Question

A block of mass $m=2\text{kg}$ is placed on a plank of mass $M=10\text{kg}$ which is placed on a smooth horizontal plane. The coefficient of friction between the block and the plank is $\mu =\frac{1}{3}$. If a horizontal force F is applied on the plank, then find the maximum value of $F$( in N) for which the block and the plank move together without toppling. (Take $g=10m/s^2)$

Answer 1

Maximum $F$ to prevent slipping,
Let, $a$ = acceleration of whole system
For block $m$,
$\therefore f_{max}=\mu mg=ma$
$\Rightarrow a=\mu g=\frac{g}{3}$
for plank, $F-f=Ma$
$\Rightarrow F=(m+M)a$
$F_{max}=(2+10)\frac{g}{3}=4g=40N$
But to prevent toppling of m block,
$F\le (M+m)\frac{g}{4}$,
$\le (10+2)\times \frac{10}{4}\le 30N$
$\therefore F_{max}=30N$
So, the maximum value of $F$ for which the block and the plank move together without toppling is $30N$