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PHYS310_homework_12_solutions

PHYS310_homework_12_solutions - Homework set 12 due...

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Homework set 12, due Wednesday, December 7, 11:30 am Following problems from Analytical Mechanics by Fowles & Cassiday 1. 10.2 (20 points) 2. 10.3 (20 points) 3. 10.4 (20 points) 4. 10.6 (20 points) 5. 10.7 (20 points) 10.2 V mgz = ( 29 2 2 2 1 2 T m x y z = + + & & & ( 29 2 2 2 1 2 L T V m x y z mgz = - = + + - & & & L mx x = & & , L my y = & & , L mz z = & & d L mx dt x = && & , d L my dt y = && & , d L mz dt z = && & 0 L L x y = = , L mg z = - From equations 10.4.5 … 0 i i L d L q dt q - = & 0 mx = && , mx const = & 0 my = && , my const = & mz mg = - && 10.3 Choosing generalized coordinate x as linear displacement down the inclined plane (See Figure 8.6.1), for rolling without slipping … x a ϖ = & 2 2 2 2 2 2 1 1 1 1 2 7 2 2 2 2 5 10 x T mx I mx ma mx a ϖ  = + = + =   & & & & For 0 V = at the initial position of the sphere, sin V mgx θ = - 2 7 sin 10 L T V mx mgx θ = - = + & 7 5 L mx x = & & , 7 5 d L mx dt x = && & sin L mg x θ =
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7 sin 5 mx mg θ = && 5 sin
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