Question

An insect crawls up a hemispherical surface very slowly (see figure below). The coefficient of friction between the insect and the surface is $\frac{1}{3}$. If the line joining the centre of hemispherical surface to the insect makes an angle $\alpha$ with the vertical, the maximum possible value of $\alpha$ is given by

• A. $\cot \alpha =3$
• B. $\tan \alpha =3$
• C. $\sec \alpha =3$
• D. $\text{cosec}\alpha =3$

Answer 1

The correct option is A $\cot \alpha =3$

To avoid slipping, friction force $f_r=mg\sin \alpha$ at maximum $\alpha$
From the FBD of insect,
$\mu N=mg\sin \alpha$
$\mu mg\cos \alpha =mg\sin \alpha$
$\mu =\tan \alpha$
$\therefore \tan \alpha =\frac{1}{3}$
$\therefore \cot \alpha =3$
The correct option is (A)