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A rod is hinged in a vertical plane. If we throw a ball at the rod's lowest point with some velocity, and they undergo elastic collision, can the total mechanical energy be conserved? What about the total Kinetic energy? I think the former cannot, since we do not know if the hinge's force on the rod is conservative or not.

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    $\begingroup$ Instead of thinking about whether or not the hinge force is conservative, how about you just think about the work that force does on the rod. $\endgroup$ – Aaron Stevens Jul 29 at 14:00
  • $\begingroup$ @AaronStevens I see. If the work done by the hinge force on rod is zero, the change in KE would be zero. Obviously the work done by the force on rod would be zero since displacement of topmost point (where the hinge force acts) is 0. So the initial KE is equal to the final KE. Is this correct? $\endgroup$ – PhysicsMonster_01 Jul 29 at 17:09
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    $\begingroup$ Yes, at least right before and right after the collision, since it is an elastic collision $\endgroup$ – Aaron Stevens Jul 29 at 17:10
  • $\begingroup$ @AaronStevens Thank you. My doubt is cleared, how do I close this discussion? $\endgroup$ – PhysicsMonster_01 Jul 29 at 17:11
  • $\begingroup$ You could answer your own question for future readers if you would like $\endgroup$ – Aaron Stevens Jul 29 at 17:13
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Instead of trying to check if the hinge force is conservative (which can be done, of course, by finding the curl of the force), let us try looking at the work done by the hinge force on the rod. Work done by the hinge force would simply be $\int{\mathbf F_\text{hinge}\cdot \text d\mathbf r}$, where $\text d\mathbf r$ is the displacement of the point of action of force. Now, the hinge force acts on the topmost point, for which $\text d\mathbf r$ is zero. So the work done by hinge force on the rod is zero. Since the ball collides with the rod elastically, we can safely say that the kinetic energy of the system just before collision is equal to the kinetic energy of the system just after it(the collision).

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