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 Main Online Paper (Held On 12 May 2012)

A) \[\cot \alpha =3\]           

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

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

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

Correct Answer: A

Solution :

The insect crawls up the bowl upto 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) \[\tan \alpha =\frac{1}{\cot \alpha }=\frac{{{F}_{1}}}{R}=\mu \left[ As\,{{F}_{1}}=\mu R \right]\] \[\Rightarrow \]\[\tan \alpha =\mu =\frac{1}{3}\left[ \because \mu =\frac{1}{3}(\text{Given}) \right]\] \[\therefore \]\[\cot \alpha =3\]