JEE Main 2023 — Gravitation Question with Solution

A satellite in orbit around a planet is subject to two main forces: gravitational force, which is trying to pull it towards the planet, and its own kinetic energy or inertia, which is trying to keep it moving in a straight line. The balance of these two forces results in the satellite moving in a circular or elliptical orbit.

The gravitational potential energy ($U$) of the satellite is given by the formula:

$U=-\frac{GMm}{R}$

where $G$ is the gravitational constant, $M$ is the mass of the Earth, $m$ is the mass of the satellite, and $R$ is the radius of the orbit. The negative sign indicates that work would have to be done to remove the satellite from the Earth's gravitational influence.

The kinetic energy ($K$) of the satellite is given by the formula:

$K=\frac{GMm}{2R}$

This is obtained from the fact that for a satellite in stable orbit, the gravitational force must be equal to the centripetal force required to keep the satellite moving in a circle. From this, we can derive an expression for the velocity of the satellite, and hence its kinetic energy.

The total mechanical energy ($E$) of the satellite, which is the sum of its kinetic and potential energy, is therefore:

$E=K+U=\frac{GMm}{2R}-\frac{GMm}{R}=-\frac{GMm}{2R}$

So the potential energy $U$ is $-2E$, and the kinetic energy $K$ is $-E$. Thus, the statement "If $E$ be the total energy of a satellite moving around the earth, then its potential energy will be $2E$" is correct, and the statement "The kinetic energy of a satellite revolving in an orbit is equal to the half the magnitude of total energy $E$" is incorrect.