Question

A small block of mass m kept at the left end of a large 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 coefficient between the road and bigger block is $\mu$ and that between the blocks is $\mu /2$. Find the time elapsed before the smaller block separates from the bigger block.

Answer 1

Let both are two move, then block will be the first acted upon $f_1$. This $f_1$ will reduce its velocity and try to move ahead of $m$ so $f_2$ will again act on $m$ in left direction and on the right.

$f_1=\mu (m-1m)g$

$f_2=\frac{\mu }{2}\left(mg\right)$

$f_2=m_{a_2}\Rightarrow a_2=\frac{\mu g}{2}$

$f_1-f_2=m_{a_1}\Rightarrow \mu (m+m)g-\frac{\mu mg}{2}=m_{a_1}$

$O_1=\frac{\mu Mg+\mu \frac{Mg}{2}}{m}=\mu g+\frac{\mu mg}{2m}$

w.r.t. m

and= $a_1-a_2=\mu g+\frac{\mu mg}{2M}-\frac{Mg}{2}=\frac{\mu g}{2}\left(\frac{M+m}{M}\right)$

Find= $v-v=0$

then, $l=V_{mt}+\frac{1}{2}{\mu }_{in}{t}^2$

$=\frac{1}{2}\left(\frac{mg}{2}\right)\left(\frac{M+m}{M}\right)t^2$

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