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Newton's Universal Gravity Equation

by Ron Kurtus (revised 29 August 2007)

Isaac Newton's Theory of Universal Gravitation states that all matter attracts other matter to it through the force of gravity. The theory's Universal Gravity Equation defines the force of gravity as a function of the masses of the objects and the distance between them. Applications of this equation can show the force of attraction between two objects. It can also be shown that the Universal Gravity Equation is the same as F = mg, if certain assumptions are made.

Questions you may have include:

• What is the Universal Gravity Equation?
• What are some applications of the equation?
• How does it relate to the gravity equation on Earth?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Useful tools: Metric-English Conversion | Scientific Calculator.

Universal 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 Gravity Equation is:

F = GMm/R²

where

• F is the force of attraction between two objects
• G is the universal gravitational constant; G = 6.67*10^-11 N-m²/kg². The units of G can be stated as Newton meter-squared per kilogram-squared or Newton square meter per square kilogram.
• M and m are the masses of the two objects
• R is the distance between the objects, as measured from their centers
• GMm/R² is G times M times m divided by R-squared

The equation makes the assumption that the mass can be considered to be concentrated at the center of the object. This is done to greatly simplify the mathematics. The equation averages out to be true when the sizes of the objects are small or the distances are great.

Applications

There are numerous applications of Newton's Universal Gravity Equation.

Boy and girl

If a 50 kg (110 lb) girl sat 0.5 m (19.7 in) from a boy who was 75 kg (165 lb), what would be the gravitational attraction between them?

Substituting the values into the equation, you get:

F = GMm/R² = (6.67*10^-11 N-m²/kg²)(50 kg)(75 kg)/(0.5 m)(0.5 m) = 10^-6 N or one-millionth of a Newton

That is a very small gravitational attraction, but it can be measured on a sensitive instrument.

Earth and Moon

What is the gravitational force felt on the Moon from the Earth?

The Earth and Moon are 3.84*10⁵ kilometers apart. Since the units of G is in meters, you need to change them to 3.84*10⁸ m. Note that the distance is from center to center. The force on the surface of the Earth would be slightly less, but since the distance to the Moon is so much greater than the radius of the Earth, the difference would be negligible.

The mass of the Earth is 5.98*10²⁴ kg and that of the Moon is 7.35*10²² kg. Thus, the force of attraction on the Earth from the Moon is:

F = GMm/R²= (6.67*10^-11 N-m²/kg²)(5.98*10²⁴ kg)(7.35*10²² kg)/(3.84*10⁸ m)² = 1.99*10²⁰ N

This considerable force is what holds the Moon in orbit around the Earth.

Effect of Moon on person

The gravitational pull from the Moon on the 50 kg (110 pound) girl is:

F = GMm/R² = (6.67*10^-11 N-m²/kg²)(50 kg)(7.35*10²² kg)/(3.84*10⁸ m)² = 1.67*10^-3 N = 0.00167 N

She would not notice the pull from the Moon, since the gravitation pull on her toward the Earth is 490 N. But still, she is attracted more toward the moon than the boy who was sitting next to her.

Gravity acceleration on Earth

We can show that the Universal Gravity Equation is the same as the standard equation for the force of gravity on Earth

F = mg

where:

• m is the mass of the object
• g is the gravitational constant or acceleration due to gravity; g = 9.8 m/s² (meters per seconds-squared) or 32 ft/s²
• mg is m times g

Special considerations

The acceleration due gravity on the Earth has been determined by experiments and measurement. It has also been shown that g is a constant.

But note that there are some special considerations in the equation F = mg. First of all, it is assumed that the object is relatively close to the Earth. This equation would not hold as well for objects higher than our weather satellites. Another assumption is that the Earth has a much greater mass than the object's mass m.

With these considerations in mind, we can show that F = GMm/R² is the same as F = mg when the object is relatively close to the Earth. The following material is the derivation of the simple gravity equation near the Earth from the Universal Gravity Equation.

Mass

Let M equal the mass of the Earth. The approximate value for M = 6*10²⁴ kilograms (6 followed by 24 zeros). Also, let m be the mass of some object near the surface of the Earth. As you will see later, we don't need to know the specific mass of the object.

Distance is radius

This takes a little stretch of the imagination, but let's assume that an object near the surface of the Earth is attracted toward the center of the Earth, as if all of the Earth's matter was compressed at that point (as per Newton's assumption, when he came up with his theory). If r is the radius of the Earth plus a few meters, then the object near the surface would be a distance of r from the center of gravity.

The approximate radius of the Earth is 6.376*10⁶ meters, so the distance between M and m is R = 6.376*10⁶ m. Also, R² = 4*10¹³ m² (meters-squared or square meters).

Change units of G

Since a Newton is a kg-m/s², we change the units of G from N-m²/kg² to m³/s²-kg (meter-cubed per second-squared-kilograms). This is done so that the units of G (m³/s²-kg) relates to the unit of m (kg) and units of g (m/s²).

Calculation

Now let's put the values into the Universal Gravity Equation:

F = GMm/R² = (6.67*10^-11 m³/s²kg) * (6*10²⁴ kg) * m / (4*10¹³ m²)

F = m*10 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.)

Compare with g

Now we know that the force of gravity near the Earth is: F = mg

Thus, from the Universal Gravity Equation calculations above, g = 10 m/s². Since we used approximate values for r and M, that value is approximately g = 9.8 m/s² and the holds for the experimental values or measurements on Earth.

Summary

The Theory of Universal Gravitation states that all matter attracts other matter to it through the force of gravity. The Universal Gravity Equation defines the force of gravity as a function of the masses of the objects and the distance between them. Applications of this equation can show the force of attraction between two objects. If certain assumptions are made, the Universal Gravity Equation is the same as F = mg, .

Answers to Readers' Questions

Think clearly and logically

Resources

The following resources provide information on this subject:

Websites

Acceleration of Gravity Calculations - from Western Washington University

Physical Science Resources

Books

Top-rated books on Simple Gravity Science

Top-rated books on Advanced Gravity Physics

Miscellaneous

Mini-quiz to check your understanding

1. What is calculated from the Universal Gravity Equation?

2. What happens to the gravitational attraction between a boy and a girl as they move closer together?

3. When is F = GMm/R² almost the same as F = mg?

If you got all three correct, you are on your way to becoming a Champion in Physical Science. If you had problems, you had better look over the material again.

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