9.3. Circular Orbits http://www.ck12.org
9.3 Circular Orbits
Objective
The student will:
- Use Newton’s Universal Law of Gravity and Kepler’s Third Law to solve problems.
Vocabulary
- geosynchronous:An orbit where the satellite remains over the same position on the Earth.
- orbit:The path of one object revolving around the other, such as the Moon around the Earth, or the Earth
around the Sun. Orbits can be circular or elliptical.
Introduction
Putting together the principles of gravity and the principles of circular motion allows us to solve problems relating
to circularorbits. The planets and their moons do not have circular orbits, but many artificial satellites do.
The first artificial satellite was the Russian satellite Sputnik, which was launched October 4, 1957 and successfully
orbited the Earth at an altitude of 228 km, circling the Earth every 98 minutes. Sputnik was considered a challenge to
American military and technological dominance, and thus started the Space Race. Since 1957, the United States has
landed twelve men on the Moon, had an active Space Shuttle program, been a partner in building the International
Space Station, and has been one of several countries to send robotic probes all over the solar system. Currently, the
US maintains numerous artificial satellites in Earth orbit for the purposes of navigation, communication, monitoring
the weather, scientific studies, and spying.
Solving problems with a circular orbit combines centripetal acceleration and universal gravity.
- An object moving at constant speed has acceleration toward the center of a circle of ac=v
2
r. - Any object has the force of gravity from another object ofF=Gmr^12 m^2.
Putting these together with Newton’s Second Law,F=ma, we can solve problems regarding orbits. When dealing
with the Earth, we can simplify gravity because we know the acceleration due to gravity at the surface of the Earth:
F=GmmRe 2 e=mg=m( 9 .81m/s^2 )→GmR (^2) ee= 9 .81m/s^2
In orbital calculations, we use the more exact figure for the acceleration due to gravityg.
Check Your Understanding
- What is the acceleration due to gravity for a satellite at the height of one-half the Earth’s radius,^12 Re?