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

A) \[\cot \,\,\alpha =3\]       

B)        \[\sec \,\,\alpha =3\]

C) \[cosec\,\,\alpha =3\]     

D)        \[cos\,\,\alpha =3\]

Correct Answer: A

Solution :

[a] 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) \[{{F}_{1}}=mg\,\,\sin \alpha \]                          ...(ii) Dividing eq. (ii) by(i) \[\tan \alpha =\frac{1}{\cot \alpha }=\frac{{{F}_{1}}}{R}=\mu [As\,\,{{F}_{1}}=\mu R]\] \[\Rightarrow \] \[\tan \alpha =\mu =\frac{1}{3}\]\[\left[ \because \mu =\frac{1}{3}(\text{Given}) \right]\]