Question

An insect crawls up a hemispherical surfaces very slowly as shown in the figure. The coefficient of 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, the maximum possible value of $\alpha$ is given by

• A. $\cot \alpha =3$
• B. $\sec \alpha =3$
• C. $\text{cosec}\alpha =3$
• D. None of these

Answer 1

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

As the insect is moving slowly up the surface, we can assume that it has negligible/zero acceleration.
From the FBD, we can see that,
For maximum $\alpha$, limiting friction $f_l$ acts as shown

$f_l-mg\sin \alpha =0$
$⟹f_l=mg\sin \alpha$
$R-mg\cos \alpha =0$
$\frac{f_l}{R}=\tan \alpha$
$\frac{\mu R}{R}=\tan \alpha$
i.e., $\mu =\tan \alpha =\frac{1}{3}$
$⟹\cot \alpha =3$