E, U and K represent total mechanical energy potential energy and kinetic energy of a satellite revolving around a planet. Which of the following quantity is not independent of orbital radius of the satellite ?

Options

A. `K+(U)/(2)`
B. K + E
C. `(U)/(2)+E`
D. `(U)/(2)-E`

Answer

To solve the problem, we need to analyze the relationships between total mechanical energy (E), potential energy (U), and kinetic energy (K) of a satellite revolving around a planet. We will determine which of these quantities is not independent of the orbital radius of the satellite.

### Step-by-Step Solution:

1. **Understanding the Energies**:
- The total mechanical energy (E) of a satellite in orbit is given by the sum of its kinetic energy (K) and potential energy (U).
- Mathematically, this can be expressed as:
\[ E = K + U \]

2. **Potential Energy (U)**:
- The gravitational potential energy of a satellite at a distance \( r \) from the center of the planet is given by:
\[ U = -\frac{GMm}{r} \]
- Here, \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( m \) is the mass of the satellite. This shows that potential energy depends on the orbital radius \( r \).

3. **Kinetic Energy (K)**:
- The kinetic energy of the satellite can be expressed as:
\[ K = \frac{1}{2}mv^2 \]
- For a satellite in a stable orbit, the gravitational force provides the necessary centripetal force. Therefore, we can relate the orbital speed \( v \) to the radius \( r \):
\[ F = \frac{GMm}{r^2} = \frac{mv^2}{r} \]
- From this, we can derive that:
\[ v^2 = \frac{GM}{r} \quad \Rightarrow \quad K = \frac{1}{2}m\left(\frac{GM}{r}\right) = \frac{GMm}{2r} \]
- Thus, kinetic energy also depends on the orbital radius \( r \).

4. **Total Mechanical Energy (E)**:
- We can express the total mechanical energy in terms of the potential energy:
\[ E = K + U = \frac{GMm}{2r} - \frac{GMm}{r} \]
- Simplifying this gives:
\[ E = -\frac{GMm}{2r} \]
- This shows that the total mechanical energy also depends on the orbital radius \( r \).

5. **Conclusion**:
- Since all three quantities (E, U, and K) depend on the orbital radius \( r \), we need to identify which one is not independent of \( r \).
- The question asks for the quantity that is not independent of the orbital radius. Since both potential energy (U) and kinetic energy (K) are dependent on \( r \), and total mechanical energy (E) is also dependent on \( r \), we conclude that the answer is:
- **Kinetic Energy (K)** is not independent of the orbital radius.

### Answer:
The quantity that is not independent of the orbital radius of the satellite is **Kinetic Energy (K)**.