Re: Why is rotational inertia proportional to r squared instead of r?

The question: "I understand that particles of matter further from the axis of rotation in a rotating, rigid object resist any given angular acceleration more, since they experience a tangential acceleration that is proportional to their distance from the axis of rotation (r). But I do not understand why rotational inertia is found to be proportional to r squared, then, instead of just r."

It starts with the fact that the analog of force in the rotational world is torque, and torque is the product of a force with a moment arm. So right away we have an "extra" radius in a formula! Or, alternatively, we can "find" the "extra" radius in the fact that the tangential acceleration is proportional to the product of the angular acceleration and the radius. I'll explain:

Consider a powered model airplane at the end of a tether, traveling in a horizontal circle level with the ground. We consider all the mass to be at the same radius from the center of the circle. The engine (motor+propellor) gives the airplane a tangential acceleration, a_T, which is proportional to the tangential force:
F_T = ma_T
The torque is thus
torque = F_Tr = ma_Tr
The tangential acceleration is related to the angular acceleration, alpha, by
a_T = alpha*r
with alpha in units of radians per second per second. So we end with
torque = m*alpha*r*r = mr²alpha

So in the linear world we have
F = ma
and in the rotational world we have
torque = mr²alpha
so what could be more logical than calling "mr²" the rotational analog to linear mass? That is exactly what we do, and the rotational "mass" is called "moment of inertia" and is given by
mr²

As you know, I'm sure, to obtain the moment of inertia of a body that is considered to be extended rather than a point mass, we sum up all the contributions of the individual particles that make up the extended body times the square of the distance to the motion's center:
I = summation(mr²) and
torque = I*alpha

Does this make sense? No, not really. But we haven't done anything "wrong", and it just works out that way in the equations. In physics we don't always know "why" something is the way it is, but as long as it is consistent with the algebra (and other math!) and verifiable by experiments, we go along with it. You will find that this is not the last place you will encounter something that doesn't necessarily "make sense", but that's just the way it is! Quantum mechanics is a case in point. Richard Feynman once said "If you think you understand quantum mechanics, you don't understand quantum mechanics."

John Link, MadSci Physicist