rotational kinematics - Origins of Moment of Inertia

Where exactly does the equation $MR^2$ for moment of inertia come from? The quantity itself seems fairly arbitrary.

Answer

The equation you are referring to is the expression for the moment of inertia of a point particle of mass $m$ at a distance $R$ away from some axis. This expression is really the definition of the moment of inertia for a point mass, so the question becomes "where does this definition come from, and why is it useful?"

Well, for simplicity's sake, suppose that such a point mass is rotating around the aforementioned axis on a string, then if a tangential force $F_t$ is applied to it, its resulting tangential acceleration $a_t$ will satisfy

$$F_t = ma_t$$ so the torque on it is $$\tau = Rma_t$$ On the other hand, recall that the tangential acceleration $a_t$ and angular acceleration $\alpha$ are related by $a_t = R\alpha$, and plugging this into the right hand side of the expression for the torque gives $$\tau = mR^2 \alpha$$ Notice that the quantity $mR^2$ has magically appeared. Well clearly it's not actually magic; the point really is that if we think of torque as a sort of rotational analog of force, and angular acceleration as the rotational analog of acceleration, then this shows that the quantity $mR^2$ is a sort of rotational analog of mass for a point mass. As a result, we give it a special name: moment of inertia.

It's important to point out that although I used the example of a point mass undergoing uniform circular motion to motivate the definition of moment of inertia, there are significantly more involved and general derivations that lead to a quantity called the inertia tensor which is the generalization of the moment of inertia for non-pointlike bodies undergoing arbitrary rotation.

See, for example, the following answer:

https://physics.stackexchange.com/a/89304/19976