Center of Mass Using Triple Integrals Question

[RJLiberator]

Homework Statement

My question is this: When finding center of mass, can you do so using spherical/cylindrical coordinates, or must you put it in cartesian coordinates?

If you can use spherical/cylindrical coordinates, how do you set up the triple integrals ?

Thank you.

Homework Equations

The Attempt at a Solution

 

[Delta2]

You can do it in spherical or cylindrical coordinates as well. Any triple integral in cartesian coordinates $\int\int\int f(x,y,z)dxdyz$ can be calculated in cylindrical coordinates(r,phi,z) as $\int\int\int f(rcos\phi,rsin\phi,z)rdrd\phi dz$ and in spherical coordinates as well http://en.wikipedia.org/wiki/Multiple_integral#Spherical_coordinates

You should take notice of 3 facts when changing from cartesian to another coordinate system

1) The function f to be integrated is transformed from function of x,y,z to a function of the new coordinates. For example in cylindrical coordinates from f(x,y,z) becomes $f(rcos\phi,rsin\phi,z)$
2) The transformed function is mupltiplied by another function of the new coordiantes which is the the Jacobian Determinant. In cylindrical coordinates Jacobian Determinant is simply r.
3) The domain of integration changes also.

 

[RJLiberator]

Hm. So I understand when calculating the center of mass it works like this:

Triple integral of x dxdydz
triple integral of y dydxdz
triple integral of z dzdydx

And then all of them over triple integral of the function (mass)

If I were to do this in cylindrical coordinates would it be the following:
triple integral of r*r drdzdtheta
triple integral of theta*r dthetadrdz
triple integral of z*r dzdthetadr

Over the mass again

Due to the added r in the cylindrical coordinates?

 

[Delta2]

I am not sure what your triple integral is but for cylindrical coordinates you replace $x=rcos\phi, y=rsin\phi, z=z$ and you multiply by r. Also you must be carefull for the new domain of integration.

 

[RJLiberator]

Yes, for sure.

My question is specifically in regards to center of mass with triple integrals.