An insect craws up a hemispherical surface very slowly (see fig.). The coefficient of friction between the insect and the surface is `1//3`. If the line joining the center of the hemispherical surface to the insect makes an angle `alpha` with the vertical, the maximum possible value of `alpha` is given by

Options

A. `cot alpha=3`
B. `tan alpha=3`
C. `sec alpha=3`
D. `cosec alpha=3`

Answer

A

Explanation

KEY CONCEPT:
For the maximum possible value of `alpha`,
`mg sin alpha` will also be maximum and equal to the frictional force.
In this case f is the limiting friction. The two forces acting on the insect are mg and N. Let us resolve mg into two components. `mg cos alpha` balances N. `mg sin alpha` is balanced by the frictional force.
`:.` `N=mg cos alpha`
`f=mg sin alpha`
But `f=muN=mu mg cos alpha`
`:.` `mu mg cos alpha=mg sin alphaimplies cot alpha = 1/muimpliescot alpha =3`