Analytical Mech Homework Solutions 147

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

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Unformatted text preview: s = a sin θ so s = aθ cos θ 2 ms mg 2 1 2 1 2 mg L= − s = ms − ks where k = 2 2a 2 2 a The equation of motion is thus k g s + s = 0 or s + s = 0 -a simple harmonic oscillator m a Let Coordinates: 10.12 x = a cos ω t + b sin θ y = a sin ω t − b cos θ x = − aω sin ω t + bθ cos θ y = aω cos ω t + bθ sin θ L = T −V = 1 m ( x 2 + y 2 ) − mgy 2 m 22 a ω + b 2θ 2 + 2bθ aω sin (θ − ω t ) − mg ( a sin ω t − b cos θ ) 2 d ∂L = mb 2θ + mbaω θ − ω cos (θ − ω t ) dt ∂θ ∂L = mbθ aω cos (θ − ω t ) − mgb sin θ ∂θ ∂L d ∂L The equation of motion − = 0 is ∂θ dt ∂θ a g θ − ω cos (θ − ω t ) + sin θ = 0 b b Note – the equation reduces to equation of simple pendulum if ω → 0 . = ( ) Coordinates: x = l cos ω t + l cos (θ + ω t ) 10.13 y = l sin ω t + l sin (θ + ω t ) ( ) x = −ω l sin ω t − θ + ω l sin (θ + ω t ) ( ) y = ω l cos ω t + θ + ω l cos (θ + ω t ) 2 1 1 m ( x 2 + y 2 ) = ml 2 ω 2 + θ + ω + ... 2 2 2 ... ml ω θ + ω sin ω t ( sin θ cos ω t + sin ω t cos θ ) + cos ω t ( cos θ cos ω t − sin ω t sin θ ) ( L =T = ( ) ) 9 ...
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This note was uploaded on 01/06/2012 for the course PHYSICS 4360C taught by Professor Johnanderson during the Fall '11 term at UNF.

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