Binding Energy of a Satellite
Binding energy (EB) is the energy required to be given to the satellite to escape the gravitational pull of the earth. It is numerically equal to the negative of the total energy possessed by the satellite.
\(\begin{array}{l}\text{Binding energy of a satellite}=\frac{GMm}{2r}\end{array} \)
The binding energy of a system of two bodies is the amount of minimum energy required to separate them by infinite distance, or it is simply the amount of energy required to make the potential energy of the system equal to zero, which is numerically equal to the kinetic energy of the body under motion.
Angular Momentum of a Satellite
When a satellite of mass ‘m’ is orbiting with an angular speed on the orbital path of radius ‘r’, its angular momentum is given by
L = mvr
L = m(rω)r
\(\begin{array}{l}L=m{{r}^{2}}\omega\end{array} \).
We know that
\(\begin{array}{l}\omega =\frac{2\pi }{T}\end{array} \)
, substituting this in the above equation, we get
\(\begin{array}{l}L=m{{r}^{2}}\frac{2\pi }{T}\end{array} \)
\(\begin{array}{l}\text{Using}\ T=\frac{2\pi }{\sqrt{GM}}{{\left( r \right)}^{{}^{3}/{}_{2}}},\ \text{we get};\end{array} \)
\(\begin{array}{l}L=m\sqrt{GMr}\end{array} \)
From the above equation, it is clear that the angular momentum of a satellite depends on both the mass of a satellite and the mass of earth. It also depends upon the radius of the orbit of the satellite.
Question-Answers-Binding Energy & Angular Momentum of a Satellite
Q1: What is Binding Energy of a Satellite?
A: Binding energy is the minimum energy required to remove a satellite from the gravitational influence of a planet or celestial body. It is given by:
\(\begin{array}{l}\text{Binding energy of a satellite}=\frac{GMm}{2r}\end{array} \)
where:
- G = Gravitational constant
- M = Mass of the planet
- m = Mass of the satellite
- R = Radius of orbit
Q2: Why is Binding Energy Negative?
A: The binding energy is negative because the satellite is bound to the planet due to gravitational attraction. A negative energy indicates that external energy must be supplied to remove the satellite from orbit.
Q3: What is the Angular Momentum of a Satellite?
A: Angular momentum (L) of a satellite is given by: L=mvR where:
- m = Mass of the satellite
- v = Orbital velocity
- R = Radius of orbit
Using the relation for orbital velocity, Angular momentum can be rewritten as:
\(\begin{array}{l}L=m\sqrt{GMr}\end{array} \)
Multiple Choice Questions (MCQs)-Binding Energy & Angular Momentum of a Satellite
Q1: Which of the following statements about binding energy is correct?
A) It is the energy required to move a satellite closer to the planet.
B) It is the energy required to remove the satellite from its orbit.
C) It is the total energy of the satellite.
D) It is always positive.
Answer: B) It is the energy required to remove the satellite from its orbit.
Explanation: Binding energy represents the energy needed to overcome gravitational attraction and move the satellite to infinity.
Q2: If the mass of the satellite doubles, how does its angular momentum change?
A) It remains the same.
B) It doubles.
C) It becomes half.
D) It becomes four times.
Answer: B) It doubles.
Explanation: Angular momentum is given by L=mGMRL = m \sqrt{GM R}. If mm doubles, LL also doubles.
Q3: What happens to the binding energy of a satellite if the radius of its orbit increases?
A) It increases.
B) It decreases.
C) It remains the same.
D) It becomes zero.
Answer: B) It decreases.
Explanation: Binding energy is inversely proportional to the orbital radius. As the radius increases, the gravitational attraction weakens, requiring less energy to escape.
FAQs-Binding Energy & Angular Momentum of a Satellite
Q1: What is the significance of binding energy in satellite motion?
A: Binding energy determines how stable a satellite’s orbit is. A higher binding energy means the satellite is more strongly bound to the planet.
Q2: Can a satellite have zero binding energy?
A: Yes, if a satellite has enough energy to reach escape velocity, its total energy becomes zero, meaning it is no longer bound.
Q3: Why is angular momentum conserved in satellite motion?
A: In the absence of external torques, angular momentum remains constant due to the law of conservation of angular momentum.
Test Your Knowledge
Binding Energy & Angular Momentum Quiz
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