What is the required altitude for geosynchronous satellites above the Earth?

[bfr]

Homework Statement

Using a development similar to Kepler's law of periods for planets orbiting the Sun, find the required altitude of geosynchronous satellites above the earth.

Homework Equations

(R^3)/T^2=(G*M)/(4pi^2)

The Attempt at a Solution

I'm honestly not even sure where to start. The correct answer is supposed to be 3.59*10^7m above the surface.

 

[Kurdt]

Well you know the period and you want to find the radius. Just rearrange the equation and plug the numbers in.

 

[Moetasim]

I can provide a response to this content by first explaining what geosynchronous satellites are and their significance in space technology. Geosynchronous satellites are man-made objects that orbit the Earth at a specific altitude, allowing them to remain stationary above a specific location on the Earth's surface. This is achieved by placing the satellite in an orbit with an orbital period that matches the rotation period of the Earth, which is approximately 24 hours.

To find the required altitude of geosynchronous satellites above the Earth, we can use Kepler's third law of planetary motion, which states that the square of a planet's orbital period is directly proportional to the cube of its average distance from the Sun. This can be expressed mathematically as (R^3)/T^2=(G*M)/(4pi^2), where R is the average distance from the Sun, T is the orbital period, G is the gravitational constant, and M is the mass of the Sun.

In the case of geosynchronous satellites, we can use this same principle by considering the Earth as the "Sun" and the satellite as the "planet". We can rearrange the equation to solve for the required altitude (R) by substituting the orbital period of 24 hours and the mass of the Earth for M. This gives us R= (4pi^2* (3.986*10^14 m^3/s^2)* (24 hours)^2)^(1/3) = 3.59*10^7 m above the surface of the Earth.

In summary, the required altitude of geosynchronous satellites above the Earth can be calculated using Kepler's third law and considering the Earth as the "Sun" and the satellite as the "planet". This altitude is approximately 3.59*10^7 m above the Earth's surface.