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

• eg2333

#### eg2333

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? :(

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.

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 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. "

Thanks for spelling that out, apparently I wasn't clear enough.