Question

Figure shows a block of mass $m$ placed on a horizontal surface. The coefficient of static friction between the block and the surface is $\mu$. The maximum force $F$ that can be applied at point $O$ such that the block does not slip on the surface is

• A. $\mu mg\sin \theta$
• B. $\mu mg\cos \theta$
• C. $\mu mg\tan \theta$
• D. $\mu mg$

Answer 1

The correct option is C $\mu mg\tan \theta$

Let $T_1$ and $T_2$ be the tensions in the string as shown in Fig.

Here frictional force $f=\mu mg$. Since the system is in static equilibrium, no net force acts at a point $O$ and on block $m$. Hence, acceleration of $O$ and $m$ is zero.
For point $O:T_2=T_1\cos \theta$
and $F=T_1\sin \theta \left(1\right)$
For block: $T_2=f$
$\Rightarrow T_1\cos \theta =\mu mg\left(2\right)$
Dividing (1) and (2) we get
$F=\mu mg\tan \theta \left(3\right)$
If force $F$ exceeds the value given by (3), the block will begin to slide on the surface. So the correct choice is (c).