A uniform rod of mass M and length L is rotating freely with an angular velocity ₀ in a horizontal plane about a vertical axis passing through its center. Two small beads, each of mass m , are initially held at the center of the rod. The beads are then released and slide outwards along the rod until they reach the ends. The ratio of the final rotational kinetic energy to the initial rotational kinetic energy of the s

Step-by-step solution

Since there is no external torque acting on the system, the angular momentum L is conserved. The rotational kinetic energy can be expressed in terms of angular momentum as K = L^2 2I . Since L is constant, the ratio of final to initial kinetic energy is K_f K_i = I_i I_f . Initially, the beads are at the center, so their distance from the axis is zero. The initial moment of inertia is just that of the rod: I_i = ML^2 12 Finally, the beads are at the ends of the rod, at a distance of L 2 from the axis. The final mom