The distance of a geo-stationary satellite from the center...

The distance of a geo-stationary satellite from the center of the earth (Radius $\mathrm{R}=6400 \mathrm{~km}$ ) is nearest to
(A) $5 \mathrm{R}$
(B) $7 \mathrm{R}$
(C) $10 \mathrm{R}$
(D) $18 \mathrm{R}$

Key Concepts

Kepler's Third Law

Kepler's Third Law provides a relationship between the orbital period of a satellite and the semi-major axis of its orbit. In the context of circular orbits, it is used to derive the distance from the center of the Earth at which a satellite must orbit to maintain a particular period, such as matching the Earth’s rotation.

Geostationary Satellite

A geostationary satellite is one that remains fixed relative to a point on the Earth's surface by orbiting with the same rotational period as the Earth. This requires it to have a specific orbital radius where its angular velocity matches the Earth's rotational speed.

Orbital Mechanics

Orbital mechanics is the study of the motions of objects in space under the influence of gravitational forces. It involves applying physical laws to understand and calculate the trajectories, speeds, and positions of satellites and other bodies in orbit.

Force Balance in Circular Motion

For a satellite in a stable circular orbit, the gravitational force acting on it must equal the necessary centripetal force to keep it in its circular path. This balance is critical in determining the orbital radius required for a geostationary orbit, linking gravitational pull with rotational motion.

Application of Mathematical Modeling in Orbital Calculations

Mathematical modeling in orbital calculations involves using equations derived from gravitational theory and circular motion dynamics to compute key parameters such as orbital radius and period. This process underpins the determination of distances in satellite orbits based on known values like the Earth's radius and rotation period.

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