# center of mass

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## center of mass

**center of mass,**the point at which all the mass of a body may be considered to be concentrated in analyzing its motion. The center of mass of a sphere of uniform density coincides with the center of the sphere. The center of mass of a body need not be within the body itself; the center of mass of a ring or a hollow cylinder is located in the enclosed space, not in the object itself. Under the action of a constant force of gravity, a body suspended or balanced at its center of mass will be stable; there will be no net moment acting on it. Sometimes a problem may be analyzed from the point of view of the center of mass of an entire system of objects, such as several colliding elementary particles or a multiple-star system. For example, the complex motions of the earth and moon about the sun become somewhat simpler when viewed from the common center of mass of the earth-moon system, located about 1,000 mi (1,600 km) below the earth's surface. It is this point that is moving in an elliptical orbit around the sun rather than the center of mass of the earth alone.

## Center of mass

That point of a material body or system of bodies which moves as though the system's total mass existed at the point and all external forces were applied at the point. The Earth-Moon system moves in the Sun's gravitational field as though both masses were located at a center of mass some 3000 mi (4700 km) from the Earth's geometric center. The function of the center-of-mass concept is to permit analysis of the motion of an entire system as distinguished from that of its individual parts.

Consider a system of mass *M* composed of *n* bodies with masses *m*_{1}, *m*_{2}, …, *m*_{n}, and radius vectors *r*_{1}, *r*_{2}, …, *r*_{n} measured from some common reference point. Define a point with radius vector *R*, such that Eq. (1) holds. Then, it is possible to derive Eq. (2),

*R*moves as though it possessed the total mass of the system and were acted upon by the total external force.

A simplification of the description of collisions can be obtained by using a coordinate system which moves with the velocity of the center of mass before collision. *See* Collision (physics), Rigid-body dynamics

## center of mass

(center of inertia) The point in a material system at which the total mass of the system may be regarded as being concentrated and that moves as if all external forces on the system could be reduced to a single force acting at that point. When a body has a center of gravity, which it does in a uniform gravitational field, this point coincides with the center of mass.Two bodies, moving under the influence of their mutual gravitation, will orbit around their center of mass, which lies on the line between them. The distances, *r *_{1} and *r *_{2}, of the bodies from this point depend on their masses, *m *_{1} and *m *_{2}, the more massive body lying closer. For circular orbits:

*r*

_{1}/

*r*

_{2}=

*m*

_{2}/

*m*

_{1}

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Center of Mass

(or center of inertia), a geometric point whose position characterizes the distribution of mass in a body or mechanical system.

The coordinates of the center of mass are

or, for a body whose mass is continuously distributed over its volume,

Here, the *m _{k}* are the masses of the material particles that form the system; the

*x*,

_{k}*y*, and

_{k}*z*are the coordinates of the particles;

_{k}*M*= Σ

*m*is the mass of the system; ρ is the density; and

_{k}*V*is the volume.

In English “center of mass” and “center of gravity” are sometimes used synonymously. Strictly speaking, however, the concept of center of gravity is meaningful only for a body located in a uniform gravitational field. By contrast, the concept of center of mass is not associated with any force field and has meaning for any mechanical system. The center of mass and center of gravity of a body are located at the same point.

When a mechanical system moves, the motion of its center of mass is the same as that of a material particle that has a mass equal to the mass of the system and that is acted on by all the external forces applied to the system. Furthermore, if the motion of a mechanical system or a body is described with respect to axes that move translationally with the center of mass and for which the origin is at the center of mass, some of the equations of motion have the same form as for motion with respect to an inertial frame of reference. Because of these properties, the concept of center of mass plays an important role in the dynamics of systems and bodies.

S. M. TARG