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Moment of Inertia Review


AlphaGeek

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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 miri² = ∫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!

Isolid disc = 1/2 mr2

Icylinder about its axis = 1/2 mr2

Ihollow disk/hoop = mr2

Isolid sphere= 2/5 mr2

Ihollow sphere= 2/3 mr2

Irod about it's center= 1/12 ml2

Irod 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).

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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 = ∫oLx2 dm, where x is the length of the rod from x=0 to x=L.

By substituting for dm, we then know I = ∫oLx2 (M/L) dx.

The constant comes out, leaving I = (M/L) ∫oLx2 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!

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