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 of determining which statement regarding a satellite revolving around Earth in a circular orbit of radius 3R is incorrect, we will follow these steps:

### Step 1: Identify the Forces Acting on the Satellite
The gravitational force acting on the satellite provides the necessary centripetal force for its circular motion. The gravitational force \( F \) can be expressed as:
\[ F = \frac{G M m}{d^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( d \) is the distance from the center of the Earth to the satellite.

### Step 2: Determine the Distance \( d \)
Since the satellite is at a distance of \( 3R \) from the center of the Earth, we have:
\[ d = 3R \]

### Step 3: Set Up the Equation for Centripetal Force
The centripetal force required for circular motion is given by:
\[ F_c = \frac{m v^2}{d} \]
where \( v \) is the orbital velocity of the satellite.

### Step 4: Equate Gravitational Force and Centripetal Force
Setting the gravitational force equal to the centripetal force:
\[ \frac{G M m}{(3R)^2} = \frac{m v^2}{3R} \]
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{G M}{9R^2} = \frac{v^2}{3R} \]

### Step 5: Solve for Orbital Velocity \( v \)
Rearranging the equation gives:
\[ v^2 = \frac{G M}{3R} \implies v = \sqrt{\frac{G M}{3R}} \]
This confirms that the orbital velocity is \( v = \sqrt{\frac{G M}{3R}} \).

### Step 6: Calculate Potential Energy
The gravitational potential energy \( U \) of the satellite at distance \( d \) is given by:
\[ U = -\frac{G M m}{d} = -\frac{G M m}{3R} \]

### Step 7: Calculate Kinetic Energy
The kinetic energy \( K \) of the satellite is:
\[ K = \frac{1}{2} m v^2 = \frac{1}{2} m \left(\frac{G M}{3R}\right) = \frac{G M m}{6R} \]

### Step 8: Calculate Total Energy
The total energy \( E \) of the satellite is the sum of its kinetic and potential energy:
\[ E = K + U = \frac{G M m}{6R} - \frac{G M m}{3R} \]
To combine these, we find a common denominator:
\[ E = \frac{G M m}{6R} - \frac{2 G M m}{6R} = -\frac{G M m}{6R} \]
This shows that the total energy is negative.

### Step 9: Identify the Incorrect Statement
The total energy being negative is a characteristic of bound systems. If any statement claims that the total energy is positive, that statement is incorrect.

### Summary of Findings
1. Orbital velocity: \( v = \sqrt{\frac{G M}{3R}} \) (Correct)
2. Potential energy: \( U = -\frac{G M m}{3R} \) (Correct)
3. Kinetic energy: \( K = \frac{G M m}{6R} \) (Correct)
4. Total energy: \( E = -\frac{G M m}{6R} \) (Incorrect if stated as positive)

### Conclusion
The incorrect statement is the one that claims the total energy is positive.

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