A bead of mass `m` is threaded on a smooth circular wire centre `O`, radius a, which is fixed in vertical plane. A light string of natural olength 'a', elastic constant `= (3mg)/(a)` and breaking strength `3mg` connects the bead to the lowest point `A` of the wire. The other end of the string is fixed to ring at point `B` near point `A`. The string is slaked initially. The bead is projected from `A` with speed `u`.

If `v =2 u_(0)`, the tension `T` in th elastic string when the bead is at the highest point `B` of the wire is

Options

A. `(3mu_(0)^(2))/(a)`
B. `4mg`
C. `2 mg`
D. `((4 u_(0)^(2))/(a)-g)m`

Answer

D

Explanation

When `v = 2v_(0) = 2 sqrt(7ga)`, For point `B`
`
(m(2v_(0))^(2))/(a) =3mg + mg-N`
`(4mv_(0)^(2))/(a) =4mg -N`
`N=(4-28) mg = -24 mg`
negative sign denote it acts downwards and adds to tension
total tension in string `T =3mg+N`
`T=((4v_(0)^(2))/(a)-g)m`.