Question

Two small rings, each of mass ${}^{\prime }{m}^{\prime }$, are connected to the block of same mass ${}^{\prime }{m}^{\prime }$ through an inextensible massless string of length ${}^{\prime }{l}^{\prime }$. Rings are constrained to move over smooth rod $AB$. Initially, the system is held at rest as shown in Figure. Let $u$ and $v$ be the velocities of ring and block, respectively when string makes an angle ${60}^{\circ }$ with the vertical.

• A. $u=\sqrt{\frac{gl}{5}}$
• B. $u=\sqrt{\frac{8gl}{5}}$
• C. $v=\sqrt{3gl}$
• D. $v=\sqrt{\frac{3gl}{5}}$

Answer 1

Answer: (A), (D)

The correct options are
A $u=\sqrt{\frac{gl}{5}}$
D $v=\sqrt{\frac{3gl}{5}}$

$v\cos \theta =u\sin \theta$

From conservation of energy,
$mgl\cos \theta =2\left(\frac{1}{2}m{u}^2\right)+\frac{1}{2}m{v}^2$
$u=\sqrt{\frac{2gl\cos \theta }{2+{\tan }^2\theta }}=\sqrt{\frac{gl}{5}}$, using $\theta ={60}^{\circ }$
$v=\sqrt{\frac{3gl}{5}}$