A bug in a rotating pipe

A thin massless horizontal pipe rotates about a vertical axis with angular speed \(\omega\) as shown in the figure. A small bug, located at a distance \(r\) from the axis, starts crawling inside the pipe at constant speed \(v\) (relative to the pipe). Because of the bug, the motor must develop the power \[ P_{m}= C m \omega^{2}v r \] in order to keep the system rotating at constant angular speed \(\omega\). Determine the numerical coefficient \(C\). m is the mass of the bug.

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