Moment of Inertia Review

Just thought we could benefit from some review on moment of inertia, because it was a pretty extensive topic and wasn't really mentioned in physics B. Not to mention that the variable is a different expression for each object.

The general form of the equation is I = ∑_i m_ir_i² = ∫r² dm .

Below are the moment of inertia equations for a few different objects. If you have another object in mind to share, please do add it in the comments!

I_{solid disc} = 1/2 mr²

I_{cylinder about its axis} = 1/2 mr²

I_{hollow disk/hoop} = mr²

I_{solid sphere}= 2/5 mr²

I_{hollow sphere}= 2/3 mr²

I_{rod about it's center}= 1/12 ml²

I_{rod about it's end=} 1/3 ml²

Though these shortcuts are great, make sure to know how do derive the moment of inertia of an object. For review, here's how to calculate the moment of inertia of a rod from it's end (Also in the textbook p 273 as well as in the notes packets).

The linear mass density (λ) = M/L, where M is the mass of the uniform rod with length L.

dm = M/L dx, or the mass density times the little wee bit of rod.

Using the general equation, we know I = ∫_o^Lx² dm, where x is the length of the rod from x=0 to x=L.

By substituting for dm, we then know I = ∫_o^Lx² (M/L) dx.

The constant comes out, leaving I = (M/L) ∫_o^Lx² dx.

And using calculus, we get I = (M/L) (1/3)x³ evaluated from L to 0, which leaves us with

I = (1/3) (M/L) (L³)

I= (1/3) ML^​2

Note: If you need further assistance on this topic, the unit packet for Rotation (with the frog on a unicycle in it) and the packet titled "Chapter 6: Rotation" are useful. However, for visuals and more elaborate derivations, I recommend reading Tipler p. 272 and the pages following and/or watching this video again:

http://www.aplusphysics.com/courses/ap-c/videos/MomentOfInertia/MomentOfInertia.html

...Which I always find extremely helpful. I'll probably post another unit summary again, since our midterm is looming in the near future. Best of luck, all!