Question

A triangular block of mass M rests on a smooth surface as shown in figure. A cubical block of mass m rests on the inclined surface. If all surfaces are frictionless, the force that must be applied to M so as to keep m stationary relative to M is :

• A. Mg tan ${30}^{\circ }$
• B. Mg tan ${60}^{\circ }$
• C. (M + m)g tan ${30}^{\circ }$
• D. (M + m) g cos ${30}^{\circ }$

Answer 1

The correct option is C (M + m)g tan ${30}^{\circ }$

Let's assume that a force F is applied to $M$ in the positive X direction and the system is moving with an acceleration '$a$' in the same direction.

From the above FBD

Balance forces on $M$ in X direction:

$F=$ $N_1$ $sin{30}^0=Ma$ .........(1)

Balance forces on $m$ in Y direction:

$mg-N_{2}cos{30}^0=0$ .........(2)

Balance forces on $m$ in X direction:

$N_{2}sin{30}^0$ =$ma$......(3)

From equation 2 and 3,

$a=gtan{30}^0$

Substitute equation 3 into equation 1

$F=(M+m)a$

$=(M+m)gtan{30}^0$