Resource Lesson
Universal Gravitation and Satellites
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The
Law of Universal Gravitation
states that every object in the universe attracts every other object in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the distance between their centers squared.
where
G is the gravitational constant, 6.67 x 10
^{-11}
Nm
^{2}
/kg
^{2}
M
_{1}
is the mass of the first body in kg
M
_{2}
is the mass of the second body in kg
R is the distance from the center of M
_{1}
to the center of M
_{2}
For a
satellite in circular orbit
, the unbalanced central force is supplied by gravity. That is, gravitation supplies the centripetal force to keep it in circular orbit. Note that this is an example of
uniform circular motion
since gravity is everywhere PERPENDICULAR to the satellite's path; that is, gravity never has a component in the direction of the satellite's motion to accelerate it. [Remember from geometry that a radius always meets the circumference of a circle at right angles.]
Notice that this expression for the centripetal acceleration for a satellite orbiting the earth at a radius
r
is identical to the expression for the gravitational field strength,
g
, at radius
r
.
Hence, for satellites in circular orbits, the "acceleration due to gravity" at its orbital radius supplies the necessary centripetal acceleration to maintain its orbit.
The speed of a satellite in circular orbit is constant. The minimum velocity needed to maintain a satellite’s circular orbit around the earth if it is located at a height,
h
, above the earth’s surface is calculated by
Refer to the following information for the next two questions.
Let's examine a satellite in circular orbit that just clears Earth's highest mountain, Mt. Everest, 29028 feet above sea level or 8850 meters. Adding this distance to the radius of the Earth, 6370 km, we would need an orbital radius of 6.38 x 10
^{6}
meters.
Calculate the value of the satellite's velocity.
At this speed, how long would it take a satellite to orbit the earth one time?
The general rule of thumb used when working with satellites in
low Earth orbit
is to take their orbital velocity to be 8 km/sec and their orbital period to be 90 minutes. When satellites travel faster than 8 km/sec, then their orbits are no longer circular, but instead become more and more
elliptical
(with variable speeds since their distance from the earth is constantly changing) until their speeds reach 11.2 km/sec. At this speed, called the
escape velocity
of the Earth, they can no longer orbit the Earth and their orbits become hyperbolic. Achieving this escape velocity does not release them from being members of our solar system, they are just no longer satellites of the Earth.
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