A toy globe rotates freely without friction with an initial angular velocity ω 0 . A bug starting at one pole N travel to the other pole S along a meridian with constant velocity u . The axis of rot

Question

A toy globe rotates freely without friction with an initial angular velocity $ω_{0}$ . A bug starting at one pole N travel to the other pole S along a meridian with constant velocity u . The axis of rotation of the globe is held fixed. Let M and R denote the mass and radius of the globe (a solid sphere, moment of inertia $I_{0} = 2 MR^{2} /5$ ), m the mass of the bug and T the duration of the bug's journey.

Show the during the time the bug is travelling, the globe torates through an angle

TestHub · Accepted Answer

5 2 ​ M R 2 ω = ( 5 2 ​ M R 2 + m R 2 sin 2 θ ) ​ ∫ ω 1 ​ d t = ∫ M + 2 5 ​ m 2 sin 2 θ M ω 0 ​ ​ d t Δ θ = ω 0 ​ ∫ M cot 2 θ + ( 2 5 m ​ + M ) V M cosec 2 θR d θ ​ ​ Δ θ = V R ω 0 ​ ​ ∫ cot 2 θ + M M + 2 5 M ​ ​ cosec 2 θ d θ ​ = V R ω 0 ​ ​ ∫ t 2 + M M + 5 m ​ d t ​ ​

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