, 3 min, 533 words
Tags: physics
Remember playing with tops when you were younger? It turns out that they're interesting not only to children but also to physicists. Take precession. If you spin a top, as it slows down, it starts to wobble. In the simplest case, the handle of the top just spins around tracing out a rough cone. It's called precession, and it affects all sorts of things, from electrons to the Earth.
Here's how it works. You have a top, which is spinning and thus has some angular momentum. If the top is adequately symmetrical, then the angular momentum just points along the spin axis, either upwards or downwards, depending on the direction you spun it. The direction is determined by the right-hand rule, so if the top is spinning clockwise if you look at it from above, the angular momentum vector points down. Similarly, counterclockwise means the angular momentum points up. For now, let's suppose that the angular momentum is pointing upwards. If the top is completely vertical, then the top spins merrily onwards, and nothing interesting happens. Luckily, in the real world, nothing is ever so exact as to be totally vertical. Random perturbations happen, and the top tilts slightly. In any case, once this happens, gravity tries to pull the top downward further, which (treating the top's point of contact with the table as our pivot) creates a torque according to the cross product $\vec\tau=\vec r\times\vec F$ ($\vec\tau$ is torque, $\vec r$ is the radius vector from the point of rotation to the point on which the force acts - in this case, the center of mass of the top, and $\vec F$ is the force, in this case from gravity). If the top is tilted to the right, then the torque vector points away from us. If the top wasn't spinning, it would just topple over under the influence of gravity. But because it's spinning, and because angular momentum is conserved, it can't just topple like that. So what happens?
Recall that $\vec\tau=\frac{d\vec L}{dt}$, that is, torque causes a change in the angular momentum. And it's a vector relationship - since the torque is pointing away from us, after a very short time $dt$, the angular momentum is pointed slightly more away from us than originally. But observe that the torque is always perpendicular to the angular momentum. So while the angular momentum changes direction, it does not change length. Thus the top continues spinning happily, but its axis of rotation precesses around a cone. Neat!
So why is this precession only apparent when the top slows down? The length of the angular momentum vector is proportional to the speed of rotation. When the top is spinning really fast, the angular momentum vector is so long that it dwarfs the contribution of the torque from gravity, so the wiggles are tiny. As the top spins on and on, it's rubbing against the floor or table, and friction slows down its rotation. As that happens, the torque is less tiny compared to the angular momentum, so the wiggles become more and more noticeable, and eventually, the top spins out of control and falls over.