classical mechanics - Paper rectangle spinning phenomenon

Why does a paper strip (small width and long length) spin on itself when dropped rather than move from side to side as a sheet of paper would do?

Answer

There are three reasons:

1. Pitch damping: The rotation creates an additional speed in the parts of the paper away from the rotation axis, and this speed causes an additional force which counteracts the motion. Note that the effect of pitch damping increases with the square of the distance to the rotation axis: Not only will the distance linearly increase the force, it will also linearly increase the lever arm of this force.


2. Inertia: As lightweight as the paper is, it does have inertia, and too much of it prevents it from picking up rotational speed. Again, this effect scales nonlinearly with length: The rotational inertia goes up with the cube of the length.


3. Stiffness: If you compare two sheets, one of high aspect ratio and one of low aspect ratio, you will not be able to scale their thickness. The longer paper will have considerable less bending stiffness, and the short, high aspect ratio one will be comparatively stiff, which will cause it to rotate without bending, while the low aspect ratio paper will bend and move from side to side rather than rotate. Try repeating the test with boards of balsa wood: Now also the low aspect ratio board will spin and not bend.

The sideways motion is a consequence of bending: The leading edge of the paper will bend up and create a force normal to the local inclination which will result in the side-to-side motion.