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

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SUMMARY

The center of mass is a point expressed as a displacement vector from the origin of a reference frame, and it is not a vector quantity in the traditional sense. In non-relativistic physics, it is treated as a position within an affine space rather than a vector space. The center of mass for a mass distribution can be calculated using the formula R = (1/Σmi) Σ(miri), where mi represents point masses and ri their displacement vectors. This concept is detailed in Barger & Olson's "Classical Mechanics, A Modern Perspective."

PREREQUISITES
  • Understanding of vector quantities and displacement vectors
  • Familiarity with affine spaces in physics
  • Basic knowledge of mass distribution and point masses
  • Access to "Classical Mechanics, A Modern Perspective" by Barger & Olson
NEXT STEPS
  • Study the derivation of the center of mass formula in classical mechanics
  • Explore the differences between vector spaces and affine spaces
  • Learn about applications of center of mass in various physical systems
  • Review examples of mass distributions and their center of mass calculations
USEFUL FOR

Students of physics, educators teaching mechanics, and anyone interested in the mathematical foundations of classical mechanics will benefit from this discussion.

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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:

\vec{R} =\frac{1}{\sum_{i}m_i} \sum_{i} m_i\vec{r}_i

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

AM
 
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