Question

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. $cos \,\alpha = 3$

Answer 1

Answer: (A)

$cot\, \alpha = 3$

Answer 2

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 $R = mg \,cos \,\alpha \quad\quad...\left(i\right)$ $F_{1} = mg\, sin \,\alpha\quad\quad ...\left(ii\right)$ Dividing e $\left(ii\right)$ by $\left(i\right)$ $tan\,\alpha = \frac{1}{cot\,\alpha} = \frac{F_{1}}{R} = \mu\left[As\,F_{1} =\mu R \right]$ $\Rightarrow\quad tan\,\alpha =\mu = \frac{1}{3} [\because\,\mu($Given$] )$ $\therefore\quad cot \,\alpha = 3$