# Derivation of the moment of inertia eqn for a thin spherical shell

So I've been trying to derive the moment of inertia equation for a thin spherical shell and I've slammed into a dead end algebraically. I was able to derive an equation for a hollow sphere:

I = (2/5) M (Ro^5 - Ri^5)/(Ro^3 - Ri^3)

where Ro is the distance to the very outside of the sphere and Ri is the distance to the inner thickness of the sphere if that makes sense.

I thought I could take the limit as Ri approached Ro, to yield the rotational inertia of a very thin spherical shell as I sought out for, however, I cannot evaluate the limits even if I used L'Hopital's Rule and derived the top and bottom seperately because that would not allow me to escape the cursed indeterminant form of 0/0 which results every time until the denominator goes away to zero and then i'm really in a bad situation.

I saw a website http://scienceworld.wolfram.com/physics/MomentofInertiaSphericalShell.html which reduced the (Ro^5 - Ri^5)/(Ro^3 - Ri^3) term using a series decomposition of some sort, but I have no idea what they did and it's been a while since I've meddled with Taloy Series and stuff, so any help with this or an explanation of what they did would be greatly appreciated.

THANKS! :)
Anthony

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ehild
Homework Helper
For a very thin shell, Ro=Ri(1+δ), and δ is very small compared to 1. Rewrite your equation in terms of Ri and δ. Remember the binomial theorem, http://en.wikipedia.org/wiki/Binomial_theorem and expand Ro5 and Ro3. Find the limit of I when $\stackrel{δ}{\rightarrow}$0.

But it is easy to get I by integration for a spherical surface.

ehild

Wait, why does Ro=Ri(1+δ) ? I don't get the (1+δ) part. I also tried expanding as they did on that webpage listed above, using Ro = Ri + r and expanding that, but I cant cancel stuff down like they did and the magical appearance of the factorials in the last step confuses me.

Thanks
Anthony

OH WAIT NEVERMIND haha I did some algebraic gymnastics and figured it out :)

ehild
Homework Helper
You are an excellent sportsman! Don't you try the integration?

ehild