Conservation of Linear and Angular Momentum

Some Simple Physics; Conservation of Momentum

There are no strange ideas in this entry. In fact, I might call this a boring entry. If you want to read the weird stuff, read one of my other entries.

I was sitting around reading a primer on particle physics (L.B. Okun) today, and got thinking about the conservation of linear and angular momentum.

Conservation of linear momentum means that, when you chuck something out the rear end of your spaceship, then your spaceship moves the opposite direction, so m1v1 = m2v2 . So, if you toss out a small mass of propellant from one end at a really high velocity, then you move the much larger mass the opposite direction at a much slower velocity (along with your remaining propellant). This is an exponential relationship, but that’s not what this blog is about today, so you can forget learning about that useful tidbit of knowledge.

Anyway, to increase your velocity, you have to chuck part of your mass in the opposite direction. Pretty basic. If you just move stuff around inside your ship, the ship won’t move at all (except incrementally for the duration that you move around in the ship, but you won’t acquire a continuous velocity). Anyone who’s been on one of those playground spinners and tried to throw your body one way or the other knows how that works. You throw your body forward a foot, and the disk rotates a foot and stops.

So you can’t change the momentum of an object by moving stuff around inside. Not even if you have the rocket inside an enclosed sphere. The sphere won’t move.

I was recently (foolishly) wondering if that was true of angular momentum, too, if you had a rotating planet or moon, is there some way you can get energy out of the rotation by diddling around with the insides, somehow tapping the angular momentum of the planetoid for energy. Ultimately I realized you cannot in a closed system, but it should have been obvious to me all along. However, as with a rocket, you can change the angular momentum by ejecting part of the object. You can even speed up the spin a lot or slow it down.

Satellites do this sort of thing all the time. Usually they have spin they want to get rid of, and they call the technique “momentum dumping”. Two methods known to me involve extending tethers (like ballerina arms) to slow down the satellite’s spin, then releasing the tethers, or spinning up a high-speed gyro in the opposite direction of your spin (potentially dumping the core of the gyro, though I’m not certain any spacecraft does that – usually they use the gyros to turn the spacecraft both ways, hoping the overall effect will cancel out, and when the spin in one direction gets to be too much, they finally use propellant to dump the angular momentum). These are called Control Moment Gyros, or CMGs, and they usually have a minimum of three on board to cover the 3 axes.

Carrying this concept one step further, since you can eject propellant from a ship to make it go faster in a straight line, you can similarly spin up a chunk of mass from your planetoid in the opposite direction of your planetoid’s spin to make the planetoid spin faster, then eject that spinning mass into space. What amused me about the idea is that it’s essentially the same as a rocket ejecting propellant linearly to increase linear momentum, but here you are ejecting an object with accelerated angular momentum to increase the angular momentum of your planetoid. The difference being, you don’t ever have to eject the mass; it’s rotating in place, like a CMG.

That’s it. Not really that interesting, I guess. The equivalency of the two systems and the idea of “rotational rocketry” just struck me as amusing.

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