question_answer

An insect crawls up a hemispherical surface very slowly (figure). The coefficient of friction between the insect and the surface is 1/3. If the line joining the centre of hemispherical surface to the insect makes an angle a 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)  \[\cos ec\,\alpha =3\]

Correct Answer: A

Solution :

 To avoid slipping, \[{{f}_{r}}=mg\,\sin \,\,\alpha \] at maximum \[\alpha \]\[\mu N=mg\,\,\sin \,\,\alpha \] \[\Rightarrow \]            \[\mu mg\,\,\cos \,\,\alpha =mg\,\,\sin \,\,\alpha \] \[\therefore \]    \[\mu =\tan \,\alpha \] \[\Rightarrow \]            \[\tan \,\alpha =\frac{1}{3}\]  \[\left( as\,\mu =\frac{1}{3} \right)\] \[\therefore \]    \[\cot \,\,\alpha =3\]