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An insect crawls up a hemispherical surface very slowly. The coefficient of friction between insect and surface is $\large\frac{1}{3}$. If the line joining the center of hemispherical surface to the insect makes an angle $\alpha$ with the vertical the maximum possible angle of $\alpha$ is

$a) \cot \alpha=3 \qquad b)\sec \alpha=3 \qquad c) cosec \alpha=3 \qquad d) none$

Suppose the insect can crawl upto a point P
$N=w \cos \alpha$-----(1)
$f=frictional\; force= w \sin \alpha$-----(2)
from (1) and (2)
$\tan \alpha=\large\frac{f}{N}$
In limiting case $\tan \alpha =\mu$
Therefore $\tan \alpha=\large\frac{1}{3}$
$\cot \alpha=3$
Hence a is the correct answer.

edited Feb 10, 2014 by meena.p