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

AI Thread Summary
The discussion clarifies that the center of mass of a body is a uniquely defined point, independent of the coordinate system used to describe it. While the position vector of the center of mass does depend on the chosen coordinate system and origin, the physical location remains constant. Participants noted that confusion arises from the distinction between the location of the center of mass and its position vector. An example provided illustrates that for a uniformly dense object, like a cylinder, the center of mass remains at its geometric center regardless of the coordinate setup. Ultimately, the center of mass is a fundamental property that does not vary with different coordinate systems.
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.
 
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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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