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We use the idea that if a body collides with something, then its angular momentum is conserved with respect to the point of impact. Source

It is clear to me that all impulsive forces and the colliding bodies weight gets cancelled off, but what about the weight of the body which gets hit?

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In the above rod, when it is hit by a ball at any point initially when it is perfectly vertical the statement is true but as soon as it moves even a little, it is false. Hence, it doesn't seem right to me to say that angular momentum and after is same.

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The angular momentum is conserved when there is no net torque acting on the body. In this case, the weight of the body is providing a restoring torque, and hence the angular momentum is not conserved.

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This question is the rotational dynamics equivalent of the query Can linear momentum be conserved before and after collision in the presence of an external force? but in this case it is the impulsive torque due to the gravitational force which is very much less than the impulsive torque due to the applied impulse.

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What is CP and if you are asking L would be conserved or not it would be conserved just before and after the collision but not in the subsequent motion of the rod since weight of the rod will apply a net anticlockwise torque.

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