Question

The height of the geostationary satellite is-
1. 35930 km
2. 45930 km
3. 55930 km
4. None of these

Answer 1

Correct Answer - Option 1 : 35930 km

CONCEPT:

• A geostationary satellite is an earth-orbiting satellite that revolves around the earth at the same time as the earth rotates along its axis. Due to the same time of the revolution, it appears stationary from the Earth, and hence it is called geostationary. 
• The time period of the satellite: It is the time taken by the satellite to complete one revolution around the Earth.
  • Consider a satellite orbiting the earth at a height h from the surface of the earth of radius R.

  • The circumference of orbit of satellite = 2π(R+h)

• The orbital velocity of the satellite at a height h is given by,

​\(⇒ v_0 =\sqrt{\frac{GM}{R+h}}\)

Where M is the mass of the Earth.

The time period of the satellite is given by:

\(⇒ T = \frac{circumference}{orbital \: velocity} = \frac{2\pi(R+h)}{\sqrt{\frac{GM}{R+h}}} = 2\pi\sqrt{\frac{(R+h)^3}{GM}}\)

CALCULATION:

• The time period of a geostationary satellite is 24 hours.

∴ T = 24 h = 86400 s

Taking into consideration that,

Radius of earth = 6400 km = 6400000 m

Mass of earth = 5.97 × 1024 kg

\(⇒ T = 2\pi\sqrt{\frac{(R+h)^3}{GM}}\)

\(⇒ 86400 = 2\pi\sqrt{\frac{(R+h)^3}{(6.67 × 10^{-11})(5.97 × 10^{24})}}\)

⇒ (R + h)3 = 7.53 × 1022

⇒ R + h = 42226910

⇒ h = 42226910 - R = 42226910 - 6400000 

⇒ h = 35826910 m ≈ 35930 km

Thus, the geostationary satellite is placed at about 35930 km from the surface of Earth.