Question

A ball of mass in `m= 1 kg` is hung vertically by a thread of length `l = 1.50 m`. Upper end of the thread is attached to the ceiling of a trolley of mass `M= 4 kg`. Initially, the trolley is stationary and it is free to move along horizontal rails without friction. A shell of mass `m=1 kg`, moving horizontally with velocity `v_(0) = 6 m//s` collides with the ball and gets stuck with it. As a result, the thread starts to deflect towards right.

The maximum deflection of the thread with the vertical is
A. `cos^(-1)(4/5)`
B. `cos^(-1)(3/5)`
C. `cos^(-1)(2/3)`
D. `cos^(-1)(3/4)`

Answer 1

Correct Answer - A
When shell strikes the ball and gets stuck with it, combined body of mass `2m` starts to move to the right, Let velocity of the combined body (just after collision) be `v_(1)`. According to law of conservation of momentum,
`(m+m) v_(1)=mv_(0)`
`v_(1)=(v_(0))/2=3m//s`
As soon as the combined body starts to move rig wards, thread becomes inclined to the vertical. Horizontal com tension retards the combined body while trolley accelerate rightwards due to the same component of tension.
Inclination of thread with the vertical continues to increase till velocities of both (combined body and trolley) become identical or combined body comes to rest relative to the trolley. Let velocity at that instant of maximum inclination of thread be `v`. according to law of conservation of momentum,
`(2m+M)v=2mv_(1)`
or `v=1m//s`
During collision of ball and shell, a part of energy is lost. But after that there is no loss of energy., Hence, after collision, kinetic energy lost is used up in increasing gravitational potential energy of the combined body.
If maximum inclination of threads with the vertical is `theta`, then according to law of conservation of energy.
`=1/2(2m)v_(1)^(2)-1/2(2m+M)v^(2)=2mg(l-lcostheta)`
`costheta=0.8` or `theta=37^(@)`