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 it `v gt sqrt(5) m//s`
B. normal force on bob is `19 N` when `v = 2 m//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))`
`rArr T sin 30^(@) - 3 N = 4mv_(2)`
`sqrt(3)T - 3N =4mv_(2)`....(ii)
by (i),(ii) `rArr N =(2mg-4mv^(2))/(4)`
`T = (6mg-4mv^(2))/(4 sqrt(3))`
for `N gt0 rArr v lt sqrt(5) m//s`
at `v = 2 T = (38)/(sqrt(3)) N , N = 2 N`.