## Moment of Inertia for Infinite Density (mass per area)

1. The problem statement, all variables and given/known data
Suppose you have the density of a disk given by σ=σ0*rn.
For n approaches infinity, find the limit of M.I. Interpret your result, which should be physical, reasonable and intuitively clear.

2. Relevant equations

Now i found the moment of intertia of the object to be

I am having trouble with the limit though because if I try to do Lo'Hopitals Rule on R^(n+4)/(n+4) it doesnt help and I am not sure if I can use just I = MR^2(n+2)/(n+4) and treat M as a constant but I dont think that would be possible since M is a function of n itself. When i try to figure the limit i get undefined solutions and I am not sure what that means physically. Can someone help please?

Thanks,
HK
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Hi Hells_Kitchen,

 Quote by Hells_Kitchen 1. The problem statement, all variables and given/known data Suppose you have the density of a disk given by σ=σ0*rn. For n approaches infinity, find the limit of M.I. Interpret your result, which should be physical, reasonable and intuitively clear. 2. Relevant equations Now i found the moment of intertia of the object to be I am having trouble with the limit though because if I try to do Lo'Hopitals Rule on R^(n+4)/(n+4) it doesnt help and I am not sure if I can use just I = MR^2(n+2)/(n+4) and treat M as a constant but I dont think that would be possible since M is a function of n itself.
It's true that the value of M depends on n; however the goal of a moment of inertia calculation is to get an expression in the form

I = (number) M (length)2

(or perhaps sums of those types of terms). So for purposes of taking the limit of large n, I think you can consider M to be a "given" and find out what happens to the numerical factor in front of the MR2 as n gets larger.
 In that case through Lo'Pitals rule by taking the derivative of lim n --> infinity of (n+2)/(n+4) = 1. So I = MR^2 right? So that seems like the moment of inertia of a point mass does that mean that if the mass per area of the disk is infinite it will act as if it were a point mass instead of a disk physically speaking?

Recognitions:
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## Moment of Inertia for Infinite Density (mass per area)

 Quote by Hells_Kitchen In that case through Lo'Pitals rule by taking the derivative of lim n --> infinity of (n+2)/(n+4) = 1. So I = MR^2 right?
That looks right to me.

 So that seems like the moment of inertia of a point mass does that mean that if the mass per area of the disk is infinite it will act as if it were a point mass instead of a disk physically speaking?
I'm not sure if that would be true. For a point mass the R is the distance away from some origin--and where the origin is placed is entirely up to you. For the disk, the R is set by the physical size of the disk itself.

I would think more about comparing this to a thin ring.
 Oh you mean a thin walled hollow ring which has a moment of inertia of MR^2? That does make sense to me as well. So when the mass per area is infinite the disk will act as if its mass was concentrated R distance away uniformally and hollow in the middle. Thanks a bunch for the help!

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 Quote by Hells_Kitchen Oh you mean a thin walled hollow ring which has a moment of inertia of MR^2? That does make sense to me as well. So when the mass per area is infinite the disk will act as if its mass was concentrated R distance away uniformally and hollow in the middle. Thanks a bunch for the help!

What I was actually thinking was not that the mass per area is infinite, but about how the density goes as rn. If you plot a series of curves, say r^2, r^7, r^70, etc. over a range from 0->2 (for example), at the very high n values the mass is overwhelmingly located in a smaller and smaller ring at the largest r value.

for example at n=2, it looks like:

http://img185.imageshack.us/img185/687/r2newor7.jpg

at n=7,

http://img185.imageshack.us/img185/3636/r7newnx8.jpg

and at n=70

http://img185.imageshack.us/img185/852/r70newft7.jpg

so it actually effectively becomes a ring for large n (because the inner portions are so much less dense).

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