E, U and K represent total mechanical energy. potential energy and kinetic energy respectively of a satellite revolving around a planet, then which of the following is correct?

Options

A. K = E
B. U= 2E
C. `K= abs(E)`
D. Both (2) and (3)

Answer

To solve the problem, we need to analyze the relationships between the total mechanical energy (E), potential energy (U), and kinetic energy (K) of a satellite revolving around a planet.

### Step-by-Step Solution:

1. **Define the Energies**:
- The kinetic energy (K) of the satellite is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
- The potential energy (U) of the satellite due to gravitational attraction is given by:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, \( m \) is the mass of the satellite, and \( r \) is the distance from the center of the planet to the satellite.

2. **Centripetal Force and Gravitational Force**:
- For the satellite to remain in circular motion, the gravitational force must provide the necessary centripetal force:
\[ \frac{GMm}{r^2} = \frac{mv^2}{r} \]
- Simplifying this equation by canceling \( m \) from both sides gives:
\[ \frac{GM}{r^2} = \frac{v^2}{r} \]
- Rearranging leads to:
\[ v^2 = \frac{GM}{r} \]

3. **Substituting Kinetic Energy**:
- Substitute \( v^2 \) into the kinetic energy formula:
\[ K = \frac{1}{2} m \left(\frac{GM}{r}\right) = \frac{GMm}{2r} \]

4. **Total Mechanical Energy**:
- The total mechanical energy (E) is the sum of kinetic and potential energy:
\[ E = K + U \]
- Substituting the expressions for K and U:
\[ E = \frac{GMm}{2r} - \frac{GMm}{r} = \frac{GMm}{2r} - \frac{2GMm}{2r} = -\frac{GMm}{2r} \]

5. **Relationship Between Energies**:
- From the calculations, we find:
- \( K = \frac{GMm}{2r} \)
- \( U = -\frac{GMm}{r} \)
- \( E = -\frac{GMm}{2r} \)

6. **Analyzing the Relationships**:
- The kinetic energy is positive, while the potential energy is negative.
- The total mechanical energy is negative and equal to half of the potential energy (in magnitude):
\[ U = 2E \quad \text{and} \quad K = -E \]

### Conclusion:
From the analysis, we conclude:
- \( U = 2E \)
- \( K = -E \)
- The relationships show that both kinetic and potential energies are related to the total mechanical energy.

### Final Answer:
The correct relationships are:
- Option 2: \( U = 2E \) is correct.
- Option 3: \( K = -E \) is also correct.
- Therefore, the best option is option 4, which states that both 2 and 3 are correct.