Question

A block of mass $m$ slides on a frictionless table. It is constrained to move inside a ring of radius R. At time $t=0$, block is moving along the inside of the ring (i.e., in the tangential direction) with velocity $v_0.$ The coefficient of friction between the block and the ring is $\mu .$ Find the speed of the block at time $t$.

• A. $\frac{v_0}{1+\frac{\mu v_{0}t}{R}}$
• B. $\frac{v_0}{1+\frac{2\mu v_{0}t}{R}}$
• C. $\frac{v_0}{1+\frac{\mu v_{0}t}{2R}}$
• D. $\frac{v_0}{1+\frac{\mu v_{0}t}{3R}}$

Answer 1

Answer: (A)

$\frac{v_0}{1+\frac{\mu v_{0}t}{R}}$

As block is constrained to move inside a ring, Normal force by ring on block, $N=$ centrifugal force $=\frac{M{v}^2}{R}$

Friction force $=\mu N=\mu \frac{M{v}^2}{R}\Rightarrow a=-\mu \frac{v^2}{R}$

$\frac{\delta v}{\delta t}=\mu \frac{v^2}{R}\Rightarrow \int \frac{\delta v}{V^2}=(-\mu /R)\int \delta t$

$\Rightarrow [\frac{1}{v_o}-\frac{1}{v}]=-\left(\mu /R\right).t$

On solving above, we get:

$V=\frac{v_o}{1+\frac{\mu v_{o}t}{R}}$