Question

A cylinder of mass $m$ is kept on the edge of a plank of mass $2m$ and length $12$ metre, which in turn is kept on smooth ground. Coefficient of friction between the plank and the cylinder is $0.1$. The cylinder is given an impulse, which imparts it a velocity $7m/s$ but no angular velocity. Find the time after which the cylinder falls off the plank.

Answer 1

REF.Image.

$\to f=0.1mg\Rightarrow f/m=1m/s^2$

$a_m=1m/s^2;a_{2}m=f/2m=0.5m/s^2$

$T=M{R}^2\alpha /2=f\ast r=0.1mg{R}^2$

$R\alpha =0.2g=2m/s^2$

$\therefore V_p=7-(2+1)t=7-3t$

velocity of slab $V_l=0.5t$

$\because V_p=vl$

$7-3t=0.5t$

$\therefore t=2sec$

$v_1=1m/s,V_{ball}=7-t=5m/s$

1) $d_{ball}={\int }_0^{2}7-tdt=14-2=12m$

$d_{slab}={\int }_0^{2}0.5tdt=0.5\ast 2^2/2=1m$

net distance = 11m, remaining 1m

1m to be travelled when no friction acts.

ii) $V_{ball}-V_{slab}=5-1=4m/s$

$t_2=1/4=0.25sec$

Total time taken $=2+\mathrm{0.25.4}t$

$=9$