A solid cylinder is kept on one edge of plank of same mass and length 25 m 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 20 ms^-1 towards right. Find the time (in s) after which plank and cylinder will  separate.  $[g=10m{s}^{-2}]$ 

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 travelled by cylinder with respect to plank in 1s is 
$S^{\text{'}}=-vt+\frac{1}{2}\left(2\mu g\right)t^2=-15m$ 
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  10 m will be travelled in time $t^{\text{'}}=\frac{10}{10}=1s$ 
$\therefore$ Total time = 2s