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Center of Mass Location and Motion

by Ron Kurtus (14 May 2011)

The separations of two objects from the center of mass (CM) between them are defined by their masses. This information can be used to show the location of the objects and CM on a coordinate line.

You can then establish the velocities of the objects and the CM from the equation of the CM location. Then you can determine their accelerations, provided the two objects are part of a closed system, with no outside forces acting on them.

Typically, you set the position of the CM at the zero-point on the coordinate axis. Velocities are then with respect to this fixed CM location. The only accelerations are due to gravitation along the radial axis.

Questions you may have include:

• What are the locations of the objects?
• What are their velocities?
• What are the accelerations?

This lesson will answer those questions.

Useful tool: Metric-English Conversion

Locations

In order to establish the relationship between the locations of the objects and the location of the center of mass (CM), consider the points: rm, rCM and rM along a coordinate line. Relationships are:

Rm = rCM − rm

RM = rM − rCM

mRm= MRM

where

• rCM is the position of the CM on the coordinate line
• rm and rM are the positions of the objects
• m and M are the masses of the objects
• Rm and RM are the respective separations from the CM

Locations of objects on coordinate line

Location of CM position

You can determine the equation for the CM position rCM:

mRm= MRM

m(rCM − rm) = M(rM − rCM)

mrCM − mrm = MrM − MrCM

rCM(m + M) = mrm + MrM

Therefore the general location relationship is:

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

Locations relative to CM

The general location relationship shows the position of the objects and CM with respect to the zero-point on the coordinate line. In order to locate the objects with respect to the CM, you need to set its location to the zero position on the coordinate line:

rCM = 0

(mrm + MrM)/(m + M) = 0

mrm = −MrM

This means that the locations of the objects are on opposite sides of the zero-point and are a function of their masses.

Velocities

Although you can view the motion of two objects with respect to some outside reference point, our main interest is the motion with respect to the CM, since that is where the gravitational forces are focused.

Velocity is defined as a change in position in a specific direction with respect to an increment of time. Take the derivative with respect to time of the location values to get:

vCM = 0

m(drm/dt) = −M(drM/dt)

mvm = −MvM

where

• dr is the derivative or small change in position r
• dr/dt is the derivative of r with respect to an increment of time
• vCM is the velocity the rCM position in a specific direction
• vm and vM are the velocities of the objects with respect to rCM

This means that the velocities move in opposite directions and mirror each other, as seen from the CM.

With respect to external reference

As a point of interest, finding the velocity relationship with respect to an external reference is:

vCM = (mvm + MvM)/(m + M)

An example of this is if rCM would move perpendicular to the coordinate line, then rm and rM would also move in the same direction, as seen with respect to an outside observer.

Accelerations

Acceleration is the change in velocity with respect to time. We are considering the two objects as part of closed system, such that there are no outside forces acting on the objects, except for the gravitational forces between them. In such as case, the only acceleration of the objects is toward the CM. In other words:

aCM = 0

maRm = −MaRM

where

• aRm is the radial acceleration of mass m toward the CM
• aRM is the radial acceleration of mass M toward the CM

This means that the radial acceleration vectors are in opposite directions, when viewed from the CM.

Summary

The location of the two objects and the CM between them, with respect to the zero-point of the coordinate line, follows the relationship:

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

Setting the position of the CM at the zero-point in the coordinate system, the relationship between the object locations with respect to the CM is:

mrm = −MrM

The velocities of the objects with respect to the CM are in opposite directions:

mvm = −MvM

Acceleration is only in the radial direction, due to gravitation:

maRm = −MaRM

When observed with relative to the CM, the motions of the objects mirror each other.

Resources and references

Author's Credentials

Websites

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

Center of Mass - Wikipedia

Gravitation Resources

What do you think?

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.

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