Question

A uniform rod of mass m is hinged at its upper end. A particle of mass m moving horizontally strikes the rod at its midpoint elastically. If the particle comes to rest after collision find the value of m/m

Answer 1

Let V be the velocity of the ball at which it strikes the rod at its mid point.

From conservation of momentum

L₁=L₂

L₁= initial angular momentum

L₂=final angular momentum

mv× 1/2+ 0= 0+ (I×ω)----- (1)

where I= moment of inertia, ω= angular velocity

(1)=> (mvL/2)= (ML²/3)×ω

 v= (2ML/3)×(ω)----- (2)

From conservation of energy,

(1/2)Iω²= (1/2)mv²

mv²=(ML²/3)×ω²

(Here, m= mass of the ball and M= mass of the rod)

M/m=(3v²/L²ω²)=3(v/ωL)²

From eqn (2), v/ωL= (2M/3m)

M/m=3×(2/3 M/m)²

=> M/m= (4M²/9m²)

=> M/m=3/4