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

In summary, Marion states that the location of the center of mass of a body is uniquely defined, but the position vector used to determine it depends on the chosen coordinate system. The center of mass is physically independent of the coordinate system, and is a unique point within the body. The position vector simply specifies its location within the chosen coordinate system, and may vary depending on the origin of the frame.
  • #1
HMT
9
1
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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  • #2
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?
 
  • #3
In fact this is my doubt. Are the location of center of mass & position vector of the center of mass the same thing?
 
  • #4
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.
 
  • #5
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?
 
  • #6
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.
 
  • #7
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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  • #8
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.
 
  • #9
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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FAQ: How is the center of mass determined independently of the coordinate system?

What is "center of mass invariance"?

Center of mass invariance is a principle in physics that states that the center of mass of a system remains constant regardless of external forces acting on the system. In other words, the location of the center of mass does not change as long as the system remains isolated.

Why is "center of mass invariance" important in physics?

The concept of center of mass invariance is important because it allows us to simplify complex systems and analyze them as a single point. This makes it easier to apply the laws of physics, such as Newton's laws of motion, to predict the motion of the system.

How is "center of mass invariance" related to conservation of momentum?

Center of mass invariance is closely related to the principle of conservation of momentum. Since the center of mass remains constant, the total momentum of the system must also remain constant in the absence of external forces. This allows us to use the center of mass as a reference point to analyze the motion and interactions of objects within the system.

Can "center of mass invariance" be violated?

In classical mechanics, center of mass invariance is a fundamental principle and cannot be violated. However, in relativistic physics, objects with mass can be converted into energy, which can affect the location of the center of mass. In this case, the concept of center of mass must be modified to include the concept of relativistic mass.

How is "center of mass invariance" used in practical applications?

The concept of center of mass invariance is used in a variety of practical applications, such as spacecraft design, robotics, and sports. In spacecraft design, engineers must consider the location of the center of mass to ensure stability and control during flight. In robotics, the center of mass is important for balance and movement. In sports, athletes must understand the concept of center of mass to optimize their performance, such as in diving or figure skating.

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