The angular momentum remains constant because there is a changing net torque acting on the system that averages to zero over time.
This principle states that If the external torque acting on a system is zero, then the angular momentum of the system is constant.
When the spheres-rod system is thrown straight upward and it rotates counterclockwise, the net torque acting on the system averages to zero over time.
Thus, we can conclude that, the angular momentum remains constant because there is a changing net torque acting on the system that averages to zero over time.
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