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# Center of Mass and Radial Gravitational Motion

by Ron Kurtus (revised 14 May 2011)

The motion of two objects in space can be broken into perpendicular radial and tangential vector components with respect to the center of mass (CM) between them. The radial components are along the axis through the CM and are affected by the gravitational force between the two objects, as well as any initial velocities.

Gravitational attraction will normally pull the objects toward each other along the radial axis. However, if there is an outward initial velocity, the objects may be moving away from the CM, where they can reach a maximum displacement and return toward each other or escape the gravitational pull and fly off into space.

The amount of the tangential components determine whether the objects will collide at their CM, go into orbit around each other or simply pass by and fly off into space.

If the objects seem to be moving in the same direction, the viewpoint is with respect to an outside observer. This view can be transformed to be with respect to the stationary CM.

Questions you may have include:

- What happens when both objects move toward each other?
- What happens when to objects move apart?
- What happens when objects are moving in the same direction?

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

## Both objects moving toward CM

Two objects in space may be moving with the radial components of their velocity vectors in a direction toward each other, due to the gravitational pull between them, as well as initial velocities toward the CM.

Objects moving toward CM

### Velocity relationship

The relationship between the velocities is:

mv_{Rm}= −Mv_{RM}

where

**m**and**M**are the masses of the objects**v**and_{Rm}**v**are the radial velocities_{RM}

Note: The negative sign indicates the velocity vectors are in opposite directions.

### Beyond velocity relationship

If the velocities do not appear to follow the above relationship, your point-of-view is not with respect to the CM. Instead, it may be with respect to some other point-of-view.

### Effect of tangential velocities

The amount of the tangential component determines whether the objects will collide at their CM, go into orbit around each other or simply pass by and fly off into space.

### Acceleration relationship

The relationship between the accelerations of the objects with respect to the CM is:

ma_{Rm}= −Ma_{RM}

These accelerations also are related to the gravitational force. Consider:

F = GMm/R^{2}

where

**F**is the force of attraction between two objects**G**is the Universal Gravitational Constant**R**is the separation between the centers of the objects

Compare with the force-acceleration relationship for mass **m**:

F_{Rm}= ma_{Rm}

a_{Rm}= GM/R^{2}

Likewise,

F_{RM}= Ma_{RM}

a_{RM}= Gm/R^{2}

Although the accelerations are in opposite directions toward the CM, their magnitudes are related to the mass of the attracting object and the separation of the objects.

## Objects moving away from each other

For the objects to be moving away from each other, there must have been some initial impetus or force applied to give them their velocities. That force is no longer applied when we examine the motion of the objects. although they still have their initial outward velocities.

Objects moving away from each other

Depending on their initial velocities, the objects may reach some maximum displacement and reverse directions to move toward the CM. This is similar to the effect of throwing a ball into the air and having it return to Earth.

(

See Velocity Equations for Objects Projected Upward for more information.)

If the velocities are sufficient, the objects may escape the gravitational pull of each other.

### Example of expansion of Universe

A good example of various sized objects moving away from the center of mass between them is the expansion of the Universe, where the galaxies have been measured as moving away from some center point.

Speculation on the expansion of the Universe after the Big Bang occurrence is whether the expansion is at a rate where the objects will move outward forever or whether the galaxies will reach a maximum displacement and reverse directions back toward the CM of the Universe.

Measurements on the rate of expansion have also determined that "something" is affecting the expansion—perhaps dark matter.

(

See Effect of Dark Matter and Dark Energy on Gravitation for more information.)

### Example of escape velocity by rocket

When one object is much larger than the other, the CM is close to the geometric center of the larger object. An example of this is the comparison of a rocket and the much larger Earth.

If the rocket is sent upward at a sufficient velocity, it can escape the gravitational pull between it and the Earth, such that it will go off into space.

### Effect of tangential component

In cases where the velocities are less than the escape velocity, the tangential velocities will determine whether the objects fall back into each other or go into orbit around the CM.

## Objects moving in same direction

You may observe two objects moving in the same radial direction. This means you are observing the object with respect to some other point of reference and not relative to the CM. When the objects are moving in the same direction, the CM is moving along with the objects. In such a case, the **mv _{Rm} = −Mv_{RM}** ratio does

*not*hold.

Objects moving in the same direction

If the viewpoint is changed to be with respect to the CM, the CM will appear stationary and the objects will be either both moving toward the CM or away from that point.

## Summary

The radial component of the motion of two objects in space is along the axis through the CM. The gravitational attraction normally pulls the objects toward each other along the radial axis. But if there is an outward initial velocity, the objects may be both moving away from the CM, where they can reach a maximum displacement and return toward each other or escape the gravitational pull and fly off into space.

The tangential motion component affects whether the objects will collide at their CM, go into orbit around each other or simply pass by and fly off into space.

If the objects seem to be moving in the same direction, the viewpoint is with respect to an outside observer. This view can be transformed to be with respect to a stationary CM.

Work hard to do your best

## Resources and references

### Websites

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

**Center of Mass** - Wikipedia

### Books

**Top-rated books on Gravitation**

## Questions and comments

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## Center of Mass and Radial Gravitational Motion