Derivation of moment of inertia of a solid sphere

[joe5185]

Homework Statement

I need to know why my derivation does not work. I am attempting to derive I=2/5 mR^2

Homework 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

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

 

[TSny]

Hello, and welcome to PF!

I am attempting to use shells to do this.
integral(r^2dm)=I

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

 

[joe5185]

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)

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

 

[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?

 

[joe5185]

oh I see. thank you