Question

A satellite is orbiting the earth in a circular orbit of radius R. Which one of the following statements is true ?

• A. Angular momentum varies as \( \frac{1}{\sqrt{R}} \)
• B. Linear momentum varies as \( \sqrt{R} \)
• C. Frequency of revolution varies as \( \frac{1}{R^2} \)
• D. Kinetic energy varies as \( \frac{1}{R} \)
• E. Potential energy varies as \( R \)

Hint:

A useful shortcut for satellite motion: - \( \text{Potential Energy} = -2 \times \text{Kinetic Energy} \) - \( \text{Total Energy} = -\text{Kinetic Energy} \)

Answer 1

The Correct Option is D

Concept: For a satellite of mass \(m\) orbiting a planet of mass \(M_E\) at radius \(R\), the gravitational force provides the centripetal force: \[ \frac{G M_E m}{R^2} = \frac{mv^2}{R} \quad \Rightarrow \quad v = \sqrt{\frac{G M_E}{R}} \]

Step 1: Analyze Kinetic Energy.
Kinetic Energy (K.E.) is given by \(\frac{1}{2}mv^2\). Substituting \(v^2 = \frac{G M_E}{R}\): \[ \text{K.E.} = \frac{G M_E m}{2R} \quad \Rightarrow \quad \text{K.E.} \propto \frac{1}{R} \] Statement (D) is true.

Step 2: Analyze Linear and Angular Momentum.
Linear Momentum \(p = mv = m\sqrt{\frac{G M_E}{R}}\). Thus, \(p \propto \frac{1}{\sqrt{R}}\). (B is false) Angular Momentum \(L = m v R = m \sqrt{\frac{G M_E}{R}} \cdot R = m \sqrt{G M_E R}\). Thus, \(L \propto \sqrt{R}\). (A is false)

Step 3: Analyze Potential Energy and Frequency.
Potential Energy (U) is \(-\frac{G M_E m}{R}\). Thus, \(U \propto \frac{1}{R}\). (E is false) Frequency \(f = \frac{v}{2\pi R} = \frac{\sqrt{G M_E / R}}{2\pi R} = \frac{\sqrt{G M_E}}{2\pi R^{3/2}}\). Thus, \(f \propto \frac{1}{R^{3/2}}\). (C is false)