Satellite Motion and Artificial Gravity
Physics
Objectives:
- Apply the concepts of circular motion to the paths of orbiting satellites.
- Understand how a rotating spacecraft can create artificial gravity.
Satellite Motion
- Modern man-made satellites use various orbits to satisfy different missions. Orbits are typically
elliptical or circular and are inclined at various angles to the equator depending on the purpose of
the spacecraft.
- The principles of circular motion can be applied to those satellites in a circular orbit.
North Pole
Angle of Inclination
Equatorial Plane
Satellite
Path
South Pole
- The centripetal force on the satellite is provided by ________________________ .
- Unless the satellite has some type of thrusting rocket motor, the only force keeping it on its
circular path is the centripetal force and its forward velocity vector.
- There is only one speed that a satellite can have if the satellite is to remain in an orbit with a
fixed radius.
Orbit Radius and Satellite Velocity
Recall our equations for universal gravitation and centripetal force:
Universal Gravitation: 𝐹 = 𝐺
𝑚1 𝑚2
𝑟2
Centripetal Force: 𝐹𝑐 =
𝑚𝑣 2
𝑟
Since gravity is the only force providing centripetal acceleration, we can set these two equations
equal to each other:
𝐺
𝑚1 𝑀𝐸 𝑚1 𝑣 2
=
𝑟2
𝑟
Where m1 is the mass of the satellite, ME is the mass of the earth, and r is the radius of the orbit
(from the center of the earth to the satellite).
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Solving the above equation for v gives
First equation of
satellite motion
𝐺𝑀𝐸
𝑣=√
𝑟
- If the satellite is to remain in a circular orbit with radius r, then its speed must be precisely v.
- Notice that the mass of the satellite (m1) does not appear in the above equation.
- This equation also applies to astronomical objects that follow a circular orbit.
Period of a Satellite
Recall our equation for speed around a circle:
𝑣=
2𝜋𝑟
𝑇
where T is the time of one revolution.
If we substitute this for v in our first equation of satellite motion, then
2𝜋𝑟
𝐺𝑀𝐸
=√
𝑇
𝑟
Now, solving this equation for T yields
4𝜋 2 𝑟 3
𝑇=√
𝐺𝑀𝐸
Second equation
of satellite motion
- T is the period of the satellite – the time required for one complete orbit.
- Notice that the mass of the satellite (m1) does not affect the period.
- This equation can also be used for other astronomical objects that follow a circular orbit. ME is
replaced with the mass of the object providing the gravitational centripetal force.
Apparent Weightlessness & Artificial Gravity
- In an orbiting spaceship, the astronauts experience “apparent weightlessness”.
- This can be compared to the brief feeling you get while riding an elevator and it accelerates
downward.
- In the orbiting spaceship, the ship and the astronaut both “fall” toward the center of the earth
with the same acceleration. As a result, the astronaut cannot push against the “floor” of the ship
as he could on solid ground. The result is a feeling of weightlessness.
- If a scale were available, it would not register any weight because the
astronaut would not be able to push down on it.
- However, even at these altitudes the astronaut still has weight as
defined by Newton’s 2nd law: Fg = mg
2
- In large space stations of the future, where astronauts stay for extended periods, artificial gravity
can be created by rotating the spacecraft. The rotation will cause a centripetal force directed
toward the axis of rotation.
- By selecting the proper speed of rotation, the
centripetal force can be made to feel like the force of
gravity on earth.
- To determine the speed, the centripetal force is set to
equal the force of gravity, Fg.
𝑚𝑔 =
𝑚𝑣 2
𝑟
Solving this equation for v gives
𝑣 = √𝑟𝑔
- Note that the mass of the astronaut is not a factor in the velocity.
Problems
1. In the year 2087, the international space station has evolved into a circular design with a radius
of 1,500 meters. At what speed must the surface of the space station move so that the astronauts
onboard experience a push on their feet that equals their weight on earth?
2. A satellite is placed in orbit 6.0 x 105 meters above the surface of Jupiter. Jupiter has a mass
of 1.9 x 1027 kg and a radius of 7.14 x 107 meters. Find the orbital speed of the satellite.
ans: 4.2 x 104 m/s
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3. A satellite circles the earth in an orbit whose radius is twice the earth’s radius (as measured
from the center of the earth). The earth’s mass is 5.98 x 1024 kg, and it radius is 6.38 x 106 m.
What is the period of the satellite?
ans: 1.43 x 104 s
4. In June, 2003, a low power research satellite built by students at Aalborg University in
Denmark was launched to an altitude of 900 km above the earth’s surface. The satellite was
based on CubeSat technology and had a size of 10 x 10 x 10 cm. What velocity had to be
achieved in order to maintain the desired altitude? (The earth’s radius is 6.38 x 106 m and its
mass is 5.98 x 1024 kg.)
ans: 7.4 x 103 m/s
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