Question

A solid cylinder is kept on one edge of a 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 an initial velocity of 20 m/s towards right. Find the time (in sec) after which plank and cylinder will separate. $(g=10m/s^2)$

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Answer 1

$fr=\mu mg=ma$
$a=5m/s^2$
$frR=\frac{m{R}^2}{2}\alpha$
$R\alpha =10m/s^2$
Now; t=t sec. when rolling will start [Considering length of planck is sufficient ]
$\alpha t+R\alpha t={\vartheta }_0-at$
$(2a+R\alpha )t={\vartheta }_0$
$T=\frac{20}{20}=1sec$
Now in 1 sec relative displacement of cylinder
$=20\times 1-\frac{1}{2}10\times 1=15m$
At t= 1 sec rolling will start. Centre of cylinder will move speed 5m/s. Planck will move with speed 15m/s. So extra time taken $\Rightarrow 10=10t^1$
$t^1=1sec$
Total time = 1 + 1
= 2 sec