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 $\alpha $ with the vertical, the maximum possible value of $\alpha $ so that the insect does not slip is given by
AIEEE 2012,IIT 2001, Difficult
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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 li miting condition at point $A$
$R=m g \cos \alpha.....(i)$
$F_1=m g \sin \alpha.....(ii)$
Dividingeq. $(ii)$ by $(i) \tan \alpha=\frac{1}{\cot \alpha}=\frac{F_1}{R}=\mu\left[A s F_1=\mu R\right]$
$\Rightarrow \tan \alpha=\mu=\frac{1}{3}\left[\mu=\frac{1}{3}(\right.$ given $\left.)\right]$
$\therefore \cot \alpha=3$
$R=m g \cos \alpha.....(i)$
$F_1=m g \sin \alpha.....(ii)$
Dividingeq. $(ii)$ by $(i) \tan \alpha=\frac{1}{\cot \alpha}=\frac{F_1}{R}=\mu\left[A s F_1=\mu R\right]$
$\Rightarrow \tan \alpha=\mu=\frac{1}{3}\left[\mu=\frac{1}{3}(\right.$ given $\left.)\right]$
$\therefore \cot \alpha=3$
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