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

• oreo
In summary, the center of mass is a point that is expressed as a displacement vector from the origin of the reference frame. It is not a vector quantity, but can be expressed as a sum of point masses multiplied by their displacement vectors. It is not necessarily directed towards Earth's center, but can coincide with the origin of the reference frame.

#### oreo

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

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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