Two small balls A and B , each having mass m , are rigidly connected by a light horizontal rod of length L . The rod is clamped at its centre such that it can freely rotate about a vertical axis passing through the centre. The system is set into rotation with an angular speed about this axis. A stationary particle P of mass m sticks to ball A upon collision. Determine the new angular speed of the rod.

Step-by-step solution

The initial moment of inertia of the system about the vertical axis of rotation is I_i = m ( L 2 )^2 + m ( L 2 )^2 = mL^2 2 The initial angular momentum of the system is L_i = I_i = mL^2 2 When the stationary particle of mass m sticks to ball A , the mass at A becomes 2m . The final moment of inertia of the system is I_f = 2m ( L 2 )^2 + m ( L 2 )^2 = 3mL^2 4 Since no external torque acts on the system about the axis of rotation, the angular momentum is conserved. Equating the initial and final angular momentum: L_