Question

Figure following 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?

Answer 1

Let v be the minimum velocity required to complete a circle about the ring. 

Applying the law of conservation of energy,

Total energy at point A = Total energy at point B

\[\text{ mgl } + \frac{1}{2}\text{ mv}^2 = \text{ mg(2l)} + 0\]
\[ \Rightarrow \text{ v } = \sqrt{2\text{ gl }}\]

Let the rod be released from a height h.
Total energy at A = Total energy at B

\[\text{ mgh } = \frac{1}{2}\text{ m } \nu^2 \]

\[\text{ mgh } = \frac{1}{2}\text{m} \left( 2 \text{ gl} \right)\]

So, h = l