Question

An inset 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 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 $cot\alpha =3$

Here,

From the figure,

$N=mg\cos \alpha ⟶\left(i\right)$

$f=mg\sin \alpha$

$\mu N=mg\sin \alpha$

now, $\mu =\frac{mg\sin \alpha }{N}$

$\mu =\frac{mg\sin \alpha }{mg\cos \alpha }$ (from (i))

$\mu =\tan \alpha$

$\frac{1}{3}=\tan \alpha$

$3=\cot \alpha$