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Explanation of Gravitational Force Between Two Objects by Ron Kurtus - Succeed in Understanding Physics. Key words: attraction, force, mass, separation, Earth, Moon, tides, centrifugal, physical science, astronomy, School for Champions. Copyright © Restrictions

# Gravitational Force Between Two Objects

by Ron Kurtus (revised 24 June 2012)

You can apply the Universal Gravitation Equation to show the force of attraction between two objects, provided you know the mass of each object and their separation.

The equation is: F = GMm/R2, where G = 6.674*10−11 N-m2/kg2.

With this equation, you can make calculations to determine such things as the force between the Earth and the Moon, between a boy and a girl and between the Moon and a girl.

Questions you may have include:

• What is the force of attraction between the Earth and the Moon?
• What is the force of attraction between a boy and a girl?
• What is the force of attraction between the Moon and a girl?

This lesson will answer those questions.

Useful tool: Metric-English Conversion

## Force attracting Earth and Moon

To calculate the gravitational force pulling the Earth and Moon together, you need to know their separation and the mass of each object.

### Distance

The Earth and Moon are approximately 3.844*105 kilometers apart, center to center. Since the units of G are in meters, you need to change the units of separation to meters.

R = 3.844*108 m

### Mass of each object

Let M be the mass of the Earth and m the mass of the Moon.

M = 5.974*1024 kg

m = 7.349*1022 kg

### Force of attraction

Thus, the force of attraction between the Earth and Moon is:

F = GMm/R2

F = (6.674*10−11 N-m2/kg2)(5.974*1024 kg)(7.349*1022 kg)/(3.844*108 m)2

F = (2.930*1037 N-m2)/(1.478*1017 m2)

F = 1.982*1020 N

Note: Notice how all the units, except N, cancelled out.

Attraction between Earth and Moon

### Result of force

This considerable force is what holds the Moon in orbit around the Earth and prevents it from flying off into space. Inward force of gravitation equals the outward centrifugal force from the motion of the Moon.

Also, the gravitational force from the Moon pulls the oceans toward it, causing the rising and falling tides, according to the Moon's position.

## Force attracting boy and girl

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

Note: It is common—but scientifically incorrect—to state a person's weight in kg, which is mass. Care must be taken to distinguish between units of weight and mass. Newtons or kg-force are weight and kg or kg-mass are units of mass.

Also note: The separation between the boy and girl is measured from the center of the first person to the center of the other.

In order to establish the force between the two people, their weights must be converted to units of mass. That means dividing each weight by 9.8 m/s2. Thus, the mass of the boy is:

M = 75/9.8 = 7.653 kg-mass

The mass of the girl is:

m = 50/9.8 = 5.102 kg-mass

Substituting the values into the equation, you get:

F = GMm/R2

F = (6.674*10−11)(7.653)(5.102)/(0.5)2 N

F = 260.59*10−11/0.25 N

F = 1040.361*10−11 N

This is approximately:

F = 10−8 N or 0.01 millionth of a newton

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

## Force attracting girl and Moon

What is the gravitational pull from the Moon on the 50 kg-force (110 pound) girl?

You need to find the separation between the Moon and the surface of the Earth. Since the separation between the centers of the Earth and Moon is 3.844*108 m, you subtract the radius of the Earth (6371 km) to get the distance to the Earth's surface:

6371 km = 6,371,000 m = 0.064*108 m

R = 3.844*108 m − 0.064*108 m = 3.780*108 m

Thus, the force between the girl and the Moon is:

F = GMm/R2

F = (6.674*10−11)*(5.102)*(7.349*1022)/(3.780*108)2 N

F = 250.239*1011/14.228*1016 N

F = 17.588*10−5 N

F = 1.759*10−4 N = 0.000176 N

The girl would not notice the pull from the Moon, since the gravitation pull on her toward the Earth is 50 N (50 kg-m/s2 or kg-weight), which is much larger. But still, she is attracted more toward the moon than toward the boy who was sitting next to her.

## Summary

You can apply the Universal Gravitation Equation to show the force of attraction between two objects. With this equation, you can show the force between the Earth and the Moon is F = 1.982*1020 N, between a boy and a girl is F = 10−8 N and between the Moon and a girl is F = 1.759*10−4 N.

Think clearly and logically

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