Question

A solid cylinder is kept on one edge of a plank of same mass and length $25m$ placed on a smooth surface as shown in the figure. The coefficient of friction between the cylinder and the plank is $0.5$ . The plank is given a velocity of $20m{s}^{-1}$ towards right. Find the time (in $s$ ) after which plank and cylinder will separate. [ $g=10m{s}^{-2}$ ]

Answer 1

Answer: 2

Drawing free body diagram of the cylinder with respect to plank.
$2\mu mg=ma\Rightarrow a=2\mu g$
$\left(\mu mgR\right)=\frac{1}{2}m{R}^2\alpha ;\alpha -\frac{2\mu g}{r}$
Acceleration of point of contact with respect to plank is $4\mu g$
Velocity of pure rolling starts,
$-v+4\mu gt=0$
$t=\frac{20}{4\times 0.5\times 10}=1s$
Distance traveled by cylinder with respect to plank in $1s$ is
$S^{\prime }=-vt+\frac{1}{2}\left(2\mu g\right)t^2=-15m$
At $t=1s$ , the velocity of cylinder with respect to plank is
$t=1s$
$v_{rel}=-v+2\mu gt=-20+2\times 0.5\times 10=-10m/s$
Remaining $10m$ will be travelled in time $t^{\prime }=\frac{10}{10}=1s$
$\therefore$ Total time $=2s$