Today’s advent calendar post was delayed by severe online retail issues last night and child care today, but I didn’t want to let the day pass completely without physics, so here’s the next equation in our countdown to Newton’s birthday:

This is the final piece of the story of angular momentum, the undefined symbol from the right-hand side of the angular momentum principle: torque is defined as the cross-product between the radius vector pointing out from the axis of rotation to the point where the force is applied, and the vector force that acts at that point. As with the definition of angular momentum, the simple, friendly-looking notation hides a lot of complexity: torque, like angular momentum, is a vector quantity, and points in a direction perpendicular to the force and the radius.

So, why is this important?

Torque is the rotational analogue of force: when you apply a force to an object, you cause its momentum to change in the direction of that force; when you apply a torque to an object, you cause its angular momentum to change in the direction of that torque. Torque is more complicated than force, though, as you can tell from the fact that it requires its own definition. Force is intuitive enough that you don’t need to say much more, but torque combines force and distance in a way that isn’t as obvious.

The effects of torque are all around us, though, from really obvious mechanical applications like the long handles on wrenches used to tighten big bolts– a moderate force applied at the end of the long handle produces a really big torque– to things that are so simple you would never think there was physics behind them, like the position of doorknobs. The knob on an ordinary door is always at the opposite side from the hinges, because a force applied there creates the biggest torque, and is thus the most effective for making the door swing open.

Torque is also the basis for one of the most critically important of simple machines: the lever. If you want to shift something really heavy, you do it by taking a long stick, wedging one end under the object to be moved, and prying it up, often using a small object as a fulcrum. This works because of torque: when you push down on one end of the stick, you’re trying to make it rotate, applying a torque about the fulcrum; the weight of the object you’re trying to move tries to prevent the rotation, producing a torque in the opposite direction. If the stick is long enough, though, and the fulcurm is close to the object to be moved, a relatively small force out at the end can produce a much bigger torque than a huge weight acting very close to the fulcrum. Thus, Archimedes’ immortal boast that given a place to stand, he could move the entire Earth with a lever.

The vector character or torque also has some cool consequences, such as the fun gyroscope tricks seen in this video:

(The narration is a little soporific, but can be ignored.) The trick where you suspend one end of a gyroscope, and it rotates about that point, rather than falling down, is a consequence of the vector nature of torque. The force of gravity acts downward, while the radius vector that goes into the torque points along the axis of the gyroscope. The cross product between these has to be perpendicular to both, so the direction of the torque produced by gravity is at right angles to gravity.

This doesn’t matter if the gyroscope isn’t spinning, but when it’s in motion, it has angular momentum that is directed along the axis of rotation. The torque cause by gravity causes the angular momentum to move not in the direction of the force of gravity, but in the direction of the torque. thus, the gyroscope axis moves horizontally, not vertically.

So, take a moment to appreciate the many cool properties and applications of torque, which is about so much more than just making things spin. And come back tomorrow for another equation of the season.