Question

A block of mass $m$ is placed on another block of mass $M$, which itself is lying on a horizontal surface. The coefficient of friction between two blocks is ${\mu }_1$ and that between the block of mass $M$ and horizontal surface is ${\mu }_2$. What maximum horizontal force can be applied to the lower block so that the two blocks move without separation?

• A. $(M+m)\left({\mu }_2-{\mu }_1\right)g$
• B. $(M-m)\left({\mu }_2-{\mu }_1\right)g$
• C. $(M-m)\left({\mu }_2+{\mu }_1\right)g$
• D. $(M+m)\left({\mu }_2+{\mu }_1\right)g$

Answer 1

Answer: (D)

Here the force applied should be such that friction force acting on the upper block of $m$ should not be more than the limiting friction $\left(={\mu }_{1}mg\right)$.
Let the system moves with acceleration $a$. Then for whole system,
$F-{\mu }_2(M+m)g=(M+m)a$ ...(i)
For block of mass $m$,
$f_1=ma$
or ${\mu }_{1}mg=ma$
or $a={\mu }_{1}g$ ...(ii)
From Eqs. (i) and (ii), we get
$F=(M+m)g\left({\mu }_1+{\mu }_2\right)$