Motion of Orbiting Satellites
The Moon is falling — and always missing
- The Moon is pulled toward Earth by gravity, so it is constantly "falling".
- Yet it never lands — because it also moves sideways fast enough to keep missing.
- An orbit is exactly this: falling toward a planet while sailing past it forever.
- Gravity is not fighting the orbit; gravity is what bends the path into a circle.
Gravity is the centripetal force
- For a circular orbit, gravity supplies the whole centripetal force.
- Set them equal: $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$.
- The satellite's mass $m$ cancels — orbits don't care how heavy the satellite is.
- This single equation controls every circular orbit.
Trace the orbit
Change the orbit radius and see how the geometry of the circular path changes.
What provides the centripetal force that keeps a satellite in a circular orbit?
Gravity is the centripetal force: $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$.
In $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$, the satellite's ____ cancels from both sides.
The satellite mass $m$ cancels, so orbital speed does not depend on it.
Orbital speed and radius
- Solving for speed: $v = \sqrt{\dfrac{GM}{r}}$.
- A higher orbit (larger $r$) means a slower speed — counter-intuitive but true.
- Low satellites (like the ISS) whip around in about $90$ minutes; the far-off Moon takes a month.
- Every orbit radius has exactly one speed that keeps it circular.
At an orbit where $GM/r = 25$ (scaled units), what is the orbital speed $v = \sqrt{GM/r}$?
$v = \sqrt{GM/r} = \sqrt{25} = 5$.
A satellite is moved to a higher circular orbit. Its orbital speed:
$v = \sqrt{GM/r}$: a larger $r$ gives a smaller $v$ — higher orbits are slower.
Falling forever
- A satellite is in constant free fall — its only force is gravity, pulling it inward.
- It feels "weightless" not because gravity is gone, but because it is falling with everything around it.
- Give it more sideways speed than the orbital value and it climbs to a higher orbit.
- Too little, and it spirals back down — orbits are a delicate balance of speed and pull.
A satellite in orbit is in continuous free fall, which is why astronauts feel weightless.
They fall together with everything around them, so they float — gravity is still present.
Select all true statements about a satellite in a circular orbit.
Gravity is the centripetal force, the satellite free-falls, and higher orbits are slower. Gravity is very much present.
A satellite in orbit is not beyond gravity — gravity is the very force holding it in orbit. Astronauts float because they are in continuous free fall, not because there is "no gravity" up there. At the ISS, gravity is still nearly as strong as on the ground.
At a certain orbit, $\dfrac{GM}{r} = 25\ \tfrac{\text{m}^2}{\text{s}^2}$ (in scaled units).
- Orbital speed $v = \sqrt{\dfrac{GM}{r}} = \sqrt{25} = 5$ (speed units).
Move to a higher orbit where $GM/r = 16$, and the speed drops to $\sqrt{16} = 4$ — farther out is slower.
For a circular orbit, gravity is the centripetal force: $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$, giving orbital speed $v = \sqrt{GM/r}$. Higher orbits are slower. A satellite is in constant free fall — "weightlessness" is falling, not the absence of gravity.