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: sin mg = L T sin But ) sin( 2 2 2 2 L g dt d L s mg ma dt s d m t- = =- = = mg sin Restoring force s mg sin 2 2 L g dt d- = Simple pendulum: x dt x d 2 2 2 - = SHM: The pendulum is not a simple harmonic oscillator! L g L g dt d- - = sin 2 2 2245 sin However, take small oscillations: (radians) if is small. Then This looks like L g = L g dt d- = 2 2 For small : x dt x d 2 2 2 - = , with angle instead of x . The pendulum oscillates in SHM with an angular frequency and the position is given by ) cos( ) ( o + = t t amplitude phase constant (2 / 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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