Using Calculus to Find Moment of Inertia

In calculus, we can find moment of inertia via the formula:

We went over in class how to calculate this integral for several moments of inertia.

The basic procedure is to find the relationship between r and dm. This can be accomplished by assuming uniform density in the material and using

as our basis. From there, only a simple integral remains. One thing to keep in mind is that the

Rotational Equilibrium

For something to be in rotational equilibrium, i.e., not spinning, all of its torques must cancel out. That is, the sum of all of the clockwise and counterclockwise torques must be zero.

In order for something to be completely in equilibrium, that is not moving at all, all of its up, down, and sideways forces must balance as well.

An example of this is a see-saw that is perfectly balanced. The weights of the two riders plus the weight of the board is balanced by the normal force acting from the fulcrum. The torques produced by the two riders are perfectly balanced. If there is a big kid and a little kid on the see-saw, the big kid must be sitting closer to the fulcrum for this to happen.