Solved Examples: Calculating Orbital Period of a Satellite

📚 Quick Study Guide

• 🌍 Kepler's Third Law: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. For a satellite orbiting a central body: $T^2 = \frac{4\pi^2}{GM}a^3$, where $T$ is the period, $G$ is the gravitational constant, $M$ is the mass of the central body, and $a$ is the semi-major axis.
• 🍎 Simplified Circular Orbit: If the orbit is circular, the semi-major axis $a$ becomes the radius $r$ of the orbit: $T = \sqrt{\frac{4\pi^2 r^3}{GM}}$
• 📏 Orbital Radius Calculation: The orbital radius $r$ is the sum of the central body's radius and the satellite's altitude above the surface: $r = R + h$, where $R$ is the radius of the central body and $h$ is the altitude.
• ⚖️ Gravitational Constant (G): $G = 6.674 × 10^{-11} N(m/kg)^2$
• 🛰️ Units: Ensure all values are in SI units: meters (m), kilograms (kg), and seconds (s).

🧪 Practice Quiz

1. A satellite orbits Earth at an altitude of 500 km. What additional information is needed to calculate its orbital period?
   1. (A) The satellite's mass
   2. (B) The satellite's velocity
   3. (C) Earth's mass
   4. (D) The satellite's surface area
2. A satellite orbits a planet with a mass of $6.0 × 10^{24}$ kg at a distance of $7.0 × 10^6$ m from the planet's center. What is the closest to the orbital period of the satellite?
   1. (A) 5000 seconds
   2. (B) 6000 seconds
   3. (C) 7000 seconds
   4. (D) 8000 seconds
3. If the orbital radius of a satellite is doubled, what happens to its orbital period?
   1. (A) It is halved
   2. (B) It is doubled
   3. (C) It increases by a factor of $\sqrt{2}$
   4. (D) It increases by a factor of $2\sqrt{2}$
4. A satellite's orbital period around Earth is 90 minutes. If the satellite is moved to an orbit with twice the radius, what will the new orbital period be closest to?
   1. (A) 127 minutes
   2. (B) 255 minutes
   3. (C) 360 minutes
   4. (D) 510 minutes
5. What is the relationship between a planet's mass and the orbital period of a satellite orbiting it?
   1. (A) As the planet's mass increases, the orbital period increases
   2. (B) As the planet's mass increases, the orbital period decreases
   3. (C) The planet's mass has no effect on the orbital period
   4. (D) The relationship depends on the satellite's mass
6. A satellite orbits Earth at a radius of $8.0 × 10^6$ m. If Earth's mass is $6.0 × 10^{24}$ kg, approximately what is the satellite's orbital speed?
   1. (A) 3500 m/s
   2. (B) 5600 m/s
   3. (C) 7100 m/s
   4. (D) 9200 m/s
7. Two satellites orbit the same planet. Satellite A has a smaller orbital radius than Satellite B. Which satellite has a shorter orbital period?
   1. (A) Satellite A
   2. (B) Satellite B
   3. (C) They have the same orbital period
   4. (D) Cannot be determined without more information