Question

The coefficient of static friction between the two blocks shown in figure is $\mu$ and the table is smooth. What maximum horizontal force F can be applied to the block of mass M so that the blocks move together?

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

When the maximum force F is applied, both the blocks move together towards right. The only horizontal force on the upper block of mass m is that due to the friction by the lower block of mass M. Hence this force on m should be towards right. The force of friction on M by m should be towards left by Newton's third law. As we are talking of the maximum possible force F that can be applied, the friction is limiting and hence f = $\mu N$ , where N is the normal force between the blocks.

Consider the motion of m. The forces on m are,

(a) mg downward by the earth (gravity),

(b) N upward by the block M (normal force) and

(c) f = $\mu N$ (friction) towards right by the block M.

In the vertical direction, there is no acceleration. This gives

N = mg . . . . . .(i)

In the horizontal direction, let the acceleration be a, then

$\mu N$ = m a

or, $\mu mg$ = ma

or, a = $\mu g$ . . . . . .(ii)

Next, consider the motion of M

The forces on M are

(a) Mg downward by the earth (gravity),

(b) $N_1$ upward by the table (normal force),

(c) N downward by m (normal force),

(d) $f=\mu N$ (friction) towards left by m and

(e) F (applied force) by the experimenter.

The equation of motion is

F - $\mu N$ = M a

or, F - $\mu mg$ = M $\mu g$ [Using (i) and (ii)]

or, F = $\mu g$ (M + m)