# 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?

• What is a sheet of paper if not a rectangle made of paper? It's not entirely clear what phenomenon you're describing here - can you go into a bit more detail? – ACuriousMind Sep 27 '16 at 13:40
• @ACuriousMind Isn't it the same thing? I meant a plane rectangle of paper of qbout A4 size. When you drop it, the piece of paper is not spinning. But when it is a strip, way longer than large, the piece of paper spins on itself. – Igor Mofournier Sep 27 '16 at 13:43
• Ah, I see. I was confused by your wording because you ask why a "paper rectangle" spins on itself when a "sheet of paper" moves from side to side - but to me "sheet of paper" is the same as "paper rectangle". "Strip" makes it much clearer what you're talking about. – ACuriousMind Sep 27 '16 at 13:46
• The answer is right here. Note "You can improve the performance by giving the wing a finer aspect ratio". – Mike Dunlavey Sep 28 '16 at 12:36

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.

After several minutes experimenting (The scientific method is great!) my best guess would be that this effect is dependent on the absolute size of the paper far more than its "aspect ratio".

That is to say, only the narrowest side of the paper matters (ie. an A4 sheet behaves similarly to a sheet that has one side the same as the shorter edge of an A4, and the other 3 metres long...) Try it! (not with my ridiculous hypothetical example, but cut out different shapes and see how they respond!)

If you carefully observe the motion, you will notice that the "spinning" phenomenon occurs as a result of the following process:

1. The paper "tilts" to the side, moving slightly upwards with the leading edge
2. The reactionary forces due to air resistance push it the other way, and it begins to move towards its lower edge, tilting back to horizontal and then in the other direction
3. Steps 1. and 2. repeat until (for a sufficiently small piece of paper) this "tilting" action is enough to flip the leading edge over itself, beginning the spinning motion

It appears as though the large piece of paper is subjected to too great a force from air resistance on its "top" side; if the paper's shortest dimension is above a certain critical value, it will not be able to "push over" regardless of how large or small the longer dimension is, because it is unable to force the required volume of air out the way.

It is perhaps instructive to consider this motion as a chaotic system: the librational regime of the paper (steady motion for the large sheet) represent a local minimum in phase space, but, with a sufficient "kick", can be brought over to the rotational regime: a critical "phase transition", after which it remains stuck with rotations which are, presumably, a global minimum. The small sheets make this transition easily, while the large sheets cannot naturally gather enough energy to do so.

• +1 for experimenting. You seem to be using thin, flexible paper. I suggest it might be better to experiment with stiff card, in order to avoid the complications caused by the bending of paper. – sammy gerbil Sep 28 '16 at 12:53
• @sammy gerbil The bending becomes less relevant as the paper size decreases; for small sheets I couldn't really observe bending to have a significant effect... – tusky_mcmammoth Sep 28 '16 at 13:15

To answer the part of the question about why the thin strip spins, it's a simple matter of circulation. It is not stable and it can more easily spin about its long axis. That causes lift, so it glides. The glide angle is stable because a steeper angle gives greater airspeed, giving greater spin speed, giving greater lift, giving a less-steep angle. Ordinary gliders don't have spinning wings, but they have circulation that works the same way.

(The link gives a good intuitive explanation for those who don't want to read more than a screenful. For those who want a more formal understanding, the link contains plenty more to read - too much to include here, but a very good read.)

Here is the key illustration.

To answer the part about why a simple sheet of paper follows a more chaotic motion, my guess is no better than the other guesses. There is no obvious preferred stable axis of rotation.

• Your answer should be self-contained. It should not be necessary to study a very large webpage to find the answer to which you are alluding. The question is not how circulation creates lift but why a strip of paper does not fall in the same way as a sheet of paper. – sammy gerbil Sep 29 '16 at 15:14