Understanding Center of Mass: Is it a Vector Quantity & Its Direction?

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Discussion Overview

The discussion centers around whether the center of mass is a vector quantity and its directional properties, particularly in relation to Earth's center. The scope includes conceptual clarification and technical explanation within the context of physics.

Discussion Character

  • Conceptual clarification, Technical explanation

Main Points Raised

  • Some participants propose that the center of mass is a point expressed as a displacement vector from the origin of the reference frame.
  • Others argue that the center of mass is a position, suggesting that position is an affine space rather than a vector space, particularly in non-relativistic physics.
  • A later reply provides a mathematical expression for the center of mass as a weighted sum of point masses and their displacement vectors, indicating a technical approach to the concept.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the center of mass, with some emphasizing its representation as a vector and others questioning this characterization based on the distinction between position and vector space.

Contextual Notes

There are unresolved nuances regarding the definitions of vector and position, as well as the implications of these definitions in different contexts of physics.

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Is center of mass a vector quantity. If so then how? Is it directed towards Earth's center?
 
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shayan haider said:
Is center of mass a vector quantity. If so then how? Is it directed towards Earth's center?

The centre of mass is a point. As such, it is expressed as a displacement vector from the origin of the reference frame that is being used. If it coincides with the origin, it is the vector (0, 0, 0).

AM
 
Andrew Mason said:
The centre of mass is a point. As such, it is expressed as a displacement vector from the origin of the reference frame that is being used. If it coincides with the origin, it is the vector (0, 0, 0).

AM
Thanks a lot.
 
The center of mass is a position. Technically position is an affine space, not a vector space. At least in non relativistic physics.
 
Shayan,

Just to follow up on this, the centre of mass of a mass distribution is conveniently expressed as the sum of each of the point masses in the system multiplied by their displacement vector from the origin divided by the total mass:

[tex]\vec{R} =\frac{1}{\sum_{i}m_i} \sum_{i} m_i\vec{r}_i[/tex]

See, for example, Barger & Olson, Classical Mechanics, A Modern Perspective, first ed., ch. 5-1, p. 156-160

AM
 
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