# Gravitation and Center of Mass

by Ron Kurtus (revised 14 August 2017)

The center of mass (CM) is an important concept in gravitation calculations. It is used in measuring the force between two objects and how objects move with respect to each other under gravitational influence.

The CM of a spherical object is at its geometric center. Thus, the distance between two objects—as used in the gravitational force equation—is measured from the CM of each object.

There is also a CM between two objects, such that they will orbit around that CM.

Questions you may have include:

• Where is the CM of a sphere?
• How is the CM used for gravitation between two objects?
• Where is the CM between two objects?
• What is the motion about the CM?

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

## Center of mass of a sphere

The center of mass (CM) of an object is the weighted average of the mass distribution of the body. In the case of a sphere with the material uniformly distributed, the CM is the geometric center of the object. Center of mass of sphere is at its geometric center

### Approximate center for Earth

Although objects such as the Earth are not exact spheres and do not have their mass uniformly distributed, the variations are small enough to neglect, such that you can consider the CM to be at the geometric center.

## CM used in gravitation equation

The Universal Gravitation Equation considers the mass of a sphere as concentrated at its CM. This assumption simplifies the calculation of the force between two objects, avoiding complex Calculus integration over all particles of the objects. Separation between CM of spheres

Thus the separation used in the gravitation equation is measured from the CM of each object:

F = GMm/R2

where

• F is the force of attraction between two objects
• G is the Universal Gravitational Constant
• M and m are the masses of the two objects
• R is the separation between the objects, as measured from their centers of mass

## CM between two objects

If you consider two separated objects as a single entity, there is a CM between those objects. CM between two uniform spheres

### Ration of separations and masses

The center of mass between the spheres 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
• R is the separation between masses m and M as measured from the CM of each sphere
• Rm is the separation between mass m and the CM
• RM is the separation between mass M and the CM

mRm = MRM can also be stated as the inverse ratio of the masses:

Rm/RM = M/m

### Zero gravitation force at CM

An interesting fact is that the gravitational forces from each object cancel out at their CM.

## Orbital motion around CM

The orbital motion of two objects in space is around the CM between the objects. The objects will orbit around their CM. Also, there is no gravitation at that CM.

## Summary

The center of mass (CM) is an important concept in gravitation calculations. It is used in measuring the force between two objects and how objects move with respect to each other under gravitational influence.

The CM of a spherical object is at its geometric center. Thus, the distance between two objects—as used in the gravitational force equation—is measured from the CM of each object.

There is also a CM between two objects, such that they will orbit around that 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

### Books

Top-rated books on Gravity Top-rated books on Gravitation 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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www.school-for-champions.com/science/
gravitation_center_of_mass.htm

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