Question

An insect crawls up a hemispherical surface very slowly. 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 given by

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

Answer 1

The correct option is A

$cot\alpha =3$

The two forces acting at the insects are mg and N.Let us resolve mg into two components mgcos$\alpha$ balances N

mgsin$\alpha$ is balanced by the frictional force.

$\therefore N=mgcos\alpha$

f=mgsin$\alpha$

But f =$\mu N=\mu mgcos\alpha$

$\therefore \mu mgcos\alpha =mgsin\alpha$

$\Rightarrow cot\alpha =\frac{1}{\mu }\Rightarrow cot\alpha =3$