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A sphere rolls down a track onto inside circular track

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In a vertical plane, a uniform sphere, mass m, radius r, released from rest at a point h meters above ground level, rolls without slipping down a straight section of track then around the inside of a circular track of radius R. Given R>>r.
Find the minimum initial height h, such that the sphere barely maintains contact at the highest point of the circular track. SEE ATTACHMENT for a diagram.

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Solution Summary

A sphere rolls down a track onto inside circular tracks are analyzed. The minimum initial heights to maintain contacts are given.

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Preliminary physics:
Note 1. As the sphere rolls downward, it gains translational kinetic energy,
(1) KEtrans= .5 m v^2
and also gains rotational kinetic energy,
(2) KErot= .5 I w^2 in which I is the moment of inertia of a sphere, and w is the angular velocity in rad/sec. Recall that for the sphere,
(3) I= .4 m r^2, and from motion equations,
(4) w= v/r
Note 2. While descending to ...

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