Question 98: Gravitation

Question

Given below are two statements:
Statement I: If $\mathrm{E}$ be the total energy of a satellite moving around the earth, then its potential energy will be $\frac{E}{2}$.
Statement II: The kinetic energy of a satellite revolving in an orbit is equal to the half the magnitude of total energy $\mathrm{E}$.
In the light of the above statements, choose the most appropriate answer from the options given below

Accepted Answer

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.

Answer

Both Statement I and Statement II are incorrect

Answer

Statement I is incorrect but Statement II is correct

Answer

Statement I is correct but Statement II is incorrect

Answer

Both Statement I and Statement II are correct

Gravitation

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