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I have no problem with the derivation of the moment of inertia of a cylinder but am having more trouble with a sphere. I completely understand the often referenced disk method, but, I would like to take a shell method approach. THe following work is wrong conceptually, I think mathematically it is fine, and I hope someone can point me in the right direction. I dont see what one would need to do it using discs, and am assuming that some slight variation on the following should work:

I=[tex]\int[/tex]r

^{2}dm

p=dm/dv

dvp=dm

I=[tex]\int[/tex]r

^{2}pdv

v=4/3[tex]\pi[/tex]r

^{3}

dv=4[tex]\pi[/tex]r

^{2}dr

I=[tex]\int[/tex]r

^{2}p4[tex]\pi[/tex]r^2dr

I=4[tex]\pi[/tex]p[tex]\int[/tex]r^4dr (Evluated between 0 and R)

I=4[tex]\pi[/tex]pR

^{5}/5

p=m/v=m/(4/3 [tex]\pi[/tex] r

^{3})

Plugging that in and simplifying I get

I=3/5mr

^{2}, though I need 2/5mr

^{2}.

I think the problem lies in my set up, conceptually, and that I need to set up an equation somewhere subtracting two values (R from r?) though I am not sure.

If you could help me out I would really appreciate it.

And again, I understand and have no problem with the disk method for this, I am just trying to figure out what this method is not working.

Thanks!