## Sunday, August 18, 2019

### 254: How To Annoy Your Cat

If you have a pet cat like I do, you’ve probably noticed their impressive agility:  they can jump or fall from pretty high up, and always seem to land on their feet.   This isn’t just an urban legend— it has been noticed for centuries.   Famous 19th-century scientists including George Gabriel Stokes and James Clerk Maxwell took time off from their revolutionary developments in physics and calculus to ponder the mathematics of falling cats.   For someone with a basic understanding of modern physics, cats’ ability to rotate in midair and always land on their feet might seem a bit miraculous:  due to conservation of angular momentum, you might at first think they shouldn’t be able to do this without some starting rotation being provided when they fall.   Actually, that concern is a bit of an oversimplification, since a cat is not a rigid body.    But still, before modern times, the most common thought was that cats are “cheating”— a falling cat simply pushes off against its starting point, or against the arms of whoever is dropping it, to create the proper rotation.

However, in the 1890s, French scientists Etienne-Jules Marey developed the technique of “chronophotagraphy”, where a series of photos could be taken at very high speed, exposing details of movements that were previously too rapid for human observation.   By using this method to watch falling cats, he was able to put to rest the notion that they were using their starting point as a fulcrum.    Angular momentum is the key— but rather than thinking of the cat as a single rigid body with a global momentum, it’s best to look at the front half and the rear half separately.    As later researchers pointed out, the best way to think about a falling cat is to model it as a pair of loosely connected cylinders, with one representing the cat’s front half, and the other representing its rear half.

So, how does the cat do it?   The key actually is that same angular momentum that was originally confusing the problem, but applied a bit more carefully.   You should think about how a spinning dancer or figure skater spreads their arms out to slow down, or pulls them in to speed up.   This is because the total angular momentum of a spinning body is approximately constant.   (Of course it’s only truly constant if the system is closed, and the spin will inevitably slow due to friction, etc, but let’s ignore that for now.)    Masses that are distant from the center of a spinning object have greater momentum, so if those masses are pulled inward, for the total momentum to be constant, the body must spin faster.   And likewise if those masses are pushed outward, the spinning must slow down.

Thus, the cat can use differing angular momentum of its front and rear halves to its advantage.   First it pulls in its front paws while spreading its rear paws, causing the front part to rotate faster than the rear.   Then it does the reverse to reorient its rear half, until both its front and rear are in the right position.   Cats may also use some other related techniques, exploiting their non-rigidness in various ways; some pretty complex physics papers have been published on the topic, which you can find linked at the articles in the show notes.   Apparently this work has ended up having some very useful applications, such as finding ways for astronauts to better control their motion in space while using less energy.

The most comprehensive experimental study on this topic is probably the 1998 paper from the “Annals of Improbable Research”, by an Italian scientist named Fiorella Gambali.   She claims to have dropped her cat, named Esther, 600 times from various heights from 1 to 6 feet, checking to see if it really did land on its feet.   If you’re ever traveling through Milan and see a woman covered with scratch and bite marks, that’s probably Dr Gambali.   Anyway, she concluded that a cat can stabilize itself during any drop from 2 feet or more, but can’t quite do it for a 1 foot drop.   That makes sense, since the two-phase method we described, of stabilizing the front and then the rear, probably takes a little time to execute.   Perhaps some more experimentation is needed, but I think I’ll just take her word for it rather than trying this on my cat.

And this has been your math mutation for today.

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