Problem 21 - University Physics with Modern Physics

Question

Two satellites are in circular orbits around a planet that has radius \(9.00 \times 10^{6} \mathrm{m}\) . One satellite has mass 68.0 \(\mathrm{kg}\) , orbital radius \(5.00 \times 10^{7} \mathrm{m},\) and orbital speed 4800 \(\mathrm{m} / \mathrm{s} .\) The second satellite has mass 84.0 \(\mathrm{kg}\) and orbital radius \(3.00 \times 10^{7} \mathrm{m} .\) What is the orbital speed of this second satellite?

Step-by-Step Solution

Answer

The second satellite's orbital speed is approximately 6191 m/s.

Step 1: Understand the Orbital Speed Formula

The formula for the orbital speed of a satellite in a circular orbit around a planet is \( v = \sqrt{\frac{GM}{r}} \), where \( v \) is the orbital speed, \( G \) is the gravitational constant \( (6.674 \times 10^{-11} \, \text{N}\,\text{m}^2/\text{kg}^2) \), \( M \) is the mass of the planet, and \( r \) is the orbital radius.

Step 2: Calculate the Gravitational Force or GM

We know the orbital speed formula depends on \( GM \). Since we have the velocity and radius for the first satellite, we can rearrange the speed formula: \( GM = v^2 r \). Using the first satellite's speed \( v_1 = 4800 \, \text{m/s} \) and radius \( r_1 = 5.00 \times 10^7 \, \text{m} \), calculate: \[ GM = (4800)^2 \times 5.00 \times 10^7 \].

Step 3: Compute GM Using First Satellite's Data

Calculate the gravitational parameter \( GM \) using the values from the first satellite: \[ GM = 4800^2 \times 5.00 \times 10^7 = 1.152 \times 10^{15} \, \text{m}^3/\text{s}^2 \].

Step 4: Solve for Second Satellite's Orbital Speed

Now, use the known \( GM \) to find the speed of the second satellite. Use the formula \( v = \sqrt{\frac{GM}{r_2}} \), where \( r_2 = 3.00 \times 10^7 \, \text{m} \): \[ v_2 = \sqrt{\frac{1.152 \times 10^{15}}{3.00 \times 10^7}} \].

Step 5: Calculate Final Orbital Speed for Second Satellite

Substitute the values to find: \[ v_2 = \sqrt{3.84 \times 10^7} \approx 6191 \, \text{m/s} \]. Thus, the second satellite's orbital speed is approximately 6191 m/s.

Key Concepts

Circular Orbit

A circular orbit is an idealized concept where a satellite orbits a planet in a path that forms a perfect circle with the planet at its center. This model is an excellent approximation for understanding satellite dynamics and simplifies the calculations of orbital speed. In a circular orbit, the gravitational force acting as the centripetal force keeps the satellite moving at a constant speed.
The key traits of circular orbits include:

• Uniform speed - As the satellite moves in a circle, its speed remains constant.
• Fixed orbital radius - The distance from the satellite to the center of the planet is always the same.
• Predictable motion - The satellite's motion is highly predictable, simplifying the calculation of various orbital parameters.

Studying circular orbits allows us to focus on other forces and concepts, like gravity, without the added complexity of changing speeds or orbital shapes seen in elliptical orbits.

Gravitational Constant

The gravitational constant, denoted by \( G \), is a fundamental constant in physics that appears in Newton's law of universal gravitation. Its value is \( 6.674 \times 10^{-11} \, \text{N}\,\text{m}^2/\text{kg}^2 \). This constant helps calculate the gravitational force of attraction between two masses, like a planet and a satellite.
In the context of orbital speed, \( G \) is used to determine the speed at which a satellite must travel to maintain a stable circular orbit around a planet. The formula involves the gravitational parameter \( GM \), where \( M \) is the mass of the planet. By determining \( GM \), we can simplify orbital calculations without needing to know \( M \) directly.
Importantly, \( G \) is the same everywhere in the universe, making it a cornerstone for tasks involving mass and gravitational forces, such as predicting satellite behavior or planning space missions.

Satellite Dynamics

Satellite dynamics deals with understanding and predicting the behavior and movement of satellites in space. This includes analyzing velocities, accelerations, and forces acting on a satellite.
In a circular orbit, the satellite experiences a constant centripetal force, provided by gravity, that keeps it in orbit. The dynamics of a satellite are influenced by several factors:

• Orbital speed changes - A satellite moving to a lower orbit needs to increase its speed, while moving to a higher orbit requires slowing down.
• Atmospheric drag - Though negligible at high altitudes, this can influence low Earth orbit satellites.
• Gravitational perturbations - The non-uniform distribution of mass in the planet can affect the orbit.

Understanding satellite dynamics is crucial for satellite operations, ensuring the correct alignment of orbits for communication, weather monitoring, and GPS systems. Mastery of these concepts enables us to maintain satellite constellations efficiently and effectively.

Orbital Radius

The orbital radius is the fixed distance from the center of the planet to the satellite in a circular orbit. It is a critical factor in determining both the orbital speed and gravitational forces experienced by a satellite.
Orbital radius is directly tied to the satellite's speed by the formula \( v = \sqrt{\frac{GM}{r}} \), where \( r \) is the orbital radius. A larger radius results in a slower speed, since the satellite has a longer path to cover at the same gravitational influence. Conversely, a smaller orbit requires a higher speed.
Key considerations regarding orbital radius include:

• The influence on coverage area - Higher orbits allow for greater surface coverage.
• Energy requirements - Higher altitudes need more energy to reach but often reduce speed loss due to less atmospheric resistance.
• Stability - The optimal orbital radius depends on the mission and required stability of the orbit.

Understanding and choosing the correct orbital radius ensures that satellites fulfill their intended purposes, whether it be communication, monitoring, or scientific research.