A particle of mass `m` initially at rest starts moving from point `A` on the surface of a fixed smooth hemisphere of radius `r` as shown. The particle looses its contact with hemisphere at point `B.C` is centre of the hemisphere. The equation relating `theta` and `theta'` is
.

Options

A. `3 sin alpha=2 cos beta`
B. `2 sin alpha=3 cos beta`
C. `3 sin beta=2cos alpha`
D. `2 sin beta=3 cos alpha`

Answer

C

Explanation

`(C )` Let `v` be the speed of particle at `B`, just when it is about to loose contact.
From application of Newton's second law to the particle normal to the spherical surface.
`(mv^(2))/(r)=mg sin beta ....(1)`
Applying conservation of energy as the block moves from `A` to `B`.
`(1)/(2) mv^(2)=mg(r cos alpha-r sin beta ) ....(2)`
Solving 1 and 2 we get
`3 sin beta = 2 cos alpha `