Question on center of mass (centroid)

In summary, the question being discussed is whether it is possible to calculate the centroid (C_r, C_\theta) in polar coordinates directly from the density function f(r,θ) without converting to cartesian coordinates. Some users suggest using a polar-to-cartesian transformation, but the original poster wonders if it is possible to define the center of mass in a coordinate-free manner. It is noted that Wikipedia has a coordinate-free definition for continuous distributions. However, the concept of center of mass is simply a way to describe positions and the integration process is what determines its coordinates.
  • #1
mnb96
715
5
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 [itex]f(x,y)[/itex] (Note! in Cartesian coordinates!). The x-coordinate of the center of mass is given by [tex]C_x= \frac{\int\int xf(x,y)dxdy}{\int\int f(x,y)dxdy}[/tex] 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 [itex](C_r, C_\theta)[/itex] 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 [itex]C_x(r,\theta)[/itex] and [itex]C_y(r,\theta)[/itex], and then we could just convert Cx and Cy into polar coordinates. However if you think about it, when we calculate [itex]C_x(r,\theta)[/itex] and [itex]C_y(r,\theta)[/itex] 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?
 
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  • #3
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 (Cr, Cθ) 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 :redface:
 
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  • #4
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.
 
  • #5




Hello,

Thank you for your question. The concept of center of mass (or centroid) is a fundamental principle in physics and engineering, and it is a crucial tool for understanding the motion and stability of physical systems. The center of mass is defined as the point at which the entire mass of a body can be considered to be concentrated, and it is a useful concept because it allows us to analyze the motion of a complex system as if it were a single point.

To answer your question, yes, it is possible to obtain the centroid directly in polar coordinates. This can be done by using the polar coordinate version of the formula you provided for the x-coordinate of the center of mass: C_r=\frac{\int\int rf(r,\theta)drd\theta}{\int\int f(r,\theta)drd\theta}. Similarly, the y-coordinate can be obtained using C_\theta=\frac{\int\int \theta f(r,\theta)drd\theta}{\int\int f(r,\theta)drd\theta}. These formulas are derived from the same fundamental principle of center of mass, and they can be used to calculate the centroid of a body in polar coordinates without converting to Cartesian coordinates first.

The reason we often use Cartesian coordinates when calculating the center of mass is because it simplifies the calculations and makes the process more intuitive. However, as you have correctly pointed out, we are not limited to Cartesian coordinates and can use other coordinate systems, such as polar coordinates, to obtain the centroid. The important thing to remember is that the concept of center of mass remains the same, regardless of the coordinate system used.

I hope this answers your question. Keep exploring and questioning, as it is through curiosity and curiosity that we continue to make new discoveries and advancements in science. Best of luck in your studies.
 

1. What is the center of mass (centroid)?

The center of mass, also known as the centroid, is the point at which an object's mass is evenly distributed in all directions. It is the average position of all the individual particles that make up an object.

2. How is the center of mass calculated?

The center of mass can be calculated by finding the weighted average of the individual particle positions. This involves multiplying the position of each particle by its mass and then dividing by the total mass of the object.

3. What is the significance of the center of mass?

The center of mass is an important concept in physics as it helps determine the motion and stability of an object. It is also used in engineering and design to ensure that structures and objects are balanced and stable.

4. Can the center of mass be outside of an object?

Yes, the center of mass can be located outside of the physical boundaries of an object. This can occur if the object has an irregular shape or if the mass is distributed unevenly.

5. How does the center of mass change when an object is in motion?

The center of mass remains in a fixed position when an object is in motion, unless acted upon by an external force. However, the position of the individual particles within the object may change, causing the center of mass to shift accordingly.

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