Question

General question:

I do not understand the concept of moment of inertia or how to find the equation for different uniform masses. Can anyone explain?

Answer 1

I'll try to explain moment of inertia by comparing it to mass, because they are similar concepts, and you're probably very familiar with mass as it is!

MASS is a measure of an object's TRANSLATIONAL inertia. In other words, it is a measure of how resistant an object is to moving forwards, backwards, or sideways.

MOMENT OF INERTIA is a measure of an object's ROTATIONAL inertia. In other words, it is a measure of how resistant an object is to rotating.

Mass on its own is relatively easy to understand and calculate, because when you move an object in a straight line, EVERY point on that object moves the same distance along that line, regardless of the relative locations of those points in space. This means that the shape of the object DOES NOT affect its mass.

Moment of inertia is more complicated because now the shape of the object actually does matter. Specifically, the moment of inertia is influenced by where mass is located in relation to the axis of rotation. The closer the mass is to the axis of rotation, the lower the moment of inertia, and the easier it is to spin the object. On the other hand, the farther the mass is to the axis of rotation, the greater the moment of inertia, and the harder it is to spin the object. In general, the moment of inertia, I, is equal to the sum of each mass, m, times the square of that mass's distance from the axis of rotation, r^2: I=Σmr^2. This equation can be simplified for certain types of shapes (e.g. cylinders, rings, etc).

Does that help at all?

Answer 2

that helps quite a bit thanks.

So in a uniform object you can just take an integral from 0 to r because r is the only thing changing?

and in a non uniform object you'd have to determine the mass at each point, multiply it by r^2, and sum it up, implying you'd need a computer to do this efficiently?

Answer 3

Yup!

Answer 4

awesome thanks