Problem 10 - Thomas Calculus

Question

Finding satellite speed A satellite moves around Earth in a circular orbit. Express the satellite's speed as a function of the orbit's radius.

Step-by-Step Solution

Answer

The satellite's speed is \( v = \sqrt{\frac{G M}{r}} \).

Step 1: Understanding Gravitational Force

The gravitational force acting between the Earth and the satellite keeps the satellite in orbit. This force can be expressed by \[ F = \frac{G M m}{r^2} \]where \( F \) is the gravitational force, \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the radius of the orbit.

Step 2: Relating Gravitational Force to Centripetal Force

The gravitational force provides the centripetal force needed to keep the satellite in motion along its circular path. Thus, we equate the gravitational force to the centripetal force:\[ \frac{G M m}{r^2} = \frac{m v^2}{r} \]where \( v \) is the speed of the satellite. Simplifying, we have an equation for speed.

Step 3: Solving for the Satellite's Speed

To find the speed of the satellite, solve the equation from the previous step for \( v \):\[ v^2 = \frac{G M}{r} \]Taking the square root of both sides gives:\[ v = \sqrt{\frac{G M}{r}} \]

Step 4: Expressing Speed as a Function of Radius

The final expression \( v = \sqrt{\frac{G M}{r}} \) shows that the satellite's speed is a function of the orbit's radius. As the radius increases, the speed decreases according to this inverse square root relationship.

Key Concepts

Gravitational Force

Gravitational force is a fundamental concept in physics that describes the attraction between two masses. In the context of a satellite orbiting Earth, the gravitational force is what keeps the satellite in its orbit. This force is calculated using the equation

• \( F = \frac{G M m}{r^2} \)
• where \( F \) is the gravitational force,
• \( G \) is the gravitational constant,
• \( M \) is the mass of Earth,
• \( m \) is the mass of the satellite,
• \( r \) is the radius of the orbit.

This equation tells us that gravitational force diminishes with the square of the distance. That means if you double the distance \( r \), the gravitational force becomes four times weaker.
Gravitational force ensures the satellite doesn't drift away into space. It acts as a binding force balancing other motions.

Centripetal Force

Centripetal force is the inward force required for an object to move in a circular path. When a satellite orbits Earth, this centripetal force is crucial for its circular motion. Interestingly, the centripetal force for a satellite is provided by the gravitational force itself.
This relation can be mathematically expressed as

• \( \frac{G M m}{r^2} = \frac{m v^2}{r} \)
• where \( v \) is the speed of the satellite.

The above equation shows that the gravitational force acting on the satellite is equal to the centripetal force necessary to keep it moving in a circle.
This balance ensures that the satellite remains stable in orbit without spiraling outwards or crashing towards Earth.

Circular Orbit

A circular orbit refers to the path taken by a satellite as it travels around Earth at a constant distance. This motion is unique because it is uniform, meaning the speed doesn’t change as the satellite follows this circular path. A fixed radius for the orbit ensures that this circular path remains steady.
The speed of the satellite in a circular orbit is calculated from the gravitational and centripetal force relationship, arriving at

• \( v = \sqrt{\frac{G M}{r}} \)

where \( v \) is the speed of the satellite and \( r \) is the radius of its circular path.
In this equation, as the radius \( r \) increases, the speed \( v \) decreases, indicating an inverse relationship. This ensures the satellite maintains its path with minimal energy input, given the consistent speed and path defined by circular orbits.