The period of revolution of a satellite in an orbit of radius 2R is T. What will be its period of revolution in an orbit of radius 8R ?

Explanation

To solve the problem of finding the period of revolution of a satellite in an orbit of radius 8R, given that the period in an orbit of radius 2R is T, 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.

### Step-by-Step Solution:

1. **Understanding Kepler's Third Law**: According to Kepler's Third Law, we have:
\[ T^2 \propto r^3 \]
This can be expressed as:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]
where \( T_1 \) and \( T_2 \) are the periods of the satellites in orbits of radii \( r_1 \) and \( r_2 \), respectively.

2. **Assigning Values**: In this problem:
- Let \( r_1 = 2R \) (the radius for which the period is T)
- Let \( r_2 = 8R \) (the radius for which we want to find the period)

3. **Setting Up the Equation**: Using the values assigned:
\[ \frac{T^2}{T_2^2} = \frac{(2R)^3}{(8R)^3} \]

4. **Calculating the Ratios**:
- Calculate \( (2R)^3 = 8R^3 \)
- Calculate \( (8R)^3 = 512R^3 \)
- Therefore, the equation becomes:
\[ \frac{T^2}{T_2^2} = \frac{8R^3}{512R^3} = \frac{8}{512} = \frac{1}{64} \]

5. **Finding \( T_2^2 \)**: Rearranging the equation gives:
\[ T_2^2 = 64T^2 \]

6. **Taking the Square Root**: To find \( T_2 \):
\[ T_2 = 8T \]

### Final Answer:
The period of revolution of the satellite in an orbit of radius \( 8R \) is \( 8T \).