When a ball hits a rod at its extreme, causing it to rotate about its mass center, the angular momentum is conserved with respect to the mass center of the rod. This conservation occurs even though the weight force's momentum is not zero. The reason for this can be understood by considering the principle of conservation of angular momentum.
Angular momentum is defined as the product of the moment of inertia and the angular velocity of an object. In this case, the rod is rotating about its mass center, so the angular momentum is given by:
L = I * ω,
where L is the angular momentum, I is the moment of inertia of the rod about its mass center, and ω is the angular velocity.
When the ball hits the rod at its extreme, it exerts a torque on the rod. This torque causes the angular momentum of the system to change. According to the principle of conservation of angular momentum, the total angular momentum of an isolated system remains constant if no external torques act on it.
In this situation, the weight force does exert a torque on the rod, but it acts about the mass center of the rod. The torque due to the weight force tends to change the angular momentum of the rod about its mass center. However, since the rod is fixed at its mass center, there is an equal and opposite reaction torque applied by the support at the extreme of the rod.
This reaction torque counteracts the torque due to the weight force, ensuring that the total torque acting on the system is zero. As a result, the angular momentum of the system is conserved about the mass center of the rod.
To summarize, the angular momentum is conserved with respect to the mass center because the external torques acting on the system, such as the weight force, are balanced by the reaction torques exerted by the support at the extreme of the rod.