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Finding mass of a nonuniform sphere, given a function for density

  • Thread starter Tubefox
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  • #1
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Homework Statement



A sphere with radius .175 cm has a density ρ that decreases with distance r from the center of the sphere according to

ρ= ((2.75*10^3) kg/m^3) - ((9.25*10^3) kg/m^4)r

A) Calculate the total mass of the sphere.

B) Calculate the moment of inertia for an axis along the diameter.


Homework Equations



2.75*10^3 kg/m^3 - (9.25*10^3 kg/m^4)r

Probably some sort of integral.

The Attempt at a Solution




I'm sorry, but I have absolutely no idea how to even begin to do this. I tried googling, and the only result I found contained a triple integral in spherical coordinates. Given that this is on my Calc-based physics I homework and the only prerequisite is calculus I, it seems implausible that this is actually the best solution. I've never used spherical coordinates in my life, and only vaguely know how to solve a triple integral (I've taught myself a bit online, but none of my classes have ever covered them before).
 
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Answers and Replies

  • #2
Dick
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Homework Helper
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Homework Statement



A sphere with radius .175 cm has a density ρ that decreases with distance r from the center of the sphere according to

ρ= ((2.75*10^3) kg/m^3) - ((9.25*10^3) kg/m^4)r

Homework Equations



2.75*10^3 kg/m^3 - (9.25*10^3 kg/m^4)r

Probably some sort of integral.

The Attempt at a Solution




I'm sorry, but I have absolutely no idea how to even begin to do this. I tried googling, and the only result I found contained a triple integral in spherical coordinates. Given that this is on my Calc-based physics I homework and the only prerequisite is calculus I, it seems implausible that this is actually the best solution. I've never used spherical coordinates in my life, and only vaguely know how to solve a triple integral (I've taught myself a bit online, but none of my classes have ever covered them before).

This has not been covered in the lecture at any point, and there isn't a lecture before this homework is due.
The mass of a spherical shell is ##dm=4 \pi r^2 \rho(r) dr##, surface area times density times thickness, so you can also do it as a single integral since it's spherically symmetric. Perhaps that's what they are expecting you to use.
 
  • #3
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Oh, yeah, you're absolutely correct. Thank you so much - I didn't realize that you could use the surface area, and needless to say the fact that the only similar problem I could find involved a triple integral did not exactly help me maintain confidence that I could solve it.

Thanks.
 
  • #4
rude man
Homework Helper
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mass = volume integral of density.

Divvy up your sphere into shells of thickness dr, then integrate over r.
Careful with your R given in cm, not m.
 

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