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in Physics by (11.2k points)

A smooth horizontal disc rotates about the vertical axis 0 (Fig. 4.15) with a constant angular velocity ω. A thin uniform rod AB of length l performs small oscillations about the vertical axis A fixed to the disc at a distance a from the axis of the disc. Find the frequency ω0 of these oscillations.

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Let us go to the rotating frame, in which the disc is stationary. In this frame the rod is subjected to Coriolis and centrifugal force, Fcor and Fcf, where

 

Where r is the position on an elemental mass of the rod (Fig.) with respect to point o (disc's centre) and 

As the rod is vibrating transversely, so v' is directed perpendicular to the length of the rod. Hence 2dm(v' x  vector ω) for each elemental mass of the rod is directed along PA. Therefore the net torque of Coriolis about A becomes zero. The not torque of centrifugal force about point A:

  

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