The period of revolution of a surface satellite around a planet of radius R is T. The period of another satellite around the same planet in an orbit of radius 3R is

Options

A. T
B. 3T
C. 9T
D. `3sqrt3T`

Answer

To find the period of revolution of a satellite in an orbit of radius \(3R\) around a planet of radius \(R\), we can use Kepler's Third Law of planetary motion. This law states that the square of the period of revolution \(T\) of a satellite is directly proportional to the cube of the semi-major axis \(r\) of its orbit. Mathematically, this can be expressed as:

\[ T^2 \propto r^3 \]

### Step-by-Step Solution:

1. **Identify the given information**:
- The period of revolution of a surface satellite (radius \(R\)) is \(T\).
- We need to find the period of another satellite in an orbit of radius \(3R\).

2. **Set up the relationship using Kepler's Third Law**:
- Let \(T_1\) be the period of the first satellite (at radius \(R\)) and \(T_2\) be the period of the second satellite (at radius \(3R\)).
- According to Kepler's Third Law:
\[ \frac{T_2^2}{T_1^2} = \frac{r_2^3}{r_1^3} \]
where \(r_1 = R\) and \(r_2 = 3R\).

3. **Substitute the values into the equation**:
\[ \frac{T_2^2}{T^2} = \frac{(3R)^3}{R^3} \]

4. **Simplify the right side**:
\[ \frac{T_2^2}{T^2} = \frac{27R^3}{R^3} = 27 \]

5. **Solve for \(T_2^2\)**:
\[ T_2^2 = 27T^2 \]

6. **Take the square root of both sides to find \(T_2\)**:
\[ T_2 = \sqrt{27}T = 3\sqrt{3}T \]

### Final Answer:
The period of the satellite in an orbit of radius \(3R\) is \(T_2 = 3\sqrt{3}T\).

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