A small sized mass `m` is attached by a massless string (of length `L`) to the top of a fixed frictionless solid cone whose axis is vertical. The half angle at the vertex of cone is theta. If the mass `m` moves around in a horizontal circle at speed `v` , what is the maximum value of `v` for which mass stay in contact with the comes? (g is acceleration due to gravity.)

Options

A. `sqrt(gL cos theta)`
B. `sqrt(gL sin theta)`
C. `sqrt(gL sin theta tan theta)`
D. `sqrt(gL tan theta)`

Answer

C

Explanation

At maximum velocity the mass will just loose contact with cone and will behave like free conical pendulum with time period :
`T = 2 pi sqrt((L cos theta)/(g))rArr omega =(2pi)/(T)sqrt((g)/(L cos theta))`
Hence `v_(max) = (L sin theta) omega = sqrt(gL sin theta tan theta)`.