In physics, Carroll's paradox arises when considering the motion of a rigid rod. Considered one way, the angular momentum stays constant; considered in a different way, it changes.

Consider a uniform rigid heavy rod of length l = r₂ - r₁ and two vertical concentric circles of radius r₂ and r₁. The rod is constrained so that one end remains on the inner circle and the other remains on the other circle; motion is frictionless. The rod is held so that it is horizontal, then released.

Now consider the angular momentuim:

1. The reaction force on the rod (from either circular guide) is frictionless, so it must be directed along the rod; there can be no component of the reaction force perpendicular to the rod. Taking moments about the center of the rod, there can be no moment acting on the rod, so its angular momentum remains constant. Because the rod starts with zero angular momentum, it must continue to have zero angular momentum for all time.
2. After release, the rod rotates, moving like the hands on a clock. When it gets to the six o'clock position, it has lost potential energy and, because the motion is frictionless, will be moving. It therefore posesses angular momentum.

The resolution of this paradox is not clear.

References

• Victor Namias. On an apparent paradox in the motion of a smoothly constrained rod, American Journal of Physics, 54(5), May 1986.