Conceptually Understanding Balance

Conceptually Understanding Balance
If you want to learn about balance from a physics perspective, this might be a good place to start. Whether you are a unicyclist, an academic, or an athlete, I think that learning just a bit about physics will help you to balance.
This site presents a simplified fun summary of my senior thesis on conceptually understanding balance. You can read the full version of my thesis here.
Mechanisms of Balance
My thesis starts with the bold statement that there are three mechanisms of balance, and that we can think about each one to help understand how someone balances. The picture below shows these three mechanisms of balance, and we'll talk about the first two in more detail.

So the above picture shows three mechanisms which are super cool because we can perfectly break down balance into these three mechanisms.

But before I get ahead of myself, let's make sure that you understand what's going on in the picture. First, the center black and white circle is called the center of mass. The center is mass is where we can effectively pretend all the mass of a system is concentrated. This is important, because we can take any system or any collection of objects and then analyze it like its mass is all concentrated at one point. Broadly speaking, we'll be able to use these mechanisms of balance to analyze any Newtonian system big or small.
Now in the pushing and twisting mechanisms you also notice there are forces (the solid black arrows). A force is often defined as simply a push or a pull, and in this case, the forces shown are actually special components of the net contact force or essentially the sum all forces on the system besides the force of gravity.
Deriving the Mechanisms of Balance
The first two mechanisms of balance come directly from a special decomposition of the net contact force. A decomposition for forces is when you take a force and then you split it up into other forces. To appreciate this special decomposition, let's cover:
(a) The traditional decomposition with normal and frictional forces
(b,c) My decomposition with pushing and twisting forces


The above picture shows a force with the dashed line that is the net contact force. Your physics teacher might have encouraged you to simply label the net contact force with a normal force N and a friction force F. The normal force would be perpendicular to the surface and then the frictional force is always along the surface. This traditional decomposition is usually good because you can relate the force of friction to the normal force using the coefficient of friction u (with F = u N). However, it's just a decomposition, and we could easily make up a different decomposition.
The decomposition that we will use for this analysis is the center one and then a further decomposition to the right. We take the net contact force and then split this force into two components. One component, the "pushing force," is along the axis from the center of pressure to the center of mass. The other component, "the twisting force," is whatever is perpendicular.