Question

A uniform ring of radius $R$, is fitted with a massless rod $AB$ along its diameter. An ideal horizontal string (whose one end is attached with the rod at a height $r$) passes over a smooth pulley and other end of the string is attached with a block of mass double the mass of ring as shown. The co-efficient of friction between the ring and the surface is $\mu$. When the system is released from rest, the ring moves such that rod $AB$ remains vertical . The value of $r$ is

• A. $R(1-\frac{3\mu }{2(1+\mu )})$
• B. $R(1-\frac{\mu }{2(1+\mu )})$
• C. $R(2-\frac{3\mu }{2(1+\mu )})$
• D. $R(1-\frac{3\mu }{(1+\mu )})$

Answer 1

The correct option is A $R(1-\frac{3\mu }{2(1+\mu )})$

FBD


Applying torque balance about centre of the ring
$T(R-r)=\mu mgR$
Also applying Newton's laws of motion on the block and on the ring
$2mg-T=2ma$
$T-\mu mg=ma$
On solving, we get $\therefore r=R(1-\frac{3\mu }{2(1+\mu )})$