Question on center of mass (centroid)

[mnb96]

Hello,

I was thinking of the following question: suppose we have a body in two-dimensional space and its density is described by the function f(x,y) (Note! in Cartesian coordinates!). The x-coordinate of the center of mass is given by C_x= \frac{\int\int xf(x,y)dxdy}{\int\int f(x,y)dxdy} where the domain of integration is ℝ². The y-coordinate of the center of mass is given by an analogous formula.

Now, suppose we are given the density of this body in polar coordinates, that is: f(r,θ). Is it possible to obtain directly the centroid (C_r, C_\theta) in polar coordinates from f(r,θ)?

Of course I know that by using a polar-to-Cartesian curvilinear transformation x=x(r,θ), y=y(r,θ), we can easily obtain the formulas C_x(r,\theta) and C_y(r,\theta), and then we could just convert C_x and C_y into polar coordinates. However if you think about it, when we calculate C_x(r,\theta) and C_y(r,\theta) we are essentially calculating the Cartesian coordinates of the center of mass.

Can we "bypass" this step? What's so special here with Cartesian coordinates? Aren't we supposed to be able to calculate the center of mass of a body without resorting to a special coordinate system?

 

[Simon Bridge]

Yes you can bypass the step and do the calculation entirely in polar coordinates.
https://www.physicsforums.com/showthread.php?t=496083

 

[mnb96]

Hi Simon,
I read the thread that you cited in your post. I believe the user who posted that question was essentially posing the same question as mine, but actually he did not get an answer.

I did not see any post in that thread with anyone suggesting how to "bypass" the cartesian coordinates and obtain directly the centroid in polar coordinates (C_r, C_θ) by integrations of f(r,θ).

In that thread, to the question "Are there integrals that give r-bar and theta-bar?", Mark44 replied: There might be, but I don't recall ever seeing any.

At the current state of things, it seems to me that the concept of "center of mass" was defined in cartesian coordinates. Once we have that definition we can obviously find the centroid in any other coordinate system. However I wonder if it is possible to define the "center of mass" in a coordinate-free manner (if that makes sense), or at least to give a (physically) consistent definition without using the cartesian coordinates as a starting point.

EDIT: I noticed that Wikipedia has a "coordinate-free" definition of center of mass for continuous distributions

 

[Simon Bridge]

Ah - I should have read more carefully.
Yes - the coordinates are just a way of describing positions and the com is just a position.
To work out how to do it in polar coords, you need to understand what the center of mass actually is... what you are doing when you do the integration.