Given below are two statements: Statement I: If 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 E. In the light of the above statements, choose the most appropriate answer from the options given below
Statement I is correct but Statement II is incorrect
Statement I is incorrect but Statement II is correct
Show Solution
The Correct Option isB
Solution and Explanation
To evaluate the statements regarding the energy of a satellite orbiting Earth, we need to understand the relationship between the total energy, potential energy, and kinetic energy of a satellite.
Conceptual Explanation:
Total Energy (\(E\)): The total mechanical energy of a satellite in orbit is the sum of its kinetic energy (\(K\)) and potential energy (\(U\)). The total energy for a bound satellite is always negative.
Potential Energy (\(U\)): The gravitational potential energy of a satellite at a distance \(r\) from the center of the Earth is given by \(U = -\frac{GMm}{r}\), where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(m\) is the mass of the satellite.
Kinetic Energy (\(K\)): The kinetic energy of a satellite in orbit is given by \(K = \frac{1}{2}mv^2\).
Relation of Energies for Satellite:
The total energy \(E\) of a satellite in a stable orbit is given by:
\[
E = -\frac{GMm}{2r}
\]
The potential energy \(U\) is:
\[
U = -\frac{GMm}{r} = 2E
\]
Thus, the potential energy is actually twice the total energy (magnitude-wise) but negative.
The kinetic energy \(K\) equals:
\[
K = \frac{GMm}{2r} = -E
\]
Hence, the kinetic energy equals the negative of the total energy.
Analysis of Statements:
Statement I: "If \(E\) be the total energy of a satellite moving around the earth, then its potential energy will be \(\frac{E}{2}\)." Analysis: As derived above, the potential energy \(U\) is not \(\frac{E}{2}\); rather it is \(2E\). Therefore, this statement is incorrect.
Statement II: "The kinetic energy of a satellite revolving in an orbit is equal to the half the magnitude of total energy \(E\)." Analysis: The kinetic energy \(K\) is \(-E\), which is not half the magnitude of \(E\); it is actually equal in magnitude to the total energy. Thus, this statement is incorrect.
Conclusion: Based on the analysis, both statements are incorrect. Therefore, the answer is:
