A Texas cockroach first rides at the center of a circular disk that rotates free

Question

A Texas cockroach first rides at the center of a circular disk that rotates freely like a merry-go-round without external torques. The cockroach then walks out to the edge of the disk, at radius R. The figure gives the angular speed of the cockroach-disk system during the walk. When the cockroach is on the edge at radius R, what is the ratio of the bug's rotational inertia to that of the disk, both calculated about the rotation axis? [The figure shows angular speed w=4.30 rad/s when the cockroach starts off and w=2.10 rad/s when the cockroach is on the edge at radius R]

Explanation / Answer

by conservation of momentum

initial momentum = final momentum

initial moment of inertia of cug = 0 as its on the axis so

moment of inertia of disk * 4.3 = (moment of inertia of disk + moment of inertia of bug) * 2.1

moment of inertia of disk * 2.2 = moment of inertia of bug * 2.1

moment of inertia of bug / moment of inertia of disk = 2.2 / 2.1

moment of inertia of bug / moment of inertia of disk = 22 / 21

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