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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 both and voila, obviously wrong result. What is wrong here?

Cm = 1 / M [itex]\int[/itex] [itex]\rho[/itex] r^3 sin [itex]\theta[/itex] dr d[itex]\theta[/itex] d[itex]\varphi[/itex]

M = ρ V → 1/M = 1/ρV

V = 2 [itex]\pi[/itex] R^3 / 3

Cm = 3/ (2 [itex]\pi[/itex] R^3) [itex]\int[/itex] r^3 sin [itex]\theta[/itex] dr d[itex]\theta[/itex] d[itex]\varphi[/itex]

Integrated over:

r → 0 to R

[itex]\theta[/itex] → 0 to [itex]\pi[/itex]/2

[itex]\varphi [/itex] → 0 to 2 [itex]\pi[/itex]

Cm = 3/ (2 [itex]\pi[/itex] R^3) * [itex]\pi[/itex] R^4 / 2

Cm = 3R / 4

So, what do I do wrong?

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 both and voila, obviously wrong result. What is wrong here?

Cm = 1 / M [itex]\int[/itex] [itex]\rho[/itex] r^3 sin [itex]\theta[/itex] dr d[itex]\theta[/itex] d[itex]\varphi[/itex]

M = ρ V → 1/M = 1/ρV

V = 2 [itex]\pi[/itex] R^3 / 3

Cm = 3/ (2 [itex]\pi[/itex] R^3) [itex]\int[/itex] r^3 sin [itex]\theta[/itex] dr d[itex]\theta[/itex] d[itex]\varphi[/itex]

Integrated over:

r → 0 to R

[itex]\theta[/itex] → 0 to [itex]\pi[/itex]/2

[itex]\varphi [/itex] → 0 to 2 [itex]\pi[/itex]

Cm = 3/ (2 [itex]\pi[/itex] R^3) * [itex]\pi[/itex] R^4 / 2

Cm = 3R / 4

So, what do I do wrong?

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