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 $\left(=\mu_{1} m g\right)$.
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 $m$,
$f_{1}=m a$
or $\mu_{1} m g=m a$
or $a=\mu_{1} g$ ...(ii)
From Eqs. (i) and (ii), we get
$F=(M+ m) g\left(\mu_{1}+\mu_{2}\right)$