# Stuck on a rotational dynamics related question

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.

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.

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