Homework 1 Solutions

R r r r b are there stationary horizontal circular

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Unformatted text preview: m is conserved in this system, L ≡ mr2 φ, so the equation of motion for generalized coordinate r can be rewritten as mr 1 + ¨ α R 2 cos2 r R = L2 α +m 3 mr R 2 r2 sin ˙ r α r r cos −mg cos . R R R R (b) Are there stationary horizontal circular orbits? This problem can be solved using the equation of motion or the effective potential, Ueff = U + L2 /(2mr2 ). A stationary point, r0 is defined by, ∂Ueff ∂r = 0. r=r0 For the effective potential at hand ∂Ueff L2 α r = − 3 + mg cos , ∂r mr R R which leads to the transcendental equation L2 = r0 cos(˜0 ), ˜3 r m2 R2 gα where r0 = r0 /R. Dependi...
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