# Spherical Coordinates mass

1. Nov 16, 2008

### squeeky

1. The problem statement, all variables and given/known data
Use Spherical Coordinates.
Let H be a solid hemisphere of radius a whose density at any point is proportional to its distance from the center of the base.
a) Find the mass of H.
b) Find the center of mass of H.

2. Relevant equations
$$M=\int\int_D\int\delta dV$$
$$M_{yz}\int\int_D\int x \delta dV;M_{xz}\int\int_D\int y \delta dV;M_{xy}\int\int_D\int z \delta dV$$
$$C.O.M.=(\bar{x},\bar{y},\bar{z})$$
$$\bar{x}=\frac{M_{yz}}{M};\bar{y}\frac{M_{xz}}{M};\bar{z}\frac{M_{xy}}{M}$$

3. The attempt at a solution
I think that if we place the hemisphere's center at (0,0,0), then the limit of theta is from 0 to 2pi, phi is from 0 to pi/2, and rho is from 0 to a, while the density is equal to rho. This gives me the equation:
$$M=\int^{2\pi}_0\int^{\pi/2}_0\int^a_0 \delta \rho^2 sin \phi d \rho d \phi d \theta$$
Solving this, I get a mass of $$\frac{a^4 \pi}{2}$$, $$M_{xy}=\frac{a^5\pi}{5}$$, Mxz=Myz=0. Then the center of mass is (0,0,2a/5).
Is this right?

2. Nov 16, 2008

### HallsofIvy

Staff Emeritus
You are told only that the density is proportional to the distance from the center. How can you possibly get a specific number as the mass? What happened to the "proportionality"? Other than that, I think you are correct.