I just came back from my Introduction to Rotational Kinematics class, and one of the important concepts they described was Rotational Inertia, or Moment of Inertia.
It's basically the equivalent of mass in Netwon's $F = m a$ in linear motion. The equivalent rotational equation is $\tau = I \alpha$, where $\tau$ is rotational force, $\alpha$ is rotational acceleration, and $I$ is rotational inertia.
For a point about an axis, $I$ is $m r^2$, where $r$ is the distance from the point to the axis of rotation.
For a continuous body, this is an integral -- $I = \int r^2 \,dm$.
This really doesn't make any sense to me...you have two independent variables? I am only used to having one independent variable and one constant. So I would solve this, using my experience with calculus (which encompasses a read through the Sparks Notes packet) as $ I = m r^2 $
But obviously, this is wrong? $r$ is not a constant! How do I deal with it? Do I need to replace $r$ with an expression that varies with $m$? But how could $r$ possibly vary with $m$? Isn't it more likely the other way around? But how can $m$ vary with $r$? It's all rather confusing me.
Could someone help me figure out what to do with all these substitutions for, example, figuring out the Moment of Inertia of a hoop with no thickness and width $w$, with the axis of rotation running through its center orthogonal to its plane?