Question

A block of mass $m$ rests on a rough horizontal surface as shown in the figure. Coefficient of friction between the block and the surface is $\mu$. A force $F=mg$ acting at angle $\theta$ with the vertical side of the block pulls it. In which of the following cases can the block be pulled along the surface?

• A. $\tan \theta \ge \mu$
• B. $\cot \theta \ge \mu$
• C. $\tan \frac{\theta }{2}\ge \mu$
• D. $\cot \frac{\theta }{2}\ge \mu$

Answer 1

The correct option is D $\cot \frac{\theta }{2}\ge \mu$

It is clear from the figure that vertical component of force will decrease the normal reaction, i.e., $N=mg-mg\cos \theta$

and frictional force $=\mu N$
$=\mu (mg-mg\cos \theta )$
Now block can be pulled when, horizontal component of force $\ge$ frictional force
i.e., $mg\sin \theta \ge \mu (mg-mg\cos \theta )$
or $2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\ge \mu (1-\cos \theta )$
or $2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\ge 2\mu {\sin }^2\frac{\theta }{2}$
or $\cot \frac{\theta }{2}\ge \mu$
Hence, the correct answer is option (d).