# Overview of Gravitation and Center of Mass

by Ron Kurtus (revised 2 June 2011)

The center of mass (CM) of two objects is the weighted average or mean location between their individual centers of mass. It is also called the barycenter. The separation of the objects from the CM is a function of their masses. You can define their locations on a coordinate line and show the relationship of the locations, velocities and accelerations with respect to the CM.

The importance of the CM is that the motion of the two objects is tied to gravitational attraction between them, such that the objects move in opposite directions with respect to that CM. The motion of the objects can be broken into radial and tangential components with respect to the CM. Radial motion is affected by gravitation, while tangential motion is independent of gravitation. Assuming there are no outside forces acting on them, two objects moving in space will follow curved paths past the CM, go into orbit around the CM or meet each other at the CM.

Questions you may have include:

• Where is the center of mass between two objects?
• How is motion related to the CM?
• How is motion broken into components?

This lesson will answer those questions. Useful tool: Units Conversion

## Definition

A uniform spherical object has its center of mass (CM) at its geometric center. In gravitational calculations, you can assume each object has its mass concentrated at that point.

Of greater interest is the CM between two spheres. Its location is a point that is a ratio of the separations and masses of the objects:

mRm = MRM

R = Rm + RM

where

• m and M are the masses of the two objects
• Rm is the separation between mass m and the CM
• RM is the separation between mass M and the CM
• R is the separation between masses m and M as measured from the CM of each sphere

CM between two uniform spheres

(See Center of Mass Definitions for more information.)

## Location and motion

The location of the objects on the coordinate line is defined as:

rCM = (mrm + MrM)/(m + M)

where

• rCM is the location of the CM
• rm and rM are the locations of the objects

When the CM is at the zero-point of the coordinate system the objects are on opposite sides of this reference:

mrm = −MrM

When the CM is considered the reference point, the motions of the objects are in opposite directions with each other and maintain a velocity relationship according to their masses:

mvm = −MvM

(See Center of Mass Location and Motion for more information.)

However, it is possible to view the objects with respect to an outside reference or relative to one of the objects. In those situations, the CM may be moving and the mass/velocity relationship would not hold.

(See Relative Motion and Center of Mass for more information.)

## Components

The motion of objects can be broken into radial and tangential components.

(See Center of Mass Motion Components for more information.)

Only the radial component is affected by gravitation. If there is an initial velocity away from the CM, the objects may move away from each other until they reach a maximum displacement, at which time they reverse directions and move toward each other. If the velocity is sufficient, the objects may escape and fly off into space.

(See Center of Mass and Radial Gravitational Motion for more information.)

The tangential component affects whether the objects will collide, go into orbit or also escape into space.

(See Center of Mass and Tangential Gravitational Motion for more information.)

When the mass M is much greater than m, the CM is near the center of the object of mass M. This greatly simplifies the orbital and escape velocity equations.

## Summary

The center of mass (CM) between two objects is the weighted average or mean location of their individual centers of mass. Separation of the objects from the CM is a function of their masses.

Two objects mirror the motion of the other object with respect to the CM. They then maintain the relationship with the ratio of the masses. When viewed with respect to an outside observer, the ratio of the masses may not hold.

Motion can be broken into radial and tangential components. Radial motion is affected by gravitation, while tangential motion is independent of gravitation. Assuming there are no outside forces acting on them, two objects moving in space will follow curved paths past the CM, go into orbit around the CM or meet each other at the CM.

Work hard to do your best

## Resources and references

Ron Kurtus' Credentials

### Websites

Center of Mass Calculator - Univ. of Tennessee - Knoxville (Java applet)

Center of Mass - Wikipedia

Gravitation Resources

## Questions and comments

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