- #1
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
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