmonic Motion III • Simple and Physical...

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Unformatted text preview: Harmonic Motion ( III ) • Simple and Physical Pendulum • SHM and uniform circular motion Simple Pendulum Gravity is the “restoring force” taking the place of the “spring” in our block/spring system. Instead of x, measure the displacement as the arc length s along the circular path. Write down the tangential component of F=ma: θ sin mg = L θ T θ θ θ θ sin But ) sin( 2 2 2 2 L g dt d L s mg ma dt s d m t- = ⇒ =- = = mg sin θ Restoring force s mg θ θ sin 2 2 L g dt d- = Simple pendulum: x dt x d 2 2 2 ϖ- = SHM: The pendulum is not a simple harmonic oscillator! θ θ θ L g L g dt d- ≈- = sin 2 2 θ θ 2245 sin However, take small oscillations: (radians) if θ is small. Then This looks like L g = ϖ θ θ L g dt d- = 2 2 For small θ : x dt x d 2 2 2 ϖ- = , with angle θ instead of x . The pendulum oscillates in SHM with an angular frequency and the position is given by ) cos( ) ( o φ ϖ θ θ + = t t amplitude phase constant (2 π / period) • a simple harmonic oscillator is a mathematical ‘approximation’ to the full problem • for large amplitudes, the solution that the SHO gives...
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lecture34 - Harmonic Motion III • Simple and Physical...

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