Question

A plank of mass $M$ and length $L$ is at rest on a frictionless floor. The top surface of the plank has friction. At one end of it a man of mass $m$ is standing as shown in Fig. If the man walks towards the order end, find the distance, which the plank moves(a) till the man reaches the centre of the plank, (b) till the man reaches the other end of the plank.

Answer 1

As there is no external force on the system (man + plank) and the system is initially at rest, there should not be movement of centre of mass of the system dusting the motion of the man.
The displacement of centre of mass is $\Delta x_{CM}$ and is given by
$\Delta x_{CM}=\frac{m\Delta {\overline{x}}_m+M\Delta {\overline{x}}_p}{m+M}$
$⟹\Delta {\overline{x}}_m=$ displacement of the man with respect to the ground
$\Delta {\overline{x}}_m=\Delta {\overline{x}}_{m.p}+\Delta {\overline{x}}_p=(x-X)$ where $\Delta \Delta {\overline{x}}_p=-X$ (towards left)
A $\Delta {\overline{x}}_{CM}=0$, from Eq. (i),
$0=\frac{m(x-X)-MX}{m+M}⟹X=\frac{mx}{m+M}$
a. When $x=\frac{L}{2}$, displacement of block $X=\frac{m_2^L}{m+M}$
b. When $x=L$, displacement of block $X=\frac{mL}{m+M}$