# Homework Help: Moment of inertia of a sphere

1. Apr 3, 2010

### Zamba

1. The problem statement, all variables and given/known data
calculate the moment of inertia of a sphere of mass M and radius R by integrating over thin shells

2. Relevant equations
Ishell=(2/3)mR2

3. The attempt at a solution
this is what i have so far
the sphere is decomposed into infinitesimal shells with surface area 4$$\pi$$r2
the mass of each shell is dm=$$\rho$$(4$$\pi$$r2)dr
after expanding rho and canceling terms I get
dm=(3m/r)dr

I=$$\int$$(2/3)(dm)r2dr from 0 to R.
I=2mrdr from 0 to R
this gives me I=mr2

does anyone see what I did wrong?
PS. those "pi" are not supposed to be powers. sorry, i'm not sure how to change them

2. Apr 3, 2010

### kuruman

You did not replace dm with ρ(4πr2) under the integral sign. Instead, you integrated dm into m and then integrated over dr once more.

3. Apr 3, 2010

### Zamba

i'm sorry, i didn't write it all out

I=$$\int$$(2/3)dmr2 dr
from here I replaced dm with (3m/r)dr so i got

I=$$\int$$(2/3)(3m/r)r2 dr
then i got
I=$$\int$$2mr dr from 0 to R
which gave me mr2

4. Apr 4, 2010

### kuruman

And why is that correct? What expression did you use for the density?