Question

Find the total mechanical energy of a satellite of mass \(m\) revolving in a circular orbit of radius \(a\) around the Earth (mass \(M\)).

• A. \(-\frac{GMm}{a}\)
• B. \(-\frac{GMm}{2a}\)
• C. \(\frac{GMm}{2a}\)
• D. \(\frac{GMm}{a}\)

Hint:

A negative total energy signifies that the satellite is "bound" to the Earth. If the energy were zero or positive, the satellite would have enough energy to escape the Earth's gravitational pull.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The total mechanical energy of a satellite is the sum of its kinetic energy ($K$) and its gravitational potential energy ($U$).

Step 2: Key Formula or Approach
1. Potential Energy: \(U = -\frac{GMm}{a}\) 2. Kinetic Energy: \(K = \frac{1}{2}mv^2\), where orbital velocity \(v = \sqrt{\frac{GM}{a}}\). 3. Total Energy: \(E = K + U\)

Step 3: Detailed Explanation
1. Calculate Kinetic Energy (\(K\)): \[ K = \frac{1}{2} m \left( \sqrt{\frac{GM}{a}} \right)^2 = \frac{GMm}{2a} \] 2. State Potential Energy (\(U\)): \[ U = -\frac{GMm}{a} \] 3. Sum for Total Energy (\(E\)): \[ E = \frac{GMm}{2a} + \left( -\frac{GMm}{a} \right) \] \[ E = \frac{GMm - 2GMm}{2a} = -\frac{GMm}{2a} \]

Step 4: Final Answer
The total mechanical energy is \(-\frac{GMm}{2a}\).