Constrained, uneven dumbbell

Two beads of mass \(\displaystyle 2m\) and \(\displaystyle m\), connected by a rod of length \(\displaystyle\ell\) and of negligible mass, are free to move in a smooth vertical circular wire frame of radius \(\displaystyle\ell\) as shown.

Calculate the velocity (in \(\displaystyle m/s\)) that should be given to the mass \(\displaystyle 2m\) (when the rod is in horizontal position) in counter-clockwise direction so that the rod just becomes vertical.

Details and Assumptions:
\(\bullet\) \(\displaystyle m = 250g\)
\(\bullet\) \(\displaystyle \ell = 25cm\)
\(\bullet\) \(\displaystyle g = 9.8 m/s^2\)

Note: This problem appeared in our AITS (All India Test Series) - 6


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