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Showing Gravity Equation Comes From Universal Gravitation
by Ron Kurtus (5 July 2009)
You can show that the standard equation for the force of gravity near the Earth, F = mg, is a special case of the universal gravitation equation: F = GMm/R2.
This derivation is done by first making sure the units correspond and then substituting the mass of the Earth for M and the radius of Earth for R. The result will show that g = GM/R2.
Questions you may have include:
- What is the gravity equation?
- What is the universal gravitation equation?
- How can you show the gravity equation comes from the universal gravitation equation?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
Force of gravity equation
Gravity is defined as the force on an object near the Earth's surface from the Earth's gravitational field. The equation for the force of gravity is:
F = mg
where:
- F is the force in newtons (N)
- m is the mass of the object in kilograms (kg)
- g is the acceleration due to gravity; g = 9.8 m/s²
- mg is m times g
Note: In this derivation, keep the units of measurement in the metric or SI system.
Assumptions
This equation holds for objects that are relatively close to the ground. In other words, the equation would not be an accurate description of the force on objects higher than our weather satellites.
Another assumption is that the Earth has a much greater mass than the object's mass, m.
Universal gravitation equation
Newton's Theory of Universal Gravitation states that two objects will attract each other with a force proportional to their masses and inversely proportional to the square of distance between them. The universal gravitation equation is:
F = GMm/R²
where
- F is the force of attraction between two objects in newtons (N)
- G is the universal gravitational constant, where G = 6.67*10−11 N-m²/kg²
- M and m are the masses of the two objects in kilograms (kg)
- R is the distance between the objects, as measured from their centers in meters (m)
Derivation
You want to show that if M is the mass of the Earth, R is its radius and m is the mass of some object on the Earth's surface, then the universal gravitational force equals the force of gravity on Earth:
F = GMm/R2 = mg
Dividing both sides of the equation by m, results in:
g = GM/R2
Thus, you want to prove that g and GM/R2 have the same value. You do that by substitute the values into the GM/R2 term.
Mass of Earth
The approximate value for the mass of the Earth is:
M = 5.974*1024 kg
Radius of Earth
The approximate value for the radius of the Earth is R = 6.376*106 meters.
Also:
R2 = (6.376*106 m)*(6.376*106 m)
R2 = 40.65*1012 m2
R2 = 4.065*1013 m2
Change units of G
Since a newton (N) is equivalent to kg-m/s2, you want to change the units of G from N-m2/kg2 to m3/s2-kg. This is done so that the units of the items in the two equations correspond.
G = 6.67*10−11m3/s2-kg.
Calculation
Now let's put the values into the equation:
g = GM/R²
g = (6.674*10−11 m3/s2-kg)*(5.974*1024 kg)/(4.065*1013 m2)
g = (39.87*1013 m3/s2)/(4.065*1013 m2)
g = 9.808 m/s2
Note how the various units will cancel out in the multiplication and division. This is important to verify that your units and the equation are correct.
Thus F = mg is derived from F = GMm/R2.
Summary
You can show that the value of g in standard equation for the force of gravity near the Earth is a special case of the value of GMm/R2 from the universal gravitation equation. The derivation is done by simply making sure the units correspond, substituting the mass of the Earth for M and the radius of Earth for R to show that g = GM/R2.
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Resources
The following resources provide information on this subject:
Websites
Acceleration due to Gravity Calculations - from Western Washington University
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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Showing Gravity Equation Comes From Universal Gravitation