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 ${\mu }_2$. What maximum horizontal force can be applied to the lower block so that the two blocks move without separation?

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

Answer 1

The correct option is D

$(M+m)({\mu }_2+{\mu }_1)g$

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 (= mg). 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

$F=(M+m)g({\mu }_1+{\mu }_2)$