A thin rod of mass M and length L is pivoted about one end a

A thin rod of mass M and length L, is pivoted about one end. as shown in the figure. The rod is released from rest in the horizontal position, and allowed to swing downward without friction. On reaching the vertical position, it hits a pellet of mass m = M/3, which sticks to its end. The ratio wa/wb of the angular speeds of the system immediately after and before the collision is

Solution

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The initial angular velocity can be obtained by :

decrease in potential energy = increase in kinetic energy i.e mgl/2  =  .5IW_B² gives W_B = 3g/l which is obtained by I = Ml²/3.

after collision ,the conservation of angular momentum gives the answer:

initial angular momentum = Ml²/3 (W_B) .Final angular momentum = (Ml²/3 + Ml²/3)W_A I.e the moment of inertia of both the rod and the small particle .The particles is (M/3)l² gives

W_A/W_B = 1/2.