Kinetic Energy of a Satellite
Kinetic energy is the energy possessed by the body in motion (whether translational or rotational or a combination of both) in the case of the satellite orbiting around the earth. The satellite makes a pure rotational motion. In a pure rotation, the kinetic energy of the satellite is given by,
K = 1/2 mv2
Since, v = rω, put in above equation, we get
\(\begin{array}{l}K=\frac{1}{2}m{{r}^{2}}{{\omega }^{2}}\end{array} \)
Where, the angular velocity of a satellite is related to the time period of a satellite by a formula, \(\begin{array}{l}\omega =\frac{2\pi }{T}\end{array} \)
Substituting in above equation, we get
\(\begin{array}{l}K=\frac{1}{2}m{{r}^{2}}{{\left( \frac{2\pi }{T} \right)}^{2}}\end{array} \)
From Kepler’s 3rd law, we know the time period of a satellite,
\(\begin{array}{l}{{T}^{2}}=\frac{4{{\pi }^{2}}}{GM}{{\left( r \right)}^{3}}\end{array} \)
By substituting T in the above formula, we get
\(\begin{array}{l}K=\frac{GMm}{2r}\end{array} \)
Which is the formula for Kinetic Energy of a Satellite. Kinetic energy can never be negative for any force.
Potential Energy of a Satellite
Potential energy is the energy possessed by the body in a particular position. Potential energy changes when the position of the body changes. To study potential energy, we require two bodies, one is source mass (M) which provides the gravitational force, and the other is test mass (m) which experiences the gravitational force by the source mass. In our case, the satellite is the test mass, and the earth is the source mass. The potential energy possessed by the satellite at a distance ‘r’ from the centre of the earth is given by,
\(\begin{array}{l}U=\frac{-GMm}{r}\end{array} \)
Which is the formula for Potential Energy of a Satellite. Potential energy always be negative for gravitational force.
Total Energy of a Satellite
The total energy of the satellite is the sum of all energies possessed by the satellite in orbit around the earth. As only the mechanical motion of the satellite is considered, it has only kinetic and potential energies.
Total energy of the satellite = kinetic energy of the satellite + potential energy of the satellite
\(\begin{array}{l}E=K+U\end{array} \)
\(\begin{array}{l}E=\frac{GMm}{2r}+\left( \frac{-GMm}{r} \right)\end{array} \)
\(\begin{array}{l}\Rightarrow E=~-\frac{GMm}{2r}\end{array} \)
Which is the formula for Total Energy of a Satellite. Total energy always be negative for gravitational force.
Negative total energy means that the satellite is bound to the central body (like Earth). This ensures that the satellite remains in orbit and does not escape into space.
From the above equation,
Total energy of the satellite = – (kinetic energy of the satellite)
\(\begin{array}{l}\text{Total energy of the satellite} = \frac{\text{potential energy of the satellite}}{2}\end{array} \)
Virial Theorem
The virial theorem gives us the relationship between kinetic energy and potential energy. According to the virial theorem, if potential energy is proportional to the nth power of position (r), then the average kinetic energy is equal to (n/2) times potential energy at that position.
\(\begin{array}{l}\text{If}\ U\propto {{r}^{n}}\ \text{then},\ K=\frac{n}{2}\left( U \right)\end{array} \)
Q1: What is the total energy of a satellite?
A: The total energy of a satellite is the sum of its kinetic energy (KE) and potential energy (PE). It is given by: E=KE+PE. Since the satellite is in a stable orbit, its total energy is negative, indicating that it is bound to the gravitational field of the planet.
Q2: How do we calculate the kinetic and potential energy of a satellite?
Kinetic Energy (KE):
The kinetic energy of a satellite in a circular orbit is:
\(\begin{array}{l}K=\frac{GMm}{2r}\end{array} \)
Potential Energy (PE):
The gravitational potential energy is given by:
\(\begin{array}{l}U=\frac{-GMm}{r}\end{array} \)
The negative sign indicates that the satellite is bound to the Earth.
Total Energy (E):
The total energy of the satellite is:
\(\begin{array}{l}\Rightarrow E=~-\frac{GMm}{2r}\end{array} \)
This negative total energy signifies a stable orbit.
Q3: What does negative total energy indicate?
A: Negative total energy means that the satellite is bound to the central body (like Earth). This ensures that the satellite remains in orbit and does not escape into space.
Frequently Asked Questions (FAQs)
Q1: Why is the total energy of a satellite negative?
A: The negative sign indicates that work must be done to remove the satellite from orbit and send it to infinity.
Q2: What happens if the total energy becomes zero?
A: If the total energy becomes zero, the satellite will escape Earth’s gravitational pull and move indefinitely into space.
Q3: Can a satellite have only kinetic energy?
A: No, a satellite always has both kinetic and potential energy in orbit.
Multiple-Choice Questions (MCQs)
Q1: If the kinetic energy of a satellite is KE, what is its total energy?
A) 2KE
B) KE
C) −KE
D) −KE/2
Answer: (C) −KE
Explanation: The total energy is given by
\(\begin{array}{l}\Rightarrow E=~-\frac{GMm}{2r}\end{array} \)
Total energy of the satellite = – (kinetic energy of the satellite)
Q2: What happens to the total energy when the satellite moves to a higher orbit?
A) Increases
B) Decreases
C) Remains constant
D) Becomes zero
Answer: (A) Increases
Explanation: The total energy becomes less negative as the satellite moves farther from Earth.
Q3: If a satellite’s total energy is −E, what is its potential energy?
A) −2E
B) −E
C) 2E
D) E/2
Answer: (A) −2E
Explanation: Potential energy is always twice the total energy in magnitude but negative
Total energy of the satellite = – (kinetic energy of the satellite)
\(\begin{array}{l}\text{Total energy of the satellite} = \frac{\text{potential energy of the satellite}}{2}\end{array} \)
Test Your Knowledge
Satellite Energy Quiz
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