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

  • Thread starter Thread starter reminiscent
  • Start date Start date
  • Tags Tags
    Mass Sphere
reminiscent
Messages
131
Reaction score
2

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?
 
Physics news on Phys.org
reminiscent said:

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.
 
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
 
can you guys please elaborate on this?
 
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
 
  • Like
Likes Samy_A
thanks :D
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top