Section 2.4: Satellite Orbits
Satellites move in orbits due to the balance between gravitational pull and their inertia. Understanding orbital mechanics allows calculation of velocity, period, and altitude for stable orbits.
\( v = \sqrt{\frac{GM}{r}} \), where \( r \) is distance from center of Earth (or planet).
\( T = 2\pi \sqrt{\frac{r^3}{GM}} \)
Example 1
Find the velocity of a satellite orbiting 500 km above Earth's surface.
Earth radius \( R = 6370 \text{ km} \), altitude \( h = 500 \text{ km} \), so \( r = R+h = 6870 \text{ km} = 6.87\times10^6 \text{ m} \)
\( v = \sqrt{\frac{GM}{r}} = \sqrt{\frac{6.674\times10^{-11}*5.97\times10^{24}}{6.87\times10^6}} \approx 7.6 \text{ km/s} \)
Example 2
Compute the orbital period of the same satellite.
\( T = 2\pi \sqrt{\frac{r^3}{GM}} = 2\pi \sqrt{\frac{(6.87\times10^6)^3}{6.674\times10^{-11}*5.97\times10^{24}}} \approx 5670 \text{ s} \approx 1.57 \text{ hours} \)
Practice Problems
- Find the orbital velocity of a satellite at 1000 km above Earth.
- Determine the period of a geostationary satellite.
- Calculate the altitude for a satellite with period 90 minutes.
- Find the speed of a satellite orbiting Mars at 4000 km from its center (Mars mass \(6.42×10^{23} kg\)).
- Compute centripetal acceleration of a satellite at 800 km above Earth.
- Find radius of circular orbit if a satellite’s orbital speed is 3 km/s.
- Calculate gravitational force on a 500 kg satellite at 500 km altitude.
- A satellite completes an orbit in 2 hours. Find orbital radius.
- Determine the escape velocity at 1000 km above Earth.
- Find velocity and period for a satellite in low Earth orbit at 300 km.