Suppose the rod with the balls A and B of theprevious problem is clamped at the centre in such a way that it ca rotate freely about a horizontal axis through the clamp. The system is kept at rest in the horizontal position. A particle P of the same mass m is dropped from a heigh h hon the ball B. The particle collides with B and sticks to it. a. Find the angular momentum and the angular speed of the system just after the collision. b. What should be the minimum value of h so that the system makes a full rotation after the collision.

Answer

A, B, C

Explanation

a. Angular momentum `=mvr`
conservation of linear momentum
`mu=2mv-mv=mv`
`:.u=v=mvr`

Velocity =sqrt(2gh)`
and `r=L/2`
`Angular momentum =m.sqrt(2gh.L/2`
`=(mLsqrt(gh))/sqrt2`
Angular momentum =lw
`:.omega=angular velocity =L/I`
or `=I=(2mL^2)/4+(mL^2)/4=(3mL^2)/4`
`:. omega=(mLsqrtI(gh)/sqrt2)/(3mL^2/4)=sqrt(8gh)/(3L)`
b. When the mass 2m wil be at the top most position and the mass m be that the lowest point. They will automaticaly rotate.
in this position the total gaion in potential energy
`=2mgxx(L/2)-mg(L/2)`
therefore `mgL/2=((1/2x2mL^2))/4xx((8gh)/(9gL^2))`
`rarr h=(3L)/2`