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Gravitational Force Between Two Objects
by Ron Kurtus (revised 5 January 2016)
You can find the gravitational force between two objects by applying the Universal Gravitation Equation, provided you know the mass of each object and their separation.
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: Units Conversion
Universal Gravitation Equation
The Universal Gravitation Equation is:
F = GMm/R^{2}
where
- F is the force of attraction between two objects in newtons (N)
- G is the Universal Gravitational Constant = 6.674*10^{−11} N-m^{2}/kg^{2}
- M and m are the masses of the two objects in kilograms (kg)
- R is the separation in meters (m) between the objects, as measured from their centers of mass
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*10^{5} 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*10^{8} m
Mass of each object
Let M be the mass of the Earth and m the mass of the Moon.
M = 5.974*10^{24} kg
m = 7.349*10^{22} kg
Force of attraction
Thus, the force of attraction between the Earth and Moon is:
F = GMm/R^{2}
F = (6.674*10^{−11} N-m^{2}/kg^{2})(5.974*10^{24} kg)(7.349*10^{22 }kg)/(3.844*10^{8} m)^{2}
F = (2.930*10^{37 }N-m^{2})/(1.478*10^{17} m^{2})
F = 1.982*10^{20} N
Note: Notice how all the units, except N, canceled 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.
(See Circular Planetary Orbits for more information.)
Also, the gravitational force from the Moon pulls the oceans toward it, causing the rising and falling tides, according to the Moon's position.
(See Gravitation Causes Tides on Earth for more information.)
Force attracting boy and girl
If a boy who weighed 165 lb (74.8 kg) sat near a 50 kg (110 lb) girl, what would be the gravitational attraction between them, assuming the separation from their centers was 0.5 m (19.7 in)?
Substituting the values into the equation:
F = GMm/R^{2}
where
- M = 74.8 kg
- m = 50 kg
- R = 0.5 m
The result is:
F = (6.674*10^{−11} N-m^{2}/kg^{2})(74.8 kg)(50 kg)/(0.5 m)^{2}
F = 24961*10^{−11}/0.25 N
F = 99843*10^{−11} N
F = 0.99843*10^{−16} N
This is approximately:
F = 10^{−16} N
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 height of the girl is negligible compared to that distance, it is not a factor in the calculations.
Since the separation between the centers of the Earth and Moon is 3.844*10^{8} m, you subtract the radius of the Earth (6371 km) to get the distance to the Earth's surface.
Convert the Earth's radius into meters:
6371 km = 6,371,000 m = 0.064*10^{8} m
The separation from the center of the Moon to that of the girl on the Earth's surface is:
R = 3.844*10^{8} m − 0.064*10^{8} m = 3.780*10^{8} m
Thus, the force between the girl and the Moon is:
F = GMm/R^{2}
F = G*(mass of girl)*(mass of Moon)/(separation)
F = (6.674*10^{−11})*(50)*(7.349*10^{22})/(3.780*10^{8})^{2} N
F = 2452.36*10^{11}/14.228*10^{16} N
F = 172.36*10^{−5} N
F = 1.72*10^{−3} N = 0.00176 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/s^{2} 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*10^{20} N
- a boy and a girl is F = 10^{−16} N
- the Moon and a girl is F = 1.72*10^{−3} N.
Think clearly and logically
Resources and references
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Gravitational Force Between Two Objects