# What is wrong with my derivation for the moment of inertia of a sphere?

Ok, so I thought about a derivation for the moment of inertia, but my answer comes out to (3/5)MR^2

Basically, what I did was I considered the sphere as a sum of infinitesimally thin spherical shells.

The moment of inertia for one shell is dI=(r^2)*dm

where dm=(M/V)*4*pi*r^2*dr

where V=(4/3)*pi*R^3

so the equation dI=3*pi*M*r^4*dr when simplified.

Integrating this from 0 to R (Summing up the spherical shells from the center to the edge of the big sphere) gives me (3/5)*M*R^2. What is wrong with this derivation? :(

Nabeshin
According to wikipedia, moment of inertia for a spherical shell is 2/3 M R^2 , not what you use.

It is actually (2/5)*MR^2 for a sphere. The method I used clearly gives the incorrect answer, which is why I'm asking to see if anyone can tell me where the fault is.

Mute
Homework Helper
Nabeshin isn't talking about the sphere, but a spherical shell. As I just said in your identical thread,

"The moment of inertia of a thin shell is (2/3)MR^2, not MR^2, so your original dI should be (2/3)r^2 dm - there's your missing factor of 2/3. "

Nabeshin