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# Center of Mass Definitions

by Ron Kurtus (revised 31 May 2011)

The * center of mass* (CM) of an object is a point that is the average or mean location of its mass, as if all the mass of the object was concentrated at that point. A uniform sphere has its center of mass at its geometric center. The CM is sometimes called the

*barycenter*.

The CM of a group of objects is point that is the mean location of their individual centers of mass. In our gravitational studies, we are only considering the CM between two objects.

You can find the center of mass between two spheres through a simple ratio formula. When one object is much larger than the other, the CM may actually be within the larger object.

Note: Some textbooks confuse center of mass with center of gravity (CG), which is related to the effect of gravity on an object, while center of mass concerns mass distribution. Although CG is often at the same location as the CM, they are completely different concepts.(

See Center of Gravity for more information.)

Questions you may have include:

- Where is the center of mass of a sphere?
- Where is the center of mass between two spheres?
- What are some special cases for two spheres?

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 total 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.

(

See Universal Gravitation Equation for more information.)

## CM between two spheres

In calculating the CM between two spheres—such as between the Earth and the Moon—you can assume each has its mass concentrated at its geometric center. The center of mass between the spheres is then a point that is a ratio of the separations and masses of the objects:

mR_{m}= MR_{M}

R = R_{m}+ R_{M}_{}

where

**m**and**M**are the masses of the two objects**R**is the separation between mass_{m}**m**and the CM**R**is the separation between mass_{M}**M**and the CM**R**is the separation between masses**m**and**M**as measured from the CM of each sphere

**mR _{m} = MR_{M}** can also be stated as the inverse ratio of the masses:

R_{m}/R_{M}= M/m

CM between two uniform spheres

### Solve for R_{m} and R_{M}

If you solve the equations for **R _{m}**, you get:

R_{m}= MR_{M}/m

and

R_{m}= R − R_{M}

Combine the equations and solve for **R _{M}**:

MR_{M}/m = R − R_{M}

MR_{M}= mR − mR_{M}

Rearrange items:

MR_{M}+ mR_{M}= mR

R_{M}( M + m) = mR

Thus:

R_{M}= mR/(M + m)

Likewise:

R_{m}= MR/(M + m)

## Special cases

Special cases of the CM between two objects include equal-sized spheres and when one sphere much larger than the other.

### Equal sized spheres

For two objects of equal mass, the CM is the point midway between the line joining their centers. Start with the equation:

R_{m}= MR/(M + m)

Since:

m = M

R_{m}= MR/2M

Thus:

R_{m}= R/2

Center of mass is at the midpoint for equal objects

### CM when one sphere much larger

If one sphere is much larger than the other, the center of mass may even be within the larger object.

Center of mass can be inside much larger object

#### Earth and Moon example

A good example concerns the CM between the Earth and the Moon.

The mass of the Earth is

M =5.974*10^{24}kgThe mass of the Moon is

m =7.348*10^{22}kg

(M + m) =5.974*10^{24}kg + 0.073*10^{24}kg = 6.047*10^{24}kg

The separation between the Earth and the Moon is **R =** 384,403 km, which equals 3.844*10^{5} km. Substitute in values to find the separation between the Earth and the CM, **R _{M}**:

R_{M}= mR/(M + m)

R7.348*10_{M}=^{22}*3.844*10^{5}/6.047*10^{24}km

R4671 km_{M}=

Since the radius of the Earth is about 6685 km and the CM between the Earth and Moon is about 4671 km from the center of the Earth, the CM between the Earth and Moon is at about 2014 km below the Earth's surface.

## Summary

The center of mass of an object is the average or mean location of its mass. A uniform sphere has its center of mass at its geometric center. The CM of a group of objects is point that is the mean location of their individual centers of mass.

You can find the center of mass between two spheres through a simple ratio formula:

mR_{m}= MR_{M}

R = R_{m}+ R_{M}

When one object is much larger than the other, the CM may actually be within the larger object.

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**

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## Center of Mass Definitions