A bob of mass `2kg` is suspended from point `O` of a core with an inextensible string strinng of length `sqrt(3)m`. It is moving in horizontal circle over the surface of cone as shown in the figure. Then `:(g=10m//s^(2))`

Options

A. bob looses contact with cone if`vgtsqrt(5)m//s`
B. normal force on bob is `19N` when `v=2m//s`
C. tension in string is `(38)/(sqrt(3))N` when `v=2m//s`
D. normal force on bob is `(17)/(sqrt(3))N` when `v=2m//s`

Answer

A, C

Explanation

`T=cos 30^(@)+N sin 30^(@)=mg`
`rArr sqrt(3)T+N=2mg ......(i)`
`T sin 30^(@)-N cos 30^(@)=(mv^(2))/((sqrt(3)//2))rArrT sin 30^(@)-3N =4mv^(2) .....(ii)`
by `(i),(ii)= gt N =(2mg-4mv^(2))/(4) , T=(6mg-4mv^(2))/(4sqrt(3))`
for `N gt 0rArr vlt sqrt(5) m//s`
at `v=2, T=(38)/(sqrt(3))N , N=2N`.
Solution `:`
In `P,Q` and `S,` the centre of masses lies at `D//2` height from the base level. WHere as in `R(` cons `)` the complies at `D//4` height from the base.
Hence `U_(P) gt U _(Q)`
`Ans (C )`.