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Gravitation and Center of Mass
by Ron Kurtus (revised 1 September 2010)
The center of mass (CM) of an object or group of objects is the average or mean location of their mass. Gravitation from the object or group of objects can be approximated as coming from the center of mass.
A uniform sphere has its center of mass at its geometric center. You can find the center of mass of two spheres through a simple ratio formula.
Note: Some textbooks confuse center of mass with center of gravity (CG). Finding the center of gravity requires that the object is under the influence of gravity, while center of mass is the center of a mass distribution. Although CG is often at the same location as the CM, they are completely different concepts.
Objects attracted to each other will meet at the center of mass between them. The acceleration of the smaller mass will be greater than that of the larger object. Objects in orbit rotate around the center of mass between them.
Questions you may have include:
- How do you calculate the center of mass?
- How do objects accelerate toward each other?
- How do objects in orbit rotate about a center of mass?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
Determining the center of mass
Although you can determine the center of mass for complex or irregularly shaped objects, as well as multiple objects, we will concentrate on the CM between two spherical objects.
CM of a sphere
The center of mass 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 physical center of the sphere.
Center of mass is at center of sphere
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 and consider the CM at the geometric center.
CM used in gravitation equation
Considering the mass concentrated at the CM is convenient in using the Universal Gravitation Equation. Otherwise the calculation of the force between two objects would require a complex Calculus integration over all particles of the objects.
(See Universal Gravitation Equation for more information.)
CM between two spheres
The center of mass between two spheres—such as between the Earth and the Moon—is a point that is a ratio of the distances and masses of the objects:
mrm = MrM
R = rm + rM
where
- m and M are the masses of the two objects
- rm is the distance of mass m to the CM
- rM is the distance of mass M to the CM
- R is the distance between masses m and M as measured from the CM of each sphere
Calculation of CM
If you solve the equations for rm, you get:
rm = MrM/m
and
rm = R − rM
Combine equations and solve for rM:
MrM/m = R − rM
MrM = mR − mrM
MrM + mrM = mR
rM ( M + m) = mR
Thus:
rM = mR/(M + m)
Likewise:
rm = MR/(M + m)
Equal sized spheres
For two objects of equal mass, the CM is the point midway between the line joining their centers:
rm = MR/(M + m)
m = M
rm = MR/2M
Thus:
rm = R/2
Center of mass is at the midpoint for equal objects
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 = 597*1022 kg
The mass of the Moon is m = 7.36*1022 kg
(M + m) = 604.36*1022 kg
The distance between them is R = 384403 km
rM = mR/(M + m)
rM = 7.36*1022*384403/604.36*1022 km
rM = 4681 km
Since the radius of the Earth is about 6685 km and the CM between the Earth and Moon is about 4681 km from the center of the Earth, their center of mass is about 2005 km below the Earth's surface.
Acceleration toward each other
The gravitational force between two objects accelerates each toward the other until they meet at the center of mass between them. The object with greater mass will accelerate slower than the object with a smaller mass.
When one object is much larger than the other, it appears as if only the smaller object is moving toward the larger.
Difference in acceleration
The gravitational force of attraction is:
F = GMm/R2
where
- F is the force of attraction between two objects in newtons (N)
- G is the universal gravitational constant 6.67*10−11 N-m2/kg2
- M and m are the masses of the two objects in kilograms (kg)
- R is the distance in meters (m) between the objects, as measured from their centers of mass
The relationship between force and acceleration is:
F = ma
where a is the acceleration in m/s2 of an object of mass m in kg. Thus, the acceleration of each object toward each other is:
mam = GMm/R2
am = GM/R2
and
aM = Gm/R2
where
- am is the acceleration of smaller mass m
- aM is the acceleration of the larger mass M
You can see that the acceleration of the smaller object will be greater than the larger one. The objects will meet at the center or mass between the two.
If objects are similar in mass
If the objects are similar in mass, they will appear to travel toward each other.
Similar objects meet at the center of mass
If one object much larger than the other
If one object is much larger than the other, the motion of the larger object is so small that it appears that only the smaller object is being attracted to the other.
A good example of this is when you drop an object and it falls to the Earth. In reality, the Earth also moves toward the object, but the movement is so small that it is imperceptible.
Only the smaller object appears to move toward the larger
Orbiting around center of mass
Seldom do astronomical objects move directly toward each other. Usually, they have tangential velocities, such that they may be attracted toward each other but never collide. In some situations, the objects go into orbit around each other.
(See Circular Gravitational Orbits for more information.)
Double stars
In the case of two objects approximately the same mass that have tangential velocities, they may not collide but instead go into orbit around each other. They revolve around the center point between them.
Same mass objects rotate about CM
Astronomers have seen what are called double stars, where they seem to revolve around each other.
Large and small objects
In the case of a large object and a smaller object—such as the Earth and the Moon or the Sun and the Earth—they still revolved around the center of mass, except that the center may be within the larger object.
Each object orbits the center of mass
In such a case, the distance of the smaller to larger object remains constant, as they both rotate about the CM.
For example, the center of mass between the Earth and Moon beneath the Earth's surface. When seen from outer space, the Earth has a slight wobble when the Moon is orbiting it. In reality, they are both orbiting the CM.
Summary
Center of mass is the average or mean location of the mass of an object or group of objects. It is a different concept than center of gravity.
Gravitation can be approximated as coming from the center of mass. A uniform sphere has its center of mass at its geometric center. You can find the center of mass of two spheres through a simple ratio formula.
Objects attracted to each other will meet at the center of mass between them. The acceleration of the smaller mass will be greater than that of the larger object. Objects in orbit rotate around the center of mass between them.
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Resources
The following resources provide information on this subject:
Websites
Center of Mass Calculator - Univ. of Tennessee - Knoxville (Java applet)
Center of Mass - Wikipedia
Gravity and Gravitation Resources
Books
Top-rated
books on Simple Gravity Science
Top-rated
books on Advanced Gravity Physics
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
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