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Motion of
Orbiting Satellites

Explore the physics of circular orbits, understand how gravity provides centripetal force, and derive the relationships between orbital speed, radius, and period.

AP Physics 1

What Is a Satellite Orbit?

A satellite orbit is the curved path an object follows as it moves around a much more massive body. The satellite is in constant free fall toward the central body, but its tangential velocity carries it forward fast enough that the surface curves away beneath it.

Essential Vocabulary

Orbital Radius \( r \)
Distance from the center of the central body to the satellite
Orbital Speed \( v \)
Tangential speed of the satellite along its circular path
Orbital Period \( T \)
Time for one complete revolution around the central body

Gravity as Centripetal Force

\(M\) \(m\) \(v\) \(F_g\) \(r\) Orbit path

Deriving Orbital Speed

For a satellite of mass \( m \) in a circular orbit of radius \( r \) around a body of mass \( M \), gravity provides the centripetal force:

Cancel \( m \) (satellite mass drops out!) and solve for \( v \):

The orbital speed depends only on the central body's mass \( M \) and the orbital radius \( r \): not on the satellite's mass.

The Orbital Period \( T \)

The circumference of the orbit is \( 2\pi r \). Since the satellite moves at constant speed \( v \):

Speed
\( v = \sqrt{\dfrac{GM}{r}} \)
Period
\( T = 2\pi\sqrt{\dfrac{r^3}{GM}} \)

Connection to Kepler's Third Law

Squaring the period equation reveals a powerful proportionality:

This is Kepler's Third Law for circular orbits. The ratio \( T^2 / r^3 \) is the same constant for every satellite orbiting the same central body.

Comparing orbital radii

The photograph is illustrative; the companion gives the exact circular-orbit comparison for radii in a 1:4 ratio around the same central mass.

The orbital photograph is illustrative. The companion compares radii r₂ = 4r₁, giving v₂ = v₁/2, T₂ = 8T₁ and a₂ = a₁/16.
The orbital photograph is illustrative. The companion compares radii r₂ = 4r₁, giving v₂ = v₁/2, T₂ = 8T₁ and a₂ = a₁/16.
Exact 1:4 orbit-radius model and the resulting motion ratios.
Exact 1:4 orbit-radius model and the resulting motion ratios.

Orbital Radius vs Altitude

Planet \(R\) \(h\) \(r = R + h\)
Altitude \( h \)
Height above the surface of the planet
Orbital Radius \( r \)
Distance from the center of the planet: \( r = R + h \)

Energy of an Orbiting Satellite

A satellite in a circular orbit has both kinetic energy and gravitational potential energy:

Kinetic Energy
\( K = \dfrac{1}{2}mv^2 = \dfrac{GMm}{2r} \)
Potential Energy
\( U = -\dfrac{GMm}{r} \)

Apparent Weightlessness

Astronauts on the ISS appear to "float" even though gravity is only about 10% weaker than at Earth's surface. Why?

A floating wrench inside an orbiting cabin

An astronaut holds a cabin rail while a wrench and other loose objects float near a window overlooking Earth.The cabin and its loose objects fall together in orbit. Gravity remains present and points toward Earth while the orbital velocity is tangent to the path.

A floating wrench inside an orbiting cabin

A floating wrench inside an orbiting cabin. The cabin and its loose objects fall together in orbit. Gravity remains present and points toward Earth while the orbital velocity is tangent to the path.The cabin and its loose objects fall together in orbit. Gravity remains present and points toward Earth while the orbital velocity is tangent to the path.

Problem-Solving Strategy

1
Identify the central body and its mass \( M \). Find the orbital radius \( r \) (remember: \( r = R + h \) if given altitude).
2
Set gravity equal to centripetal force: \( \dfrac{GMm}{r^2} = \dfrac{mv^2}{r} \). Cancel \( m \).
3
Solve for the unknown: speed \( v = \sqrt{GM/r} \), period \( T = 2\pi\sqrt{r^3/(GM)} \), or radius from a known \( T \).
4
Check units and reasonableness. LEO satellites: \( v \approx 7{,}700 \;\text{m/s} \), \( T \approx 90 \;\text{min} \). Geostationary: \( T = 24 \;\text{hr} \).

Example 1: ISS Orbital Speed

The ISS orbits at an altitude of 408 km above Earth's surface. Earth's mass is \( M = 5.97 \times 10^{24}\;\text{kg} \) and radius is \( R = 6.37 \times 10^6\;\text{m} \). Find the orbital speed.

1
Convert altitude to orbital radius: \( r = R + h = 6.37 \times 10^6 + 4.08 \times 10^5 = 6.778 \times 10^6\;\text{m} \)
2
Apply the orbital speed formula: \( v = \sqrt{\dfrac{GM}{r}} = \sqrt{\dfrac{(6.674 \times 10^{-11})(5.97 \times 10^{24})}{6.778 \times 10^6}} \)
3
Calculate: \( v = \sqrt{5.877 \times 10^7} \approx 7{,}666\;\text{m/s} \approx \) 7.67 km/s

Example 2: Geostationary Orbit

A geostationary satellite has a period of \( T = 24 \;\text{hours} \). Find the required orbital radius around Earth.

1
Convert period: \( T = 24 \times 3600 = 86{,}400\;\text{s} \)
2
Rearrange for \( r \): \( T^2 = \dfrac{4\pi^2}{GM} r^3 \implies r = \left(\dfrac{GM T^2}{4\pi^2}\right)^{1/3} \)
3
Substitute: \( r = \left(\dfrac{(6.674 \times 10^{-11})(5.97 \times 10^{24})(86400)^2}{4\pi^2}\right)^{1/3} \approx \) \( 4.22 \times 10^7 \;\text{m} \)

Example 3: Comparing Two Orbits

Satellite A orbits at radius \( r \). Satellite B orbits at radius \( 4r \). How do their speeds and periods compare?

Common Mistakes to Avoid

Using altitude instead of radius
Plugging \( h \) directly into \( v = \sqrt{GM/r} \). Remember: \( r = R + h \).
Thinking mass matters
Satellite mass \( m \) cancels out. A bowling ball and a feather orbit at the same speed at the same radius.
Faster means higher
Higher orbits are actually slower. Increasing \( r \) decreases \( v \). Do not confuse with launching speed.

Summary of Orbital Relationships

Quantity Formula Depends on \( r \)
Orbital Speed \( v \) \( \sqrt{GM/r} \) \( v \propto r^{-1/2} \)
Period \( T \) \( 2\pi\sqrt{r^3/(GM)} \) \( T \propto r^{3/2} \)
Centripetal Accel. \( a_c \) \( GM/r^2 \) \( a_c \propto r^{-2} \)
Kinetic Energy \( K \) \( GMm/(2r) \) \( K \propto r^{-1} \)
Total Energy \( E \) \( -GMm/(2r) \) \( E \propto -r^{-1} \)

Question 1 of 5

A satellite orbits Earth in a circular orbit. If the orbital radius is doubled, what happens to the orbital speed?

Question 2 of 5

The mass of a satellite is doubled while keeping the same orbital radius. What happens to its orbital period?

Question 3 of 5

A satellite orbits a planet at an altitude of 600 km. The planet has a radius of 3,400 km. What orbital radius should be used in calculations?

Question 4 of 5

Astronauts on the ISS experience apparent weightlessness because:

Question 5 of 5

Satellite X has orbital radius \( r \) and period \( T \). Satellite Y orbits the same planet at radius \( 2r \). What is the period of Satellite Y?