Moment of Inertia of an Infinite Rod with Non-Uniform Density?

[chromium1387]

Homework Statement

A thin rod extends along the x-axis from x= +b to infinity. It has a non-uniform linear mass density of A/x$^{4}$ where A is a constant with units of kg m3. Calculate the moment of inertia of the rod for rotation about the origin.

Homework Equations

The Attempt at a Solution

Well, I know that you have to put the rod in a coordinate system, break it up into small pieces, take the sum of the those, take the limit as n goes to infinity, and integrate.
So, I basically want to $\int$x^2dx because the word "thin" suggest that the y-coordinates do not matter. I understand that by breaking the rod into small pieces and choosing one, I have a $\Delta$m which I need to relate in terms of dx. I can do this using using the $\frac{\Delta m}{M}$=$\frac{\Delta x}{A/x^{4}}$, where $\Delta m$ is my piece of mass, M is the total mass, $\Delta x$ is my small width, and then A/x$^{4}$ is my linear mass density. But I don't really know where this b and infinity and limits of integration come into play. :( If any can help me get started, that would be awesome!

 

[tiny-tim]

hi chromium1387!

Well, I know that you have to put the rod in a coordinate system, break it up into small pieces, take the sum of the those, take the limit as n goes to infinity, and integrate.

So, I basically want to $\int$x^2dx because the word "thin" suggest that the y-coordinates do not matter. I understand that by breaking the rod into small pieces and choosing one, I have a $\Delta$m which I need to relate in terms of dx. I can do this using using the $\frac{\Delta m}{M}$=$\frac{\Delta x}{A/x^{4}}$, where $\Delta m$ is my piece of mass, M is the total mass, $\Delta x$ is my small width, and then A/x$^{4}$ is my linear mass density.

yes

except that's not the way linear density works …

the mass of a length L is linear density time L, so the mass of [x , x+∆x] is A∆x/x⁴

But I don't really know where this b and infinity and limits of integration come into play.

you're adding (integrating) the moment of inertia for every [x , x+∆x]

so you start at one end of the rod, and go to the other … ∫_b^∞

 

[chromium1387]

Ohhh.. My bad. Silly algebra mistake.
And that makes sense. I'm just used to placing one end at the origin.
Sooo, after I integrate and everything, I get $\frac{MA}{b}$?
Thanks for your reply!

 

[tiny-tim]

Sooo, after I integrate and everything, I get $\frac{MA}{b}$?
Thanks for your reply!

erm … why do you keep talking about M ?

apart from that, yes ​

 

[chromium1387]

I don't know.. haha. It's just incorporated into the density, right? :P

 

[tiny-tim]

it doesn't exist!

 

[chromium1387]

okay. :)