Finding Mass of Non-Uniform Density

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The discussion revolves around calculating the mass and moment of inertia of a solid with non-uniform density defined by the equation ρ=x²+y+z, within a sphere of radius one. Participants clarify that the density should remain inside the integral during calculations, and it must be converted into spherical coordinates for proper integration. There is confusion regarding the limits of integration for the angles, with suggestions to adjust them to avoid redundancy. One participant reports spending considerable time on the problem and arrives at a solution of (4π/15) - (π/4), questioning its accuracy due to the complexity of the integrals. Overall, the conversation emphasizes the challenges of integrating non-uniform densities in spherical coordinates.
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This is just a practice problem, not actual homework. I'm studying for my final but am having a bit of difficulty in understanding this concept.

Homework Statement


Consider a solid of non-uniform density ρ=x2+y+z, consisting of all points inside the sphere x2+y2+z2=1
a) Find the mass of the solid (use spherical coordinates.)
b) Find the moment of inertia of the solid with respect to the z-axis (use spherical coordinates.)

Homework Equations


<br /> M=\int \rho dV<br />
<br /> dV= r^{2}sin\theta dr d\theta d\phi<br />

The Attempt at a Solution


I am unsure if since the density equation is given, should I bring it out infront of the integral as if it's a constant and just integrate the spherical part of dV. Or do I also integrate the density?
My textbook has no examples of this, only uniform densities where rho is considered a constant and brought out infront of the integral
 
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Hi Mr LoganC! :smile:

(have an integral: ∫ and a theta: θ and a phi: φ :wink:)
Mr LoganC said:
I am unsure if since the density equation is given, should I bring it out infront of the integral as if it's a constant and just integrate the spherical part of dV. Or do I also integrate the density?
My textbook has no examples of this, only uniform densities where rho is considered a constant and brought out infront of the integral

You have to keep it inside the ∫ …

you can only take actual constants outside (including functions of variables other than the one belonging to that ∫) :wink:
 
Okay, well then I'm confused again, because I would have this: (Using Spherical coordinates)
<br /> <br /> M=\int_{\phi=0}^{2\pi} \int_{\theta=0}^{2\pi} \int_{r=0}^{1} (x^{2}+y+z)r^{2}sin\theta dr d\theta d\phi<br /> <br />

But if I'm only integrating with respect to radius, theta and phi, then the x, y, and z would be the same, acting like a constant as if I were to bring it out front of the integral. Again, my textbook is not helping at all as there is no example with the density inside the integral
 
Last edited:
ah, you need to convert x2 + y + z into r θ and φ, and then integrate :wink:

(btw, only one of the integrals is to 2π)
 
Rightttt! It's been a while since doing spherical!
So I need to convert those.
Also, does it matter which I change from 2pi to pi? Forgot about that too, having both at 2pi is just like sweeping it out twice. So should I only let phi go from 0-pi, or does it not matter which one I choose?
Thanks again, you've been very helpful!
 
So I worked through the problem and got an answer. Took me a good 30-40mins. There's no way he would give us one question that takes 40mins to do on the final. So I must have either done something wrong or did it a very difficult way. Not only that, the integrals were very difficult!
I ended up getting an answer of
\frac{4\pi}{15} - \frac{\pi}{4}

Is there any easy way to check this answer to see if it's right? Unfortunately, the practice problems do no have solutions for them.
 
Hi Mr LoganC! :smile:

(just got up :zzz: …)
Mr LoganC said:
So I worked through the problem and got an answer. Took me a good 30-40mins. There's no way he would give us one question that takes 40mins to do on the final. So I must have either done something wrong or did it a very difficult way. Not only that, the integrals were very difficult!

hmm … as soon as I saw ρ = x2 + y + z, I thought "I wouldn't like to try to integrate that!" :redface:
I ended up getting an answer of
\frac{4\pi}{15} - \frac{\pi}{4}

Is there any easy way to check this answer to see if it's right?

(have a pi: π :wink:)

Nope. :biggrin:
 
So if this question were to show up on the exam, (This is a practice question from last years exam), How should I go about doing it? And I still have to use spherical coordinates!
 
Do it the same way!

Apart from the tediousness, what's wrong with that? :smile:
 
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Nothing I guess. I'm a terribly slow test writer, so perhaps I'll leave that one 'till the end, but at least I can show my work and show that I do know how to go about the problem!

Thanks again Tiny-Tim! A thumbs up to you, sir!:biggrin:
-LoganC
 
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