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 solve the problem of finding 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. Here’s a step-by-step solution:

### Step 1: Understand Kepler's Third Law
Kepler's Third 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:
\[ T^2 \propto r^3 \]
This means:
\[ \frac{T_2^2}{T_1^2} = \frac{r_2^3}{r_1^3} \]

### Step 2: Define the Variables
- Let \( T_1 = T \) (the period of the satellite at radius R).
- Let \( r_1 = R \) (the radius of the first satellite).
- Let \( r_2 = 3R \) (the radius of the second satellite).
- We need to find \( T_2 \).

### Step 3: Apply Kepler's Third Law
Using the relationship from Kepler's Third Law:
\[ \frac{T_2^2}{T^2} = \frac{(3R)^3}{R^3} \]

### Step 4: Simplify the Equation
Calculating the right side:
\[ \frac{(3R)^3}{R^3} = \frac{27R^3}{R^3} = 27 \]
Thus, we have:
\[ \frac{T_2^2}{T^2} = 27 \]

### Step 5: Solve for \( T_2 \)
Now, we can express \( T_2 \) in terms of \( T \):
\[ T_2^2 = 27T^2 \]
Taking the square root of both sides:
\[ T_2 = \sqrt{27}T = 3\sqrt{3}T \]

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