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A ball of mass $\frac M 2$ strikes the bottom point $P$ of a rod of mass $M$ and length $L$ hinged at the top point O with velocity $u$. What is the angular velocity of the rod-ball system just after collision?

I went about solving the question through conservation of angular momentum since external torque is zero and got the correct answer by conserving it about the hinge with $w=\frac{3u}{5l}$. Then I thought about conserving angular momentum about the point $P$ but couldn't since the $L_i$ equals zero since $L_i=\frac M 2 ur_⊥=0$ due to $r_⊥$ being zero while $L_f = Iw + \frac M 2 L^2w$ where $Iw$ can be written as $ML^2/3 w$ Moreover I can't get the same value of $w$ via conservation of translational and rotational kinetic energy where I took into account rotational kinetic energy of the rod-ball system about the hinge and the ball's kinetic energy at point $P$. I'm stumped for now.

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Then I thought about conserving angular momentum about the point $P$ . . . .
You cannot as the hinge exerts an external torque on the rod.

. . . . . I can't get the same value of w via conservation of translational and rotational kinetic energy . . . .
Not surprising as the collision between the rod and the ball is inelastic (ie kinetic energy is not conserved.

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  • $\begingroup$ OH RIGHT I forgot about inelastic collision they're literally stuck together, my bad. This was dumb. $\endgroup$
    – Random
    May 28, 2022 at 10:56

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