# top physics

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considering a top at rest on a table. it is not in stable equilibrium and therefore rests on its sides. spin it, and now the top is able to remain upright. exactly what forces are at work here and how is it that adding some rotational inertia to the system keeps it aloft? if possible, im looking for a detailed or at least physical explanation with supporting equations

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Hmmm. OK, I typed in one response, then hit that magical keyboard "reload" shortcut that deleted the whole damned thing. Crap. I HATE when I do that. Well, I'll try to type it again.

So, it is possible to understand this situation intuitively, at least at the beginning of the top's motion. And I'll try to help you do that.

Lets look first at the situation when the top is not spinning. If you place the top on the table, technically it is possible to get the thing to balance on its pin-point. But due to manufacturing imperfections/asymmetries, hand shake, table wobble, slight breeze, surface roughness, etc., its not likely that you can get it to balance. I did see a guy on TV one time balance 11 bowling balls one on top of the other, but that's superhuman balancing intuition + maybe he took a file and flattened the balls slightly before hand. Don't count on it.

So, you put the top on end and it doesn't balance. Why does it fall? The weight of the object acts at its center-of-mass, which lies above the table. The center-of-mass is not directly above the pin-point, and so the weight causes a torque about the pin-point that causes the thing to experience rotational acceleration and fall over.

Now lets look at the angular momentum of the top, when it is not given an initial spin. Initially, when you first place it, its angular momentum is basically zero. After it begins rotating as it falls over, it has nonzero angular momentum. It actually is now rotating about an axis through the pin-point that is parallel to the direction of the torque caused by the object's weight. What happened is this. The torque caused a change in angular momentum, which in turn caused a change in angular acceleration. torque and angular momentum are analogous to force and linear momentum. For force, you have F = ma, with m being a constant mass. That ma is the change in linear momentum, e.g., Force = change in linear momentum. Same thing happens with rotation. Torque is equal to the change in angular momentum. For rotation, mass is replaced by a 3x3 matrix called the inertia tensor. For angular, nonsymmetric objects, the inertia tensor is fully populated and can cause very weird and nonintuitive things to happen. But, for a top, which is axisymmetric, the inertia tensor is the diagonal matrix, which simplifies the gyroscopic effects. The weight acts straight down, in the negative z direction. The resulting torque is horizontal, e.g., lies in the xy plane. Initially the top is nearly vertical and so the x and y diagonal components of the world space inertia tensor are equal. The z component is different, but doesn't matter. The fact that the torque lies in the xy plane and the x and y components of the diagional inertia tensor are equal causes the initial change in angular velocity to be parallel to the torque applied by the weight. Meaning the top initiall starts falling over by rotating about an axis through the pin-point and parallel to the torque axis.

Read Baraff's articles and do the math to work this bit about the change in angular velocity out for yourself. This bit is actually only intuitive at the instant the top begins to rotate, and does get weird as the top falls over. But the top is already unstable after it begins to fall over. The really important thing here is that torque equals a change in angular momentum. Starting from zero angular momentum, any torque applied is relatively large and so the net effect on the top is large, noticeable. It is that torque due to weight that is entirely responsible for determining the axis of spin, initially.

Angular momentum determines the change in angular velocity, the direction of rotation or spin. If angular momentum does not change much, the direction of angular velocity also does not change much. This is where the spin helps.

If the top is spinning, it already has a large angular momentum vector, pointing initially in the nearly vertical direction. The torque applied due to the weight of a spinning top causes a change in angular momentum that is small relative to the initial angular momentum vector. If you add a small vector (the torque) to a large vector (the initial angular momentum due to spin), the new updated angular momentum vector is not much different from the old one. The small torque just pushed it over slightly. The angular momentum vector is still pointing nearly vertical, and the result is that the top continues to spin about roughly the vertical axis. The slight tilting of the spin axis due to the torque due to weight is the cause of gyroscopic precession, which causes the spin axis to move in a circular pattern----this is the wobble of the top. Now friction eventually slows the spin, and as it does the magnitude of the angular momentum vector gets smaller. When this happens, the torque due to weight has a greater contribution to the change in angular momentum. Initially, the wobble gets larger, and eventually the top becomes unstable as it winds down.

Hmmm. That's a long post. I hope it helps. The situation really is fairly intuitive, whether the top is spinning or not, when the top is nearly vertical. When it is at a large angle from vertical, it is much harder to understand, and you basically need to go back and just trust the equations.

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I haven't taken any tensor calculus classes yet and the only things I know about them are what I have already researched on my own... but I know for sure that I wont be able to come up with a mathematical model anytime soon D=

Thanks for the assistance though =D!

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