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Explanation of Gravitation and Center of Mass - Succeed in Understanding Physics. Also refer to center of gravity, force, acceleration, Earth, Moon, Sun, orbit, rotate, physical science, Ron Kurtus, School for Champions. Copyright © Restrictions
Gravitation and Center of Mass
by Ron Kurtus (16 November 2009)
Two objects are attracted toward each other by the force of gravitation. Each moves toward the center of mass of the two objects at acceleration proportional to the mass of the other object. Objects of equal mass will meet in the center between them. If one object is much larger than the other, it will appear as if the larger object was not moving at all. Also, the center of mass may even be within the larger object.
Objects in orbit rotate around the center of mass between them. Double stars have about the same mass and rotate about a center point. The Earth and Moon revolve around a center of mass point between each other.
Questions you may have include:
- What is the center of mass between two objects?
- 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.
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Center of mass
The center of mass between two objects is the point between them where the average of the masses times the distances of both objects is equal. When there is a uniform gravitational field, it is often called the center of gravity.
For two objects of equal mass, it is the point midway between the line joining their centers.
Center of mass is at the midpoint for equal objects
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
For example, the radius of the Earth is about 6685 km. The center of mass between the Earth and Moon is about 4630 km from the center of the Earth. That means the center of mass is 2055 km below the Earth's surface.
Acceleration toward each other
The gravitational force between two objects accelerates each toward the other. 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 the object of mass m is accelerated toward the object of mass M is:
am = GM/R2
and the acceleration of the object of mass M is accelerated toward the object of mass m is:
aM = Gm/R2
The objects will meet at the center or mass or center of gravity.
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.
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.
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 or close to its surface.
For example, the center of mass between the Earth and Moon is about 2055 km 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 orbiting each other.
Summary
Two objects that are attracted toward each other by the force of gravitation move toward their center of mass. Objects of equal mass will meet in the center between them. If one object is much larger than the other, it will appear as if the larger object was not moving at all. Also, the center of mass may even be within 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
Acceleration due to Gravity Calculations - from Western Washington University
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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