A small heavy block is attached to the lower end of a light rod of length l which can be rotated about its clamped upper end. What minimum horizontal velocity should the block be given so that it moves in a complete vertical circle?

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Solution

Let the velocity of the body at L is ' $ν$ ' .
If the body is moving in a vertical plane then we need to find the minimum horizontal velocity which needs to be given to the body (velocity at L).
Also as point H is the highest point in the vertical plane so horizontal velocity at H will be zero.

Applying law of conservation of energy at points L and H,

$\frac{1}{2} m ν^{2} = m g h \frac{1}{2} m ν^{2} = m g (2 L) \Rightarrow ν = \sqrt{(4 g L)} = 2 \sqrt{g L}$

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A small heavy block is attached to the lower end of a light rod of length l which can be rotated about its clamped upper end. What minimum horizontal velocity should the block be given so that it moves in a complete vertical circle? Figure

A small heavy block is attached to the lower end of a light rod of length l which can be rotated about its clamped upper end. What minimum horizontal velocity should the block be given so that it moves in a complete vertical circle?

Figure