Question

A man of mass $m$ is standing on a plank kept on ground having mass $m$. There is no friction between the plank and the ground. If the man walks with an acceleration a w.r.t. the plank then calculate the acceleration of the plank w.r.t. the ground. Also calculate friction force exerted by the plank on man.

Answer 1

We use the notation,

$a_{xy}$ to denote acceleration of $x$ with respect to $y$.

[g$=$ground, p$=$plank, m$=$man].

${\overrightarrow{a}}_{mp}={\overrightarrow{a}}_{mg}-{\overrightarrow{a}}_{pg}$

Since, there is no external force, ${\overrightarrow{a}}_{com}=0$.

$\Rightarrow {\overrightarrow{a}}_{com}={\overrightarrow{a}}_{mg}-{\overrightarrow{a}}_{pg}=0$.

$\Rightarrow {\overrightarrow{a}}_{mg}=-{\overrightarrow{a}}_{pg}$

$\Rightarrow {\overrightarrow{a}}_{mp}=-2{\overrightarrow{a}}_{pg}$

$\Rightarrow {\overrightarrow{a}}_{pg}=\frac{-{\overrightarrow{a}}_{mp}}{2}$

$\Rightarrow {\overrightarrow{a}}_{pg}=\frac{-a}{2}$.

$\Rightarrow \mid {\overrightarrow{a}}_{pg}\mid =\frac{a}{2}$.

Force$=mass\times acceleration$.

$\therefore$ Force exerted by man on plank$=m\times \frac{a}{2}=\frac{ma}{2}$.