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. $\frac{v_0}{1+\frac{\mu {\nu }_{0}t}{R}}$
• B. $\frac{v_0}{1-\frac{\mu v_{0}t}{R}}$
• C. $\frac{2v_0}{1-\frac{\mu {\nu }_{0}t}{R}}$
• D. None of these

Answer 1

Answer: (A)

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

REF.Image

By newton's law

$m\frac{dv}{dt}=\frac{-\mu m{V}^2}{r}$

$\frac{dV}{V^2}=\frac{-\mu }{R}dt$

Integrating for $t=0\to t=t$ & $V_0\to V$

$\int \frac{dV}{V^2}=\frac{-\mu }{R}{\int }_0^{t}dt$

$[\frac{-1}{V}{]}_{V_1}^V=\frac{-\mu }{R}[t{]}_0^t$

$\frac{-1}{v}+\frac{1}{v_0}=\frac{-\mu t}{R}$

$\frac{1}{V}=\frac{1}{V_0}+\frac{\mu t}{R}\Rightarrow V=\frac{V_{0}R}{R+\mu V_{0}t}\Rightarrow V=\frac{V_0}{1+\frac{\mu V_{0}t}{R}}$