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I have a situation that is as follows:

A rod of mass $M$ is hinged at one of its end A on a smooth horizontal surface and can rotate about A without friction. A particle of mass $m$ moving on the horizontal plane strikes the rod and comes to rest just after collision.

The question asks to comment on the direction of impulsive hinge reaction at A. It's pretty clear to me that the reaction can be in the forward to backward direction depending on where the mass strikes the rod.

But the book says that the hinge reaction cannot be zero during the collision. Why is this? What if the mass hits the rod at its center of percussion (COP)? Isn't the hinge reaction zero at that case?

https://sciencedemonstrations.fas.harvard.edu/presentations/center-percussion

This clearly shows three cases where the hinge reactions will be in forward, backward direction in the first two as well as zero if hit at center of percussion. What am I missing here?

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Your link showed the direction of the reaction at the hinge for three situations, where the point of impact is above, below and at the Center of Percussion.

However these refer to forces at the hinge perpendicular to the rod.

Even when the rod is struck at the COP, there will be a rotation of the rod caused by the impact (due to conservation of momentum). The rod will rotate about the hinge, with the point at the hinge stationary.

There will then need to be a force to provide the centripetal acceleration, this force acts at the hinge and is parallel to the rod.

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  • $\begingroup$ Ahh that makes sense. And can I comment on the direction of this hinge force (which acts parallel to the rod)? According to me it should be acting in the positive y direction (taking the rod to be lying in the horizontal xy plane) along the rod towards the hinge. Can it ever act away from the hinge? $\endgroup$
    – Techie5879
    Commented Sep 25, 2021 at 11:03
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    $\begingroup$ @Techie5879 Yes the force parallel to the rod (on the rod) would always be towards the hinge (upwards) on the diagrams in the link, (at least after a short time) as it must provide the centripetal force on the rod (some amount of rotation of the rod around the hinge is bound to happen) - one thing that isn't clear yet is if 'during the collision' the parallel force can act the other way?? Will think about that one...Perhaps best to assume not at the moment $\endgroup$
    – John Hunter
    Commented Sep 25, 2021 at 11:22

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