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

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$ω \sqrt{L^{2} - a^{2}}$

$\frac{v_{0}}{1 + \frac{{μv}_{0} t}{R}}$

$ω \sqrt{L^{2} - a^{3}}$

$v_{0}$

answer is B.

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Detailed Solution

$N = \frac{{mv}^{2}}{R}$ $f_{\max} = μN = \frac{{μmv}^{2}}{R}$

$∴ Retardation a = \frac{f_{\max}}{m} = \frac{{μv}^{2}}{R}$

$∴ (- \frac{dv}{dt}) = \frac{{μv}^{2}}{R}$

$or \int_{v_{0}}^{v} \frac{dv}{v^{2}} = - \frac{μ}{R} \int_{0}^{t} dt$

$or v = \frac{v_{0}}{1 + \frac{{μv}_{0} t}{R}}$

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