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What do I do next after finding the linear velocities?

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A rod of length $l$ falls on two metal pads of same height from a height $h$. The coefficients of restitution of the metal pads are $e_1$ and $e_2$ $e_1>e_2$. The angular velocity of the rod after it recoils is?

Attempt:

I have calculated the velocities after collision to be $e_1\sqrt{2gh}$ and $e_2 \sqrt{2gh}$.
I feel there's certainly going to be rotation but I am not able to calculate the angular velocity. Will the rod rotate about the centre of mass? If yes, why is it so?

asked Mar 22 in Physics Problems by Reststack (404 points)

1 Answer

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Immediately after the collision the line joining the 2 points of contact A, B is initially horizontal. Because end A is moving upwards faster than end B the line AB rotates. The angular velocity $\omega$ of the rod is the rate at which line AB rotates. You can find this from $\omega=(v_A-v_B)/l$ assuming the distance AB is $l$.

The angular velocity of a rigid body is the same whichever 2 points are chosen as reference. Pick any two points and measure how fast the line joining them rotates.

Any rigid body motion can be resolved into the translation of a reference point and a rotation about that point. Often we use the CM as the reference point, for convenience. In this case we know the CM follows a parabolic trajectory while gravity is the only external force acting on the rod. But you can use any other point as reference (eg A or B), even one outside of the body.

answered Mar 23 by sammy gerbil (26,096 points)
edited Mar 26 by sammy gerbil
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