Question

A small block of mass m slides on a frictionless horizontal table. It is constrained to move inside a ring of radius l which is fixed to the table. At $t=0$, the block has a tangential velocity $V_o$. The coefficient of friction between the block and the ring is $\mu$. The velocity of the block at time is?

• A. $\frac{V_o}{1+\frac{\mu V_{o}t}{l}}$
• B. $\frac{V_o}{1+\frac{2\mu V_{o}t}{l}}$
• C. $\frac{V_o}{1-\left(\frac{\mu V_{o}t}{l}\right)}$
• D. $\frac{V_o}{\left(\frac{\mu V_{o}t}{l}\right)}$

Answer 1

Answer: (A)

$\frac{V_o}{1+\frac{\mu V_{o}t}{l}}$

Frictional force = $\mu$ * centrifugal reactive force = $\mu m{v}^2/r$

$mdv/dt=-\mu m{v}^2/r$

$dv/v^2=-\left(\mu /l\right)dt$

As time changes from 0 to t, velocity v changes from V0 to V

${\int }_{V_0}^{V}dv/v^2=-{\int }_0^t\left(\mu /l\right)dt$

$[-1/v{]}_{V_0}^v=-(\mu /l\left)\right[t{]}_0^t$

$-1/V+1/V0=-\left(\mu /l\right)t$

Solving,

$V=\frac{V_0}{1+\left(\mu /l\right)V_{0}t}$

So, Option A is correct.