Question

Develop an algebraic relationship that describes a satellite moving in a circular orbit around a planet if the speed of the satellite is v, the mass of the planet is M, the mass of the satellite is m, the radius of the planet is R, and the altitude of the orbit is h. (You may include the appropriate constants as needed.) Assume that the frictional force is negligible. What will eventually happen to the satellite if the frictional (drag) force is not negligible? What could be done to compensate for a non-negligible drag force that would allow the satellite to maintain its orbit?

Answer 1

Step-by-step explanation:

For this problem we must use Newton's second law where force is gravitational attraction

F = m a

Since movement is circular, acceleration is centripetal.

a = v2 / r

Let's replace

G m M / r² = m v² / r

G M r = v²

The distance r is from the center of the planet

r = R + h

v = √ GM / (R + h)

If the friction force is not negligible

F - fr = m a

Where the friction force must have some functional relationship, for example

Fr = b v + c v² +…

Suppose we are high enough for the linear term to derive the force of friction

G m M / r - (m b v + m c v2) = m v2

G M / r - b v = v²

We see that the solution of the problem gives lower speeds and that change over time.

To compensate for this friction force, the motors should be intermittently suspended to recover speed.