If the height of a satellite from the earth is negligible in...

If the height of a satellite from the earth is negligible in comparison of the radius of the earth $\mathrm{R}$, the orbital velocity of the satellite is
(A) $\mathrm{gR}$
(B) $(\mathrm{gR} / 2)$
(C) $\sqrt{(g} / \mathrm{R})$
(D) $\sqrt{(g R)}$

Key Concepts

Approximations in Low Earth Orbits

When the satellite's height is negligible compared to the Earth's radius, the gravitational acceleration does not vary significantly with altitude. This approximation allows one to replace the varying gravitational force with a constant value (g), simplifying the derivation of orbital velocity to a form that depends only on Earth's radius (R) and the gravitational acceleration (g).

Centripetal Force

Centripetal force is the necessary inward force that keeps an object moving in a circle. In the context of orbital motion, the gravitational force acts as the centripetal force, maintaining the satellite in a circular orbit by constantly redirecting its velocity toward the center of the Earth.

Gravitational Force

Gravitational force is the attractive force between two masses as described by Newton's law of universal gravitation. For a satellite orbiting Earth, this force can be approximated using the acceleration due to gravity (g) when the satellite’s altitude is negligible compared to Earth’s radius, simplifying the force to mg.

Orbital Velocity

Orbital velocity refers to the constant speed required for an object to maintain a stable circular orbit around a central body. In a circular orbit, the gravitational force provides the necessary centripetal force to keep the object moving along its curved path. By equating the gravitational pull to the centripetal force requirement, one can derive the formula for orbital velocity.

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