Here, *F*_{r} is the horizontal component of tension, and you can see that this is a central force. Wherever the ball is in its elliptical path, *F*_{r} will point toward that center point, which is right below the pivot point of the string.

I also showed two other things above. One is an arrow representing the *linear* *momentum* of the ball (*p*) at a given instant, which is the product of its mass and velocity. Linear momentum is always tangent to the orbital path. (Why *p* for momentum? I guess *m* was already taken for mass.)

Second, I’m describing the *position* of the ball relative to the center point with an arrow labeled *r*, for radius. Note that *r* points *away* from the center; you’ll see why that matters later. With these I can calculate the *angular momentum* of the ball, which is the whole key to this carny game.

### What Is Angular Momentum?

Angular momentum is a measure of rotational motion. We can calculate it as the vector product of an object’s position and its linear motion. (And for angular momentum we use the symbol *L*, because … to be honest, I have no idea.) That gives us the first equation below:

The arrows show that these are vector variables, meaning they have more than one dimension. Specifically, three: for the x, y, and z axes of the 3D space we live in. This lets them describe direction and location. An example would just look like this: (1, 5, 2). Not too scary, right?

Multiplying vectors is complicated, but in our case we can skip the work, because we really only need the *magnitude* of angular momentum, which is a scalar. And we can get that from the magnitudes of the *p* and *r* vectors, along with the sine of the angle θ between them. (Yes, I’ve used θ twice—sorry about that.) This gives us the equation on the right above.

Now *that* is pretty slick, because if you look at the orbital diagram again, you’ll see that the *r* and *p* vectors are always perpendicular, and the sine of a 90 degree angle is 1. So *L* = *r* × *p*. No arrows, nice and simple!

### Let’s Talk Torque

You know about torque, right? You use it every time you push on something to rotate it. For instance, when you open a door, the amount of torque you create depends on three things: (1) the *force* (*F*) that you apply (i.e., how hard you push), (2) the *distance* (*r*) from the door’s axis of rotation (the hinges) to the spot you push on, and (3) the *angle* (θ) between those force and distance vectors.