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.
• B.
• C.
• D.

Answer 1

The correct option is A

Step 1,

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.

We can draw the given situation as

Step 2,

We can write

$\therefore N=mgcos\alpha$

Frictional force is given as

f=mgsin$\alpha$

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

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

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

Hence the value of $\alpha$ is $cot\alpha =3$