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Analytical Mech Homework Solutions 147

Analytical Mech Homework Solutions 147 - s = a sin so s = a...

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Let sin s a θ = so cos s a θ θ = ± ± 2 2 2 1 1 2 2 2 2 ms mg L s ms a = = ± ± 2 ks where mg k a = The equation of motion is thus 0 k s s m + = ±± or 0 g s s a + = ±± -a simple harmonic oscillator 10.12 Coordinates: cos sin x a t b ω θ = + sin cos t b y a ω θ = sin cos x a t b ω ω θ = − + ± ± θ cos sin y a t b ω ω θ = + ± ± θ ( ) 2 2 1 2 L T V m x y mgy = = + ± ± ( ) ( 2 2 2 2 2 sin sin cos 2 m a b b a t mg a t b ) ω θ θ ω θ ω ω + ± ± θ = + ( ) ( ) 2 cos d L mb mba t dt θ ω θ ω θ ω θ = + ±± ± ± ( ) cos sin L mb a t mgb θ ω θ ω θ = ± θ The equation of motion 0 L d L dt θ θ = ± is ( ) cos sin 0 a g t b b θ ω θ ω θ + ±± = Note – the equation reduces to equation of simple pendulum if 0 ω . 10.13 Coordinates: ( ) cos cos x l t l t ω θ ω = + +
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