How is the center of mass determined independently of the coordinate system?

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Discussion Overview

The discussion revolves around the concept of the center of mass, particularly how its determination is affected by the choice of coordinate system. Participants explore the relationship between the location of the center of mass and its position vector, as well as the implications of coordinate systems on these concepts.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how to determine the center of mass without a coordinate system, seeking clarity on Marion's statement regarding the uniqueness of the center of mass.
  • Another participant suggests that the location of the center of mass and the position vector are essentially the same, as the position vector specifies the location.
  • There is a discussion about whether Marion implies that there is only one center of mass for a body, which some participants confirm.
  • A participant points out that while a vector itself does not depend on the basis chosen, its components do, leading to a clarification about the dependence of the position vector on the choice of the origin of the frame.
  • One participant asserts that the center of mass is physically independent of the coordinate system, using the example of a cylinder to illustrate that its center of mass remains constant regardless of the coordinate setup.

Areas of Agreement / Disagreement

Participants generally agree that the center of mass is a unique point for a body, but there is some disagreement regarding the implications of coordinate systems on the position vector and its components. The discussion remains unresolved on certain nuances of these concepts.

Contextual Notes

Some participants express confusion over terminology and the implications of coordinate systems, indicating potential limitations in understanding how these concepts interrelate.

HMT
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Hi! I have been reading about the position of the center of mass in the Marion's Classical Dynamics book, in some point of the section he states that: "The location of center of mass of a body in uniquely defined, but the position vector R(of the center of mass ofcourse) depends on the coordinate system chosen". My query is: If we need a coordinate system to dermine the position of center of mass, how can I determine "just" the center of mass? What is Marion trying to say?

Thanks for reading my doubt.
 
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HMT said:
how can I determine "just" the center of mass?
Not sure what you mean. You want to specify the center of mass without using a coordinate system?
 
In fact this is my doubt. Are the location of center of mass & position vector of the center of mass the same thing?
 
HMT said:
In fact this is my doubt. Are the location of center of mass & position vector of the center of mass the same thing?
Yes, I would say so. The position vector (of anything) specifies its location.
 
Doc Al said:
Yes, I would say so. The position vector (of anything) specifies its location.
But when Marion says : "The location of center of mass of a body in uniquely defined..." Is trying to say that there is only one center of massin one body?
 
HMT said:
But when Marion says : "The location of center of mass of a body in uniquely defined..." Is trying to say that there is only one center of massin one body?
That's right. And, of course, the location of that unique point can be specified using a position vector.
 
Doc Al said:
That's right. And, of course, the location of that unique point can be specified using a position vector.
I think I get it, perhaps It was a language confusion (english is not my idiom); anyway thank you! Regards
 
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This is a bit misleading, because a vector does not depend on the basis chosen, but of course its components do. Of course, the position vector depends on the choice of the origin of your frame. For a rigid body usually you even work with two frames, namely an inertial frame and a non-inertial frame fixed in the body.
 
vanhees71 said:
This is a bit misleading, because a vector does not depend on the basis chosen, but of course its components do.
Good clarification.

vanhees71 said:
Of course, the position vector depends on the choice of the origin of your frame.
I think that was the issue here.
 
  • #10
The main point is the center of mass physically is independent of the coordinate system. For example, if the object (constant density) is shaped like a cylinder, the center of mass is in the center of the cylinder no matter how you set up the coordinate system.
 
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