Mass of a Sphere: Solve w/ Triple Integrals & Spherical Coordinates

[reminiscent]

Homework Statement

Find the mass M of a sphere of radius a, if its mass density is proportional to the distance
from the center of the sphere.

Homework Equations

Triple integrals using spherical coordinates

The Attempt at a Solution

The only place where I am stuck is if the density is KpcosΦ or just Kp. So is it integrating KpcosΦp²sinΦ or Kp³sinΦ?

 

[Samy_A]

Homework Statement

Find the mass M of a sphere of radius a, if its mass density is proportional to the distance
from the center of the sphere.

Homework Equations

Triple integrals using spherical coordinates

The Attempt at a Solution

The only place where I am stuck is if the density is KpcosΦ or just Kp. So is it integrating KpcosΦp²sinΦ or Kp³sinΦ?

Let me emphasize a part of the question: its mass density is proportional to the distance from the center of the sphere.

 

[Buzz Bloom]

Hi reminiscent:

You only need to integrate with respect to r. What is the mass of a shell of thickness dr at radius r?

Regards,
Buzz

 

[princessp]

can you guys please elaborate on this?

 

[HallsofIvy]

Samy_A's point is that the problem said that the mass is proportional to the distance to the center of the sphere. That distance is the variable \rho.

BuzzBloom's point is that, since the mass is given by \int_0^a\int_0^\pi\int_0^{2\pi} K\rho (\rho^2 sin(\theta) d\phi d\theta d\rho)= \int_0^a\int_0^\pi\int_0^{2\pi} K\rho^3 sin(\theta) d\phi d\theta d\rho

 

[princessp]

thanks :D