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

AI Thread Summary
The discussion focuses on deriving the moment of inertia equation for a thin spherical shell, with initial attempts leading to an algebraic dead end. The derived equation for a hollow sphere is presented, and the challenge arises when trying to evaluate the limit as the inner radius approaches the outer radius, resulting in an indeterminate form. Suggestions include using the binomial theorem and rewriting the equation in terms of the inner radius and a small parameter, δ. One participant successfully resolves the issue through algebraic manipulation, while another encourages the use of integration for a spherical surface. The conversation highlights the complexities of mathematical derivation in physics.
cubejunkies
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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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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 can't 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 :)
 
You are an excellent sportsman!:wink: Don't you try the integration?

ehild
 

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