Question

The mean radius of earth is $R$, its angular speed on its own axis is $\omega$ and the acceleration due to gravity at earth's surface is $g$. What will be the radius of the orbit of a geostationary satellite ?

• A. $(R^2 g / {\omega}^2 ) ^{1/3}$
• B. $(R g / {\omega}^2 ) ^{1/3}$
• C. $(R^2 {\omega}^2 /g ) ^{1/3}$
• D. $(R^2 g / \omega) ^{1/3}$

Answer 1

The Correct Option is A

To find the radius of the orbit of a geostationary satellite, we need to use the concept of gravitational force providing the necessary centripetal force required to keep the satellite in its orbit. A geostationary satellite orbits the Earth such that it remains fixed relative to a point on the equator. This means it must have the same angular speed as the Earth's rotation.

The necessary formulas to solve this problem are:

1. The gravitational force acting on the satellite provides the necessary centripetal force: F_{\text{gravity}} = \frac{G M m}{r^2}
2. The centripetal force required for circular motion is: F_{\text{centripetal}} = \frac{m v^2}{r} = m \omega^2 r

Where: G is the gravitational constant, M is the mass of the Earth, r is the radius of the orbit of the satellite, m is the mass of the satellite, and v is the velocity of the satellite.

For a geostationary satellite, equate the gravitational force to the centripetal force:

\[ \frac{G M m}{r^2} = m \omega^2 r \]

Solving for r:

\[ r^3 = \frac{G M}{\omega^2} \]

To express this in terms of g (acceleration due to gravity at Earth's surface), use the relation: g = \frac{G M}{R^2}, where R is the Earth's radius.

Substitute G M = g R^2 into the equation for r^3:

\[ r^3 = \frac{g R^2}{\omega^2} \]

Thus, the radius of the orbit becomes:

\[ r = \left( \frac{g R^2}{\omega^2} \right)^{1/3} \]

This matches the correct answer, so we can conclude:

The radius of the orbit of a geostationary satellite is (R^2 g / {\omega}^2 )^{1/3}.

Other options can be ruled out because they do not align with the derived formula, ensuring no contradictions to the law of gravitation and the conditions for geostationary orbit.