Question

An insect crawl up a hemispherical surface very slowly (see Fig.). The coefficient of friction between the insect and the surface is $\frac{1}{3}$. If the line joining the center of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical, the maximum possible value of $\alpha$ is the 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 $\cot \alpha =3$

It is clear from Fig. that
$f=mg\sin \alpha$ ... (i)
$R=mg\cos \alpha$ ... (ii)
Dividing (i) and (ii),
$\frac{f}{R}=\tan \alpha$
or $\mu =\tan \alpha =\frac{1}{3}$
or $\cot \alpha =3$
So, the correct choice is (a).