Physics / Unit 4: Circular Motion and Gravity
4.6 Orbits
Notes
A satellite in a circular orbit is constantly falling toward the planet while moving sideways fast enough to keep missing it. Gravity provides the centripetal force.
- Set $\displaystyle \frac{GMm}{r^2} = \frac{mv^2}{r}$ to get the orbital speed $\displaystyle v = \sqrt{\frac{GM}{r}}$.
- The orbital speed does not depend on the satellite's mass.
- Satellites farther out move slower and have longer periods (Kepler's third law: $T^2 \propto r^3$).
- Example: a satellite about 300 km above Earth moves at about $7.7 \text{ km/s}$ and circles Earth in about 90 minutes.
Practice
What provides the centripetal force that keeps the Moon in orbit around Earth?
Gravity pulls the Moon toward Earth's center.
Two satellites orbit Earth at the same height. One has twice the mass. How do their speeds compare?
$\displaystyle v = \sqrt{\frac{GM}{r}}$ has no satellite mass in it.
Compared with a low orbit, a satellite in a high orbit has:
$v$ falls as $r$ grows, and the path is longer, so the period increases.
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