Section 3.7: Orbital Motion
Objects such as planets, satellites, and moons move in orbits under the influence of gravitational forces. Orbital motion can be circular or elliptical, with the central mass at one focus for elliptical orbits.
Gravitational Force Provides Centripetal Acceleration:
\[
F_g = \frac{G M m}{r^2} = m \frac{v^2}{r} \implies v = \sqrt{\frac{G M}{r}}
\]
where:
- \(M\) = mass of central body (e.g., Earth)
- \(m\) = mass of orbiting body
- \(r\) = orbital radius
- \(v\) = orbital speed
Orbital Period:
\[
T = \frac{2\pi r}{v} = 2 \pi \sqrt{\frac{r^3}{G M}}
\]
The period is independent of the orbiting mass.
Example 1 — Satellite Speed
A satellite orbits Earth at an altitude where the orbital radius is \(7 \times 10^6\) m. Find its orbital speed.
\(v = \sqrt{\frac{G M}{r}} = \sqrt{\frac{6.674\times10^{-11} \cdot 5.97\times10^{24}}{7\times10^6}} \approx 7.35 \times 10^3 \text{ m/s}\)
Practice Problems
- Find the orbital speed of a satellite 300 km above Earth’s surface.
- Determine the period of a satellite orbiting at 4 × 10^7 m from Earth’s center.
- A planet orbits a star at 1 × 10^11 m. Compute the period given the star’s mass is \(2 \times 10^{30}\) kg.
- Calculate the speed of the Moon in its circular orbit around Earth.
- A satellite must orbit at 1 × 10^6 m. Determine the centripetal acceleration it experiences.