A particle is describing circular motion in a horizontal plane in contact with the smooth surface of a fixed rigid circular cone with its axis vertical and vertex down. The heigh of the plane of motion above the vertex is h`h` and the semi-vertical angle of the cone is `alpha` The period of revolution of the particle

Options

A. increases as `h` increases
B. decreases as `h` decreases
C. increases as `alpha` increases
D. decreases as `alpha` decreases

Answer

A, D

Explanation

`R=hcotalpha`
`Ncosalpha=mg`
`Nsinalpha=(mv^(2))/(R)` ltbr. Solving these two equations, wwe get ltbr. `v=sqrt(Rgtanalpha)` ltbr. `=sqrt((hcotalpha)(gtanalpa)`
`=sqrt(gh)`

Now,
`T=(2piR)/(v)=(2pihcotalpha)/(sqrt(gh))`
`=2pisqrt((h)/(g))cotalpha`
:. `Tpropsqrt(h)`
`Tporpcotalpha`