121  122 Newton's Law of Universal Gravitation and the
Attraction of Spherical Bodies
Gravitation is the phenomenon that between every two objects there is a force of attraction.
Newton’s law of universal gravitation
describes the behavior of this force. Between any two
point masses
m
1
and
m
2
, the magnitude of the gravitational force on each mass due to the other
is given by
where
r
is the distance between the two masses and
G
is a constant called the
universal
gravitational constant
. The value of this constant is
The force on each mass points directly at the other mass because each mass attracts the other
toward it.
For situations involving more than two masses we apply the
principle of superposition
:
The net gravitational force on any given mass due to two or more other masses is the vector sum
of the gravitational forces due to each of the other masses individually.
Exercise 121 Earth and Venus
On average Earth and Venus are separated by about 4.14 x
10
10
m. What is the magnitude of the gravitational force between them at this separation?
Solution
:
In this problem we are given only the distance between Earth and Venus. To calculate the
gravitational force between them we will need to look up their masses. The masses are as
follows:
M
E
= 5.97 x 10
24
kg;
M
V
= 4.87 x 10
24
kg
Having all the data we need, the force can now be calculated.
Remember, this is the magnitude of the force exerted on both Earth and Venus.
The above law of universal gravitation is stated for point masses. The detailed calculations for
extended bodies can become complicated, but Newton figured out that the final result for
spherical bodies becomes simple again. Basically, Newton showed that two completely
separated, uniform spherical bodies attract each other as if they were point masses with all their
mass located at their respective centers. This fact explains why the pointmass formula given
above works so well for large objects like Earth and the Moon.
Treating the Earth as a point mass
M
E
located at the center of the Earth provides further insight
into the acceleration due to gravity that we measure near Earth’s surface. Objects near the
surface are a distance from the center roughly equal to the radius of the Earth,
R
E
. Using
Newton's law of gravity we can conclude that the acceleration due to gravity near the surface is
given by
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With this result you can also see that the higher you go above the surface, the farther you are
from Earth’s center, and therefore the weaker the effect of gravity, resulting in a smaller
acceleration.
Example 122 Martian Gravity
What is the acceleration due to gravity on the surface of Mars?
Picture the Problem
The sketch is a representation of the planet Mars, with its radius extending
from the center to the surface.
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 Spring '08
 RAO
 Physics, Force, Wavelength

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