Question

If R is radius of the earth then approximate altitude of a geosynchronous satellite is
(1) 4R
(2) 6R
(3) 8R
(4) 9R

Answer 1

Hint: The binding energy of a satellite is defined as the minimum amount of energy required for a satellite to escape from the earth’s gravitational influence. To find the solution of the given question, Kepler’s third law which is a relation between the period of an orbit to the radius of an orbit.

Formula used:
$T = 2\pi \sqrt {\dfrac{{{r^3}}}{{GM}}} $

Complete step by step answer:
A geosynchronous satellite is known as a satellite which is placed in a geosynchronous orbit, and whose orbital period is the same as that of the earth’s rotation period.
Height of geostationary satellite
According to Kepler’s third law,
 $T = 2\pi \sqrt {\dfrac{{{r^3}}}{{GM}}} $
Where ‘r’ denotes the semi – major axis for elliptical orbits.
Here, it is given that R is the radius of the earth.
$ \Rightarrow 2\pi \sqrt {{{\dfrac{{\left( {R + h} \right)}}{{GM}}}^3}} = 24hr$
Now, substituting the value of ‘G’ and ‘M’ we get,
R + h = r = 42000km = 7R
Thus, the height of the geostationary satellite from the surface of the earth is given by,
h = 6R = 36000km
Thus, the altitude of a geosynchronous satellite is 6R.
Hence, option (2) is the correct answer.

Additional Information:
A geosynchronous satellite which is placed at a fixed angle to the equatorial plane is known as a geostationary satellite. A geostationary satellite is placed in a geostationary orbit. A geostationary satellite is used for the purpose of weather reports about a particular region, for weather forecasting and for terrestrial reports of a geographical area.

Note:
These concepts are based on Kepler’s law of planetary motion. Kepler’s first law states that all the planets revolve around the sun in elliptical orbits with the sun at one focus. Kepler’s second law states that the line which joins the earth or any other planet to the sun sweeps out equal areas in equal intervals of time. Kepler’s third law states that the square of the time period of the planet is directly proportional to the cube of the semi - major axis of its orbit.