First slide

Vertical circular motion

Question

A small sphere of mass m is suspended from a fringe by a light rigid rod of length l. The rod can swing in a vertical circle with what minimum velocity should the sphere be projected horizontally for which it will complete a vertical revolution ?

Moderate

A

$\sqrt{5 g l}$
B

$\sqrt{\frac{g l}{2}}$
C

$2 \sqrt{g l}$
D

$\sqrt{g l}$

Solution

(3)
For the minimum velocity of projection, the velocity of projection at the top most position of its trajectory will be zero.
Conserving Energy,
$\begin{array}{l} \frac{1}{2} m u_{\min}^{2} + 0 = 0 + m g . 2 l \\ \Rightarrow u_{\min} = 2 \sqrt{g l} \end{array}$

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