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