Oh boy. Linking to this opened a real can of worms for my productivity. The guy in the video had a whole channel of similar content and then goes ahead to link to even more content in the fields of chemistry and math.
Thank you for submitting this. You made my weekend!
Also, it's really cool to see videos like this that go such great lengths at explaining difficult stuff in such an interesting way, providing enough detail to get you started, but little enough so that you have to look it up in detail if you really want to know.
One nitpick is that he never answers his own question about how this doesn't violate conservation of angular momentum. The simple answer is that the angular momentum is zero before and after the cat flips over and is thus conserved. There is no conservation of angular orientation principle.
Practice question: imagine you're floating in space and facing away from the earth. How would you turn around to face it?
Isn't this essentially the same basic principle as rotating yourself in an office-chair without touching anything[1]? I'm not sure how the friction of the seat bearing or air resistance contributes, but my understanding is it's negligible.
[1] Usually by making sudden lateral movements with outstretched limbs whilst bringing them inward, much the same as the cat.
The trick is that the cat can create and destroy angular inertia at will by moving its legs towards or away from the axis of rotation. Think of the front and back of the cat as being separate rotational systems with the same axis. The first thing the cat does is increase the angular inertia of the back half and decrease the angular inertia of the front half. Now the system is unbalanced. The cat twists the front half around while the back half, with a much higher angular inertia, moves very little. Next, the cat reverses the roles of front and back by decreasing the angular inertia of the back half and increasing the angular momentum of the front half. Now the back half moves rotates all the way around while the front half moves very little.
A linear analogy would be a two balls connected by a spring oscillating in a vacuum. If one ball is much lighter than the other -- like a football connected to a cannonball -- it moves a lot, while the cannonball moves very little. Let's say the football and the cannonball are on a football field (a frictionless, vacuum-filled football field, of course.) The cannonball is on the 50 yard line, staying roughly stationary, and the football is oscillating back and forth between the 49 yard line and the 47 yard line. The system can't go anywhere if it starts with zero linear momentum, because the mass (linear inertia) of the balls remains fixed. But imagine you can change the masses of the balls at will. When the football reaches the 47 yard line, you swap the masses of the balls, so the football becomes very heavy and the cannonball becomes very light. Now the football stays almost stationary at the 47 yard line, and the cannonball oscillates between the 48 yard line and the 50 yard line. When the cannonball reaches the 48 yard line, you swap the masses so that the cannonball is once again very heavy and the football is very light. Now the football is oscillating between the 47 yard line and the 45 yard line. By swapping the masses repeatedly you can move down the field all the way to the end zone.
Because of conservation of mass/energy, you can't actually do that trick with linear inertia, but you can do it with angular inertia. The front and back half of the cat are like the football and the cannonball, and the cat controls the angular inertia of each half by tucking and extending its legs. When the cat is upside down, it increases the angular of inertia of its back half by extending its back legs, and it decreases the inertia of the front half by tucking its front legs. Its back half acts like the cannonball, and its front half acts like the football. That allows it to twist its front half around while the back half stays almost stationary. Once the front half is right side up, it extends its front legs and tucks its back legs, effectively swapping the football and the cannonball. In zero gravity, a cat could twist itself around that way as many times as it wanted: just like the football and the cannonball working their way down the football field, except that it wouldn't violate any physical laws.
It would be interesting to compare long tailed vs. short tailed cats on natural falls.
In the video, the cats started their falls with no angular momentum, and so they finish their falls with no angular momentum. The problem the cat has to solve is to get itself in an upright position.
In natural falls, they won't always start with no angular momentum. For example, consider a cat walking along a branch and a gust of wind shakes the branch and the cat tips off. There's a good chance the cat will get angular momentum in that kind of accident.
In that situation, merely getting upright is not sufficient as the initial angular momentum remains. There are several approaches that cat could take to deal with this.
1. Wait until near the ground to do the "get upright" operation, so that there isn't time between that and landing for the rotation to take the cat too far out of position.
2. Get upright, and then whenever the rotation takes the cat too far out of position, repeat the uprighting operation.
3. Get upright, and then use a constant counterrotation of the tail to maintain the upright position.
I would guess that the long tailed cats would do better on natural falls that are high enough to need angular momentum management.
Is it wrong of me to see #2 and #3 to be the same?
The constant counterrotation is basically the same thing as constantly trying to get to the upright position?
By the uprighting operation in #2 I mean the operation that involves using extension and contraction of the legs to give the front and rear different moments of inertia, so the cat can first bring one half of the body upright and then the other half.
It's actually a very neat trick, kind of similar to the way ice dancers speed up or slow down their rotation. Why aren't we using this instead of (or in addition to) flywheels for attitude control in spacecraft? It seems a cat-like mechanism of shifting angular momentum around could help reduce fuel costs by lessening the need to load/unload energy through maneuvering thrusters.
I worked on designing such a thing while at university using differential angular momentum - kind of like a cube with a high-speed rotating wheel embedded in each face. The wheels on opposite faces cancelled out each other's angular momentum - but if you vary the speed of one of the wheels slightly, it would give a small amount of angular momentum about any desired axis.
but i think what you're asking is why do we have fairly compact flywheels rather than longer, more extended "tails"? and the answer is probably that it makes more engineering sense. you can increase the angular momentum stored in something by making it "bigger" or by spinning it faster. i imagine it's a lot easier to spin a small flywheel to thousands of rpm than to make a "tail" (spinning at a very low speed) thousands of times larger.
also - unless i missed it - this video doesn't really explain all of it. by itself a cat's tail isn't nearly large enough or rotating fast enough to spin a cat's body. the cat is also very "intelligently" (presumably "designed" through natural selection) doing much of the rotation while in a U shape. that reduces the net angular momentum considerably (the two "arms" of the U effectively cancel).
and, from what i remember elsewhere, they also reduce the amount of rotation needed by rotating to only get one pair of legs perfectly aligned. as long as those touch down first there is a little time (i admit i am not sure it is sufficient) to then rotate the other legs while "holding on" to the ground.
>also - unless i missed it - this video doesn't really explain all of it. by itself a cat's tail isn't nearly large enough or rotating fast enough to spin a cat's body.
There's a whole section where they discuss how it's not the tail at all, and even bobtail cats can land on their feet. So you definitely missed it!
Oh, I can chip in here regarding big vs small flywheels.
If you look at the purpose of a flywheel - ie storing kinetic energy - a bigger/smaller wheel can serve the same purpose. The reason we don't have "large" flywheels is twofold:
1. size (obvious)
2. Torsional/shear stress - most materials are bad at tolerating shear stress. A bigger flywheel would require immense amounts of torque to get it moving. Generating a higher torque would impose huge amounts of torsional stress on the shaft driving the wheel.
Differential control of spinning wheels is used to control attitude in spacecraft. Wertz's book describes the attitude control of the SEASAT satellites that used two canted reaction wheels in the pitch/yaw axis for momentum bias, and actuated them differentially to correct pointing errors. Read the chapter on momentum bias.
I lost it here
"The cat and gluing dispositive will actually come together, but the cat spins so fast due to the invariance in the center of gravity that, following the law of time travel, it goes backward in time.
This has led to another theory stating that at the beginning of time there is an infinite number of antigravitatory cats, and therefore, causing the creation of the universe in the first place, due to the excessive gravitational forces produced by the infinite number of cats, they cause the big-bang. "
At first I was skeptical as well, but it turned out there is no reason to worry. He picked the right cat for the job. She was totally cooperative and it seems like she thought it was a game of some sort. It's obvious from the video that the cat was neither frightened nor angry.
In fact towards the end of the video, another cat makes a brief appearance (the guy just wanted to present her to the camera, no throwing or falling involved at all) and it serves as a good demonstration what it looks like when they're not in the mood for human shenanigans.
And they were given plenty of time to right themselves. My cats could land on their feet when dropped with about two inches clearance between their backs and the couch. And to do this experiment without thick gloves, the cat has to be treated respectfully. In short, this kind of exercise is at worst mildly annoying to the cats, and well worth it if followed by payment ham.
Thank you for submitting this. You made my weekend!
Also, it's really cool to see videos like this that go such great lengths at explaining difficult stuff in such an interesting way, providing enough detail to get you started, but little enough so that you have to look it up in detail if you really want to know.