Question

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

• A. $\cot \alpha =3$
• B. $\tan \alpha =3$
• C. $\sec \alpha =3$
• D. $\cos \alpha =3$

Answer 1

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

Given that the insect is crawling very slowly i.e its acceleration =$0$. Hence, we can apply the condition of equilibrium for the insect in the tangential direction.

$N=mg\cos \alpha$.........(1)
Equilibrium condition in radial direction gives
$\mu N=mg\sin \alpha$..........(2)


On dividing equation (i) and (ii), we get:
$\cot \alpha =\frac{1}{\mu }=3$