OK I got it, I skipped a step:
Cm = 1/V\int r (r hat) dV
As I know it's on the z axis:
Cm = 1/V\int z (z hat) dV
z = r cos\theta
dV = r^2 sin\thetadr d\theta d\varphi
Cm = 1/V\intr cos\theta (z hat) r^2 sin\thetadr d\theta d\varphi
I can pull out z hat from the integral.
With V =...
ri is the vector position of a mass in some direction. I know that by symmetry it should be on the z axis. Should I multiply the whole thing be z hat in spherical (cos\theta - sin\theta). That does not make much sense, and it does not work.
1/M \sum mi ri
1/M\int \rho(r) r dV - this is a volume integral.
1/M\int [\rho(r) r] r^2 dr sin \theta) d\theta d\varphi - this is a volume integral in spherical coordinates.
My idea was that a simple volume integral for a hemisphere, going from 0 to r, 0 to \pi/2 and 0 to 2\pi would be...
Hello,
I am struggling with what was supposed to be the simplest calc problem in spherical coordinates. I am trying to fid the center of mass of a solid hemisphere with a constant density, and I get a weird result.
First, I compute the mass, then apply the center of mass formula. I divide...