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Torque on a Dipole

A block of ma...

Question

A block of mass $m$ is constrained to move inside a ring of radius $R$ which is fixed on a horizontal smooth table. 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 $μ$ . Then the speed of the block at time $t$ is

\frac{v_{0}}{R + μ v_{0} t}

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\frac{v_{0}}{(1 + \frac{μ v_{0} t}{R})}

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\frac{v_{0}}{1 - \frac{μ v_{0} t}{R}}

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\frac{v_{0} R}{1 + μ v_{0} t}

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Solution

The correct option is B $\frac{v_{0}}{(1 + \frac{μ v_{0} t}{R})}$
At some instant $t$ ,

$f_{k} = - m a$
$μ N = - m a (N = \frac{m v^{2}}{R})$
$μ (\frac{m v^{2}}{R}) = - m a$
$\Rightarrow a = - \frac{μ v^{2}}{R}$

$\frac{d v}{d t} = - \frac{μ v^{2}}{R}$

$\frac{d v}{v^{2}} = - \frac{μ}{R} d t$

Integrating,
$\int_{v_{0}}^{v} \frac{d v}{v^{2}} = - \frac{μ}{R} \int_{0}^{t} d t$

$[- \frac{1}{v}]_{v_{0}}^{v} = - \frac{μ}{R} t$

$\frac{1}{v} - \frac{1}{v_{0}} = \frac{μ t}{R}$

$\frac{1}{v} = \frac{μ t}{R} + \frac{1}{v_{0}}$

$\frac{1}{v} = \frac{v_{0} μ t + R}{v_{0} R}$

$v = \frac{v_{0}}{1 + \frac{μ v_{0} t}{R}}$

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