Center of mass problem

[dawgs1234]

I'm having problems solving this. Any help would be appreciated! I know that xbar and ybar are 0

Find the center of mass of the solid inside the sphere ? = 2, below the cone
? = ?/3 and above the plane z = 0, if the density is proportional to the distance from
the origin. Use symmetry where possible.

 

[galactus]

Did you manage to get your triple integral set up?.

$\displaystyle \int_{0}^{2\pi}\int_{0}^{\frac{\pi}{3}}\int_{0}^{2}{\rho}^{2}sin({\phi})d{\rho}d{\phi}d{\theta}$

The mass is $\displaystyle M=\int\int\int k{\rho}dV$

As you stated, because of symmetry $\displaystyle \overline{x}=\overline{y}=0$.

So, $\displaystyle M_{z=0}=k\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{3}}\int_{0}^{2}\underbrace{{\rho}cos(\phi)}_{\text{z}}\cdot {\rho}\cdot {\rho}^{2}sin(\phi)d{\rho}d{\phi}d{\theta}$