Here, Fr is the horizontal component of tension, and you can see that this is a central force. Wherever the ball is in its elliptical path, Fr 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.