question_answer

An insect 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 centre of the hemispherical surface to the insect makes an angle a with the vertical, the maximum possible value of a so that the insect does not slip is given by [JEE ONLINE 12-05-2012]

A) \[\cot \alpha =3\]

B) \[\sec \alpha =3\]

C) \[\cos ec\,\alpha =3\]

D) \[\cos \,\alpha =3\]

Correct Answer: A

Solution :

[a]

The insect crawls up the bowl up to a certain height h only till the component of its weight along the bowl is balanced by limiting frictional force.

For limiting condition at point A

\[R=mg\cos \alpha \]                               ...(i)

\[{{F}_{1}}=ma\sin \alpha \]                               ...(ii)

Dividing eq.(ii) by (i)

\[\tan \alpha =\frac{1}{\cot \alpha }=\frac{{{F}_{1}}}{R}=\mu \left[ As\,{{F}_{1}}=\mu R \right]\]

\[\Rightarrow \]\[\tan \alpha =\mu =\frac{1}{3}\left[ \because \mu =\frac{1}{3}(\text{Given}) \right]\]

\[\therefore \]\[\cot \alpha =3\]