A satellite is revolving around earth in a circular orbit of radius 3 R. Which of the following is incorrect? ( M is mass of earth, R is radius of earth m is mass of satellite)

Options

A. Its orbital velocity is `sqrt((GM)/(3R)`
B. Its potential energy is `(-GMm)/(3R)`
C. Its total energy is `(GMm)/(6R)`
D. Its kinetic energy is `(GMm)/(6R)`

Answer

To solve the problem, we need to analyze the statements regarding a satellite revolving around the Earth in a circular orbit of radius \(3R\). We will derive the necessary equations and identify which statement is incorrect.

### Step-by-Step Solution:

1. **Understanding the Forces**:
The satellite is in circular motion, which means the gravitational force acting on it provides the necessary centripetal force. The gravitational force \(F_g\) acting on the satellite is given by:
\[ F_g = \frac{GMm}{(3R)^2} \]
where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, \(m\) is the mass of the satellite, and \(3R\) is the distance from the center of the Earth to the satellite.

2. **Centripetal Force**:
The centripetal force \(F_c\) required to keep the satellite in circular motion is given by:
\[ F_c = \frac{mv^2}{3R} \]
where \(v\) is the orbital velocity of the satellite.

3. **Setting Forces Equal**:
Since the gravitational force provides the centripetal force, we can set them equal:
\[ \frac{GMm}{(3R)^2} = \frac{mv^2}{3R} \]
Canceling \(m\) from both sides (assuming \(m \neq 0\)):
\[ \frac{GM}{9R^2} = \frac{v^2}{3R} \]

4. **Solving for Orbital Velocity**:
Rearranging gives:
\[ v^2 = \frac{GM}{3R} \quad \text{(1)} \]

5. **Kinetic Energy**:
The kinetic energy \(K\) of the satellite can be calculated as:
\[ K = \frac{1}{2} mv^2 \]
Substituting \(v^2\) from equation (1):
\[ K = \frac{1}{2} m \left(\frac{GM}{3R}\right) = \frac{GMm}{6R} \quad \text{(2)} \]

6. **Potential Energy**:
The gravitational potential energy \(U\) of the satellite is given by:
\[ U = -\frac{GMm}{3R} \quad \text{(3)} \]

7. **Total Mechanical Energy**:
The total mechanical energy \(E\) is the sum of kinetic and potential energy:
\[ E = K + U \]
Substituting equations (2) and (3):
\[ E = \frac{GMm}{6R} - \frac{GMm}{3R} \]
To combine these, we find a common denominator:
\[ E = \frac{GMm}{6R} - \frac{2GMm}{6R} = -\frac{GMm}{6R} \quad \text{(4)} \]

8. **Identifying the Incorrect Statement**:
Now we can evaluate the statements provided in the question. We have derived:
- Kinetic Energy: \( \frac{GMm}{6R} \)
- Potential Energy: \( -\frac{GMm}{3R} \)
- Total Mechanical Energy: \( -\frac{GMm}{6R} \)

If any statement contradicts these results, it is the incorrect one.

### Conclusion:
After evaluating the derived equations, we can conclude that the incorrect statement is the one that misrepresents the total mechanical energy of the satellite.