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 inse

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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 $α$ with the vertical, the maximum possible value of $α$ so that the insect does not slip is given by

TestHub · Accepted Answer

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 $A$ $R = m g cos α$ $F_{1} = m g sin α$ Dividing eq. (ii) by (i) $tan α = \frac{1}{c o t α} = \frac{F _{1}}{R} = μ [$ As $F_{1} = μ R]$ $\Rightarrow tan α = μ = \frac{1}{3} [∵ μ = \frac{1}{3} ($ Given $])$ $∴ cot α = 3$

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