Figure (8-E14) shows a light rod of length $$l$$ rigidly attached to a small heavy block at one end and a hook at the other end. The system is released from rest with the rod in a horizontal position. There is a fixed smooth ring at a depth $$h$$ below the initial position of the hook and the hook gets into the ring as it reaches there. What should be the minimum value of $$h$$ so that the block moves in a complete circle about the ring?
The rod is rigidly attached to the block , At the highest point ring will equal to $$l$$ .
Now
Total energy = Kinetic energy + Potential energy.
$$ 1/2mv² + mgl ----(i)$$
At height $$h$$. The Potential energy due to total energy
$$P.E = mgh -------(ii)$$
Now equating these two equations and get,
$$mgh = 1/2mv² + mgl$$
will be the minimum when $$v =0$$ , $$∵ l$$ is fixed
Now,
putting $$v$$ value in above equation
Hence, the minimum value is $$l$$ .
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