Paradox in physics
In physics, Carroll's paradox arises when considering the motion of a falling rigid rod that is specially constrained. Considered one way, the angular momentum stays constant; considered in a different way, it changes. It is named after Michael M. Carroll who first published it in 1984.
Explanation.
Consider two concentric circles of radius formula_1 and formula_2 as might be drawn on the face of a wall clock. Suppose a uniform rigid heavy rod of length formula_3 is somehow constrained between these two circles so that one end of the rod remains on the inner circle and the other remains on the outer circle. Motion of the rod along these circles, acting as guides, is frictionless. The rod is held in the three o'clock position so that it is horizontal, then released.
Now consider the angular momentum about the centre of the rod:
An apparent resolution of this paradox is that the physical situation cannot occur. To maintain the rod in a radial position the circles have to exert an infinite force. In real life it would not be possible to construct guides that do not exert a significant reaction force perpendicular to the rod. Victor Namias, however, disputed that infinite forces occur, and argued that a finitely thick rod experiences torque about its center of mass even in the limit as it approaches zero width.