Question

A satellite is orbiting the Earth and dissipates energy due to some resistive forces. Its initial total mechanical energy is \(E\) (negative). If the radius of its orbit becomes half of the original value, what is the new total mechanical energy of the satellite?

• A. \(E/2\)
• B. \(E/4\)
• C. \(2E\)
• D. \(4E\)

Hint:

In satellite dynamics, total energy is always half of the potential energy (\(E = \frac{U}{2}\)) and equal to the negative of the kinetic energy (\(E = -K\)). If the radius decreases, the satellite actually speeds up (K increases) even though its total energy decreases!

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
For a satellite in a circular orbit, the total mechanical energy is a function of the orbital radius. When a satellite loses energy due to resistive forces (like atmospheric drag), it "falls" into a lower orbit with a smaller radius.

Step 2: Key Formula or Approach
The total mechanical energy (\(E\)) of a satellite of mass \(m\) orbiting a planet of mass \(M\) at a radius \(r\) is: \[ E = -\frac{GMm}{2r} \] From this, we see that \(E \propto \frac{1}{r}\).

Step 3: Detailed Explanation
1. Let the initial radius be \(r_1\) and the initial energy be \(E_1 = E\). 2. The new radius is \(r_2 = \frac{r_1}{2}\). 3. Using the proportionality: \[ \frac{E_2}{E_1} = \frac{r_1}{r_2} \] 4. Substitute the values: \[ \frac{E_2}{E} = \frac{r_1}{r_1/2} = 2 \] \[ E_2 = 2E \]

Step 4: Final Answer
The new total mechanical energy of the satellite is \(2E\). Note that since \(E\) is negative, \(2E\) is a "more negative" value, representing a loss of energy.