Deriving Common Moments of Inertia: Sphere I=\frac{2}{5}mr^{2}

[amcavoy]

Could someone direct me to a site that explains how the common moments of inertia were arrived at? My physics professor put up on the board today that for a uniform sphere:

$$I=\frac{2}{5}mr^{2}.$$

He said it was just the anti-derivative of something, but he didn't want to go into it because there is a table in our book with all of the common moments of inertia.

Does anyone know? Maybe someone could show me how the above moment (for the sphere) was derived and I could try it on something else? Thanks, I'd appreciate it.

 

[Astronuc]

The general form of the moment of inertia involves an integral of the mass distribution and moments of the mass.

http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html#mi

The fourth and fifth plates provide an example of the integration ('anti-derivative') used to determine the moment of inertia.

Think about how a center of mass is defined.

 

[Tide]

You need to integrate $$r^2 \sin^2 \theta$$ over the volume of the sphere. Note that this represents the square of the perpendicular distance of a point in the sphere from the axis of rotation. Also, note that $$dV = r^2 dr d\phi \sin \theta d\theta$$.