Fizyx for Felines: A Physics Textbook for the Curious Cat

Chapter 2 – Excerpt 6

Posted in Uncategorized by skonabrittain on 30 April, 2010

Last Friday was the first one on which I didn’t post an excerpt since this project began, 14 weeks ago. The reason why is that I was unusually busy, because a major project (an opportunity to sell my math clocks at a festival) was suddenly superimposed on an already very full weekend. However, no matter how busy I am, I am sufficiently obsessive-compulsive that I would have squeezed it in, except that I completely forgot about it. Ironically, this lapse of memory was probably due to lack of sleep, which in turn was due to my staying up late to watch a movie (because it had to be returned that morning) that – here’s where the irony comes in – was about a literately successful blogger, namely Julie and Julia. Perhaps hearing about her attentions from vast quantities of blog readers caused me to repress thinking about the contrast with my small set of readers and even smaller set of commenters. And even when I did belatedly remember this project, I was not inspired to get back on track.

So I really want to encourage comments on the blog. For those of you who know me, who keep giving me comments in person or by email, please put them on the blog instead. And for those felines among you, I realize it’s hard to type when your paws are wider than the keys, but know that I really appreciate your efforts.

Meanwhile, I am making up for this lapse it by including a double excerpt today, with lots of pictures, including a couple of cartoons. Although today is National Hairball Awareness Day, I cannot think of any way to relate that to the topics in today’s excerpt. Maybe that’s a good thing – there are no gross hairballs forthcoming. Rather, you will see graceful ice skaters and ballet dancers, both human and feline, and learn some of the secrets behind their motions. Also revealed are the related but more complex directions for the even more graceful maneuvering required for you to always land on your feet.

Moments of Inertia of Extended Irregular Objects

Your body has three principal axes of rotation. They are the three most symmetric mutually perpendicular axes through your center of mass (see Figure 2-18(a)). Human bodies also have three such axes (see Figure 2-18(b)). In both cases, it is easiest to rotate about the longitudinal axis, because the mass is concentrated closer to it. Human acrobats do spins, flips, and cartwheels around their longitudinal, transverse and median axes respectively, in order of increasing difficulty due to increasing moments of inertia.

Figure 2-20. The moment of inertia is smallest around the longitudinal axes. For felines this axis is horizontal, while for humans, it’s vertical.

The longer your legs, the greater your rotational inertia, the harder it is to run quickly. This you find it easier to run fast than a horse or giraffe does but more difficult than a dachshund or a mouse. Regardless of their leg length, anyone can always reduce their rotational inertia by bending their legs. Thus, as we run, we naturally tend to bend our legs, since we can tell that it’s a lot harder to swing them when they’re straight.

Sometimes you actually want to increase your rotational inertia, such as when you are walking along the top of a thin fence. To maintain your balance, you need to keep adjusting your center of gravity. Your tail comes in very useful for this, giving you extra resistance to rotational motion and hence allowing you more time to adjust. Since human tightrope walkers don’t have tails, they hold long poles instead to obtain this increased stability.

Consider a spinning ice skater or ballerina. (See Figure 2-21.) By extending her arms, she can triple her rotational inertia, and by extending one of her legs as well as both arms, she can increase it six-fold.

Figure 2-21. The skater on the right has even more than six times as much rotational inertia around her longitudinal axis, because she also is extending her torso to the side.

Angular Momentum

Angular momentum, symbolized by the non-Greek letter L, is the rotational analog of linear momentum, which was defined as mass times velocity; hence it is defined analogously as the rotational inertia times the angular velocity:

(6) L = Iω

Thus the angular momentum of an object depends on how the mass of the object is distributed around the axis of rotation, as well as the rate at which the object is rotating.
Q. What is the angular momentum of the earth about its axis of rotation?
A. Combining the results of the previous exercises in this chapter, we have L = Iω ≈ (9.7 ×1037 kg m2)x(7.3×10-5 rad/s) ≈ 7.1 kg m2/s.

Recall the principle of the Conservation of [Linear] Momentum: if there is no net external force on a system, then its total amount of momentum is a constant. Analogously, there is an equally momentous principle of the Conservation of Angular Momentum, which was also developed by Isaac Newton. It states that if there is no net external torque on a [rotating] system, then there will be no change to its angular momentum.

Mathematical Digression

Mathematically, this principle is just a reformulation of the rotational analog of Newton’s Second Law. Note that just as force is the time derivative of linear momentum, torque is the time derivative of angular momentum. And, to be ultra-precise, angular momentum is actually the moment of momentum, and hence should be defined as L = r x p. It follows that L = Iω holds in scalar form but the vector equation (6) holds only for symmetrically rotating objects, which are the only kind ever studied in physics classes. In those cases, L and ω point in the same direction. Since L must be constant, it obviously cannot be in the constantly changing direction of motion, but must, in fact, be perpendicular to it, as given by the Right Paw rule.

People often take advantage of the principle of Conservation of Angular Momentum for recreational purposes, such as ball games. See Figure 2-22. It explains why spinning footballs are more stable than non-spinning ones.

Chan on Ping Pong table

Figure 2-22.

Above we mentioned that a dancer could increase her moment of inertia six-fold by extending three of her limbs. Alternatively, if she starts a spin with the three limbs extended, and then pulls them all in, this move decreases her moment of inertia six-fold. By the Conservation of Angular Momentum principle, her rotational speed, or spin rate, must simultaneously increase by the same factor of six, which looks very impressive (see the schematic figures in Figure 2-23(a) and the authentic one in Figure 2-23(b)).

Figure 2-23. (a) The skater on the right is spinning 3 times as fast as the one on the left.
(b) The English choreographer Christopher Wheeldon demonstrates a low moment of inertia pose.
(c) Loki, a “male ballerina”, lives in New York City with his photographer Odd Todd.

Although cats engage in similar recreational activities (see Figure 2-21(c)), the primary feline application of the principle of Conservation of Angular Momentum is far more serious, sometimes even life-saving.

Practical Application

How to Always Land on Your Feet

At first thought, it may appear that, if a nasty human drops you head-first without any spin, you would land head-first on the ground, since, by the principle of conservation of angular momentum, you cannot create spin where there is none. Once let go, you are effectively an isolated system, so your angular momentum is conserved. However, there are ways of getting around this predicament, so to speak. Since there are no external forces on you other than gravity, you must exert internal forces to accomplish your task. You have probably already learned how to do this in a previous course, if not from your mom in kittenhood, but, just in case, the technique will be described here.

You need to execute a sequence of twists and turns without ever changing your angular momentum. Thus, if you turn one part of your body one direction, you need to simultaneously turn another part in the other direction. Such a turn is called a zero-angular-momentum rotation. However, if that was all you did, you wouldn’t make any net progress. In order to keep from making a series of contortions that just undo each other, you need to repeatedly modify your moment of inertia.

To be explicit, one method is to first pull in your legs and tail while arching your back, then turn your head and legs through a large angle toward the ground while rotating your torso slightly in the other direction. Then extend your legs and tail and straighten your spine, so that your torso can catch up to your legs while they backtrack only a small angular distance. Repeat if necessary. For additional assistance, use your stiffened tail as a propeller, and cushion your landing with all four paws. See Figure 2-24.5

Figure 2-24. This is a time-lapse photo sequence of one falling cat during a fraction of a second, not seven cats.

While you presumably find these instructions to be quite clear, non-feline animals are not as adept at following them. See Figure 2-25.

Figure 2-25. (a) Far Side cartoon by Gary Larson, February 23, 1988 (b) Garfield cartoon by Jim Davis, September 21, 1989.

Humans also don’t have this skill, but in their case it’s more understandable. Being normally upright creatures, their principal axis of rotation about which it is easiest to twist is one that has no effect on changing their orientation with respect to the ground. And they are significantly less agile, partly because they have only twenty-five vertebrae, five fewer than felines. Furthermore, they don’t have the advantage of tails. Thus, although humans can perform similar maneuvers, they cannot do so fast enough to avoid landing head-first if they are dropped head-first. However, professional astronauts, after much practice, have managed to attain the skill of zero-angular-momentum twists about each of their principal axes in zero-gravity environments.

On the other hand, some inanimate objects manage to achieve what canines, rhinos and humans cannot. For example, according to Murphy’s Law, buttered toast always lands buttered side down. This lead us to suggest, but not recommend, the following experiment.

Lab Exercise – Testing Murphy’s Conjecture

To test whether your survival instinct is powerful enough to supersede Murphy’s Law, strap a piece of buttered toast to your back, then try falling from various heights. Note that it should be buttered side up, so that you and it have the same preferred orientation. Hence, you won’t get your fur oily from doing this laboratory work.
However, if Murphy’s Conjecture turns out to be correct, not only will the floor get oily, but you may sustain bodily injury. Thus, we suggest performing it on very plush carpet and, in the event of such unfortunate results, hiding well when your person discovers the mess.

While many people know Newton discovered gravity from watching an apple fall, far fewer realize that he discovered the conservation of angular momentum by watching his cat fall. He noticed that Spitface always landed on her feet and by repeated observation he deduced how she conserved angular momentum while doing so.

As recent studies have shown, survival from a fall depends on height in a surprising way. Cats falling from the seventh floor have a survival rate approximately 30% lower than that of cats falling from the 20th floor. Apparently this is because it “takes about eight floors for the cat to realize what is occurring, relax and correct itself.” 6 Thus, had he lived in the modern age of skyscrapers, Descartes may have concluded that cats do have souls!

5 – Zoologist Desmond Morris calls these maneuvers a “righting reflex” in Catwatching © 1986 Crown Publishers, Inc. He theorizes that the front legs are brought in close to the face in order to protect the vulnerable throat area. However, if this was the case, they would remain drawn in, which would be disastrous. <return-to-text>

6 -Source: TBD<return-to-text>


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