Moment of Inertia for Infinite Density (mass per area)

[Hells_Kitchen]

Homework Statement

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.

Homework Equations

Now i found the moment of intertia of the object to be
http://img233.imageshack.us/img233/84/problemou0.jpg

I am having trouble with the limit though because if I try to do Lo'Hopitals Rule on R^(n+4)/(n+4) it doesn't 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 don't 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

 

[alphysicist]

Hi Hells_Kitchen,

Homework Statement

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.

Homework 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 doesn't 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 don't 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.

 

[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? 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?

 

[alphysicist]

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.

 

[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!

 

[alphysicist]

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!

Glad to 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).