Torque, moment of intertia for extended bodies

You've seen the rotational analog of simple 1-D kinematics, which answer questions like

"how fast is it spinning?"
"when will it stop?"
"how many revolutions will it make in the next 29 seconds?"

But what about rotational dynamics, which addresses deeper questions, such as

"WHY is it spinning faster and faster?"
"what must I do to make the wheel slow down?"

Once again, we'll use your knowledge of physics in the linear world to guide us through the rotational realm.

Looking at the variables involved in Newton's Second Law in rotation: torque

A torque, it turns out, is a bit more complicated than a force; in order to compute the torque on an object, you need to know not only the size of the force involved, and the location at which it acts, but also the DIRECTION of the force.

How the direction of a force affects its torque

Stop the Earth!

Computing moments of inertia

Problems involving rotation and torque usually require one to know the moment of inertia of objects. Now, under certain simple circumstances, we know how to compute it:

if the object is a few discrete point masses, just use
if the object is a simple shape, and rotates around its center, look up the moment of inertia in this table of moments of inertia (PDF) or this copy of the table in PNG format.

But what if the situation is not quite so simple? For example, suppose that the object IS a basic shape, but is NOT rotating around its center?

In this case, you can use the

parallel-axis theorem together with table of moments of inertia

Finally, what if the object in question is just not simple at all. Perhaps it's an unusual or asymmetric shape, or maybe it's made up of material which has a non-uniform density. When all other methods fail, there is one foolproof (although perhaps difficult) method for computing the moment of inertia:

Integrating to find the moment of inertia