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For mass that is distributed according to a density ρ(x) over a body V in space, the center of mass is
Here is the total mass, given by
In , the components of the center of mass are given by:
For a system of point masses m1, m2, ..., the integrals are replaced by sums:
Where the total mass M is the sum of the constituent masses.
The origin from which positions are calculated has no effect on the physical position of the result. As long as the same unit is used for all the points, any length and mass unit can be used.
The following equations of motion assume that there is a system of particles governed by internal and external forces. An internal force is a force caused by the interaction of the particles within the system. An external force is a force that originates from outside the system, and acts on one or more particles within the system. The external force need not be due to a uniform field.
For any system with no external forces, the center of mass moves with constant velocity. This applies for all systems with classical internal forces, including magnetic fields, electric fields, chemical reactions, and so on. More formally, this is true for any internal forces that satisfy the weak form of Newton's Third Law.
The total momentum for any system of particles is given by
Where M indicates the total mass, and vcm is the velocity of the center of mass. This velocity can be computed by taking the time derivative of the position of the center of mass.
An analogue to the famous Newton's Second Law is
Where F indicates the sum of all external forces on the system, and acm indicates the acceleration of the center of mass.
The angular momentum vector for a system is equal to the angular momentum of all the particles around the center of mass, plus the angular momentum of the center of mass, as if it were a single particle of mass :
When talking about celestial bodies, the center of mass has a special relevance: when a moon orbits around planetA planet (from the Greek , planetes or "wanderers") is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. Prior to the 1990s only nine were known (all of them in our own solar system); as of 3, or a planet orbits around a starFor alternate meanings see star (disambiguation Hubble Space Telescope of the Sagittarius Star Cloud in the Milky Way Galaxy. A star is any massive gaseous celestial body in outer space. Stars appear as shining points in the nighttime sky that twinkle bec, both of them are actually orbiting around their center of mass, called the barycenterThe barycenter is the center of mass of two or more bodies which are orbiting each other, and is the point around which both of them orbit. In the case where one of the two objects is much larger and more massive than the other, the barycenter will be loc. There are some interesting consequences:
See also: Center of gravity, Pappus's centroid theorem
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