# Derivation of moment of inertia of a solid sphere

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1. Dec 18, 2015

### joe5185

1. The problem statement, all variables and given/known data
I need to know why my derivation does not work. I am attempting to derive I=2/5 mR^2

2. Relevant equations
I have seen people derive it using disks but my question is why do the shells not work? Where in my set up did I go wrong??? Thanks

3. The attempt at a solution
I am attempting to use shells to do this.
integral(r^2dm)=I
Ro=p=m/v so pdv=dm
dv=4(pi)r^2dr
moment of inertia (I) =definite integral from r=0 to r=R (4p(pi)r^4dr)

When I solve this integral it becomes (3/5)mR^2 and not (2/5) mR^2

2. Dec 18, 2015

### TSny

Hello, and welcome to PF!

What is the moment of inertia of a single thin shell of mass dm and radius r? (Hint: It is not r2dm)

3. Dec 18, 2015

### joe5185

Well in general moment of inertia R^2dm and I am using the thin shells to eventually make up the entire volume so I can sub out for dm in terms of r

4. Dec 18, 2015

### TSny

If all the mass in the element dm is at the same distance r from the axis of rotation, then the moment of inertia about that axis will be r2 dm. For a spherical shell, is all of the mass in the shell at the same distance from the axis?

5. Dec 19, 2015

### joe5185

oh I see. thank you