Question

A solid sphere of mass m and radius R is placed on a plank of equal mass, which lies on a smooth horizontal surface. The sphere is given a sharp impulse in the horizontal direction so that it starts sliding with a speed of $v_0$. Find the time taken by the sphere to start pure rolling on the plank. The coefficient of friction between plank and sphere is $\mu$.

Answer 1

Velocity of COM $\to v_0$

Angular velocity of sphere $\to 0$

Now, $f=-\mu mg$ [ $f\to$ Frictional force ]

$\Rightarrow ma=-\mu mg$

$\Rightarrow a=-\mu g$ [ $a\to$ retarding acceleration ]

Now, torque, $\tau =fR=I\alpha$ [ $\alpha \to$ angular acc ]

$\Rightarrow \alpha =\frac{fR}{I}$

$=\frac{5\mu g}{2R}$ $[I_{sphere}=\frac{2M{R}^2}{5}]$

Now, after time ${}^{\prime }{t}^{\prime }$

$\Rightarrow V=v_0+at$

$=v_0-\mu gt⟶\left(1\right)$

Let at this time, angular velocity is $\omega$

$\Rightarrow \omega ={\omega }_0+\alpha t$ [ Initial $\omega =0$ ]

$\therefore \omega =\frac{5\mu gt}{2R}⟶\left(2\right)$

Now, if rolling has stared $v_0-\mu gt=\frac{5\mu gt}{2}$

$\Rightarrow t=\frac{2v_0}{7\mu g}$

Hence, the answer is $\frac{2v_0}{7\mu g}.$