Question

An insect crawls up a hemispherical surface very slowly. The coefficient of friction between the insect and the surface is 1/3. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical, find the maximum possible value of $\alpha$.

Answer 1

Given,

Coefficient of friction is $\mu =\frac{1}{3}$

Angle made from vertical axis, $\alpha$

Condition for not slipping

Force due to gravitation component $\le$ Friction force

$mg\sin \alpha \le \mu mg\cos \alpha$

$g\sin \alpha \le \mu g\cos \alpha$

$\tan \alpha \le \mu$

$\alpha \le {\tan }^{-1}\mu$

$\alpha \le {\tan }^{-1}\left(\frac{1}{3}\right)=18.43{}^o$

Maximum angle can be, $\alpha =18.43{}^o$