Solving Moment of Inertia Homework: Deriving Disk Formula

[revres75]

Homework Statement

This was an exam question that I got wrong, my teacher tried to explain it but it only left me more confused. I found some websites that also had explanations but they were also confusing.

"Derive the moment of inertia of a uniform disk which rotates along a central axis , radius R , disk thickness d, mass M , density p
I= R^2 dm"

Homework Equations

I= 1/2*M*R^2 ?

The Attempt at a Solution

mass M = Pi*R^2*d*p

My teacher mentioned something about a third integral.

 

[Doc Al]

Presumeably you are to derive the moment of inertia by evaluating $\int r^2 dm$. So what did you do?

 

[revres75]

If I derive $\int r^2 dm$ I get 2r but I not sure what to do with the dm

 

[Doc Al]

If I derive $\int r^2 dm$ I get 2r but I not sure what to do with the dm

By "derive" I didn't mean "take the derivative". You need to evaluate that integral, which is the definition of moment of inertia. Start by expressing "dm" in terms of density and distance from the axis. (Hint: Think in terms of concentric rings.)