Question

Figure( 6-E12) shows a small block of mass m kept at the left end of a larger block of mass M and length l. The system can slide on a horizontal road. The system is started towards right with an initial velocity v. The friction coefficientbetween the road and the bigger block is $\mu$ and that between the block is $\frac{\mu }{2}$. Find the time elapsed before the smaller blocks separates from the bigger block.

Answer 1

Let $a_1$ and $a_2$ be the accelerations of m and M respectively.
Here,$a_1>a_2$, so taht m moves on M Suppose, after time 't' m is separated from M.
In this time, m covers $vt+\frac{1}{2}{a}_1{t}^2$ and
$S_v=vt+\frac{1}{2}{a}_1{t}^2$
For m to separate from M,
$vt+\frac{1}{2}{a}_1{t}^2=vt+\frac{1}{2}{a}_2{t}^2+1$
Again from free body diagram,
$m{a}_1+\frac{\mu }{2}R=0\Rightarrow m{a}_1=-\left(\frac{\mu }{2}\right)mg=\left(\frac{\mu }{2}\right)m\times 10\Rightarrow a_1=o-5\mu$

Again, $M{a}_2+\mu (M+M)g-\left(\frac{\mu }{2}\right)mg=0\Rightarrow 2M{a}^2+2\mu (M+m)g-\mu mg=02M{a}_2=\mu mg-2\mu mg-2\mu mg\Rightarrow a_2=\frac{-\mu mg-2\mu Mg}{2M}$
Putting values of $a_1$ and $a_2$ wwe get,

$t=\sqrt{\frac{4Ml}{(M+m)\mu g}}$