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A particle is describing circular motion...

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 height of the plane of motion above the vertex is `h` and the semi-vertical angle of the cone is `alpha` The period of revolution of the particle

A

increases as `h` increases

B

decreases as `h` decreases

C

increases as `alpha` increases

D

decreases as `alpha` decreases

Text Solution

Verified by Experts

The correct Answer is:
A, D
`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`
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