(adsbygoogle = window.adsbygoogle || []).push({}); 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

[tex]M=\int\int_D\int\delta dV[/tex]

[tex]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[/tex]

[tex]C.O.M.=(\bar{x},\bar{y},\bar{z})[/tex]

[tex]\bar{x}=\frac{M_{yz}}{M};\bar{y}\frac{M_{xz}}{M};\bar{z}\frac{M_{xy}}{M}[/tex]

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:

[tex]M=\int^{2\pi}_0\int^{\pi/2}_0\int^a_0 \delta \rho^2 sin \phi d \rho d \phi d \theta[/tex]

Solving this, I get a mass of [tex]\frac{a^4 \pi}{2}[/tex], [tex]M_{xy}=\frac{a^5\pi}{5}[/tex], M_{xz}=M_{yz}=0. Then the center of mass is (0,0,2a/5).

Is this right?

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# Homework Help: Spherical Coordinates mass

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