Question

An insect crawls up a hemispherical surface very slowly (see the figure). The coefficient of friction between the insect and the surface is 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. $\tan \alpha = 3$
• C. $\sec \alpha = 3$
• D. $\operatorname { cosec } \alpha = 3$

Answer 1

Answer: (A)

$\cot \alpha = 3$

Answer 2

From the above expression, for the equilibrium $R = m g \cos \alpha$ and $F = m g \sin \alpha$ .

Substituting these value in $F = \mu R$ we get $\tan \alpha = \mu$ or $\cot \alpha = \frac { 1 } { \mu } = 3$ .