A satellite is launched into a circular orbit of radius R around the earth. A second satellite is launched into an orbit of radius 4R. The ratio of their respective periods is

Options

A. `4:1`
B. `1:8`
C. `8:1`
D. `1:4`

Explanation

To solve the problem of finding the ratio of the periods of two satellites in different orbits around the Earth, we can use Kepler's Third Law of planetary motion, which states that the square of the orbital period (T) of a satellite is directly proportional to the cube of the semi-major axis (r) of its orbit.

### Step-by-Step Solution:

1. **Identify the Radii of the Orbits**:
- Let the radius of the first satellite's orbit be \( R \).
- The radius of the second satellite's orbit is \( 4R \).

2. **Apply Kepler's Third Law**:
- According to Kepler's Third Law, we have:
\[ T^2 \propto r^3 \]
- For the first satellite (radius \( R \)):
\[ T_1^2 \propto R^3 \]
- For the second satellite (radius \( 4R \)):
\[ T_2^2 \propto (4R)^3 = 64R^3 \]

3. **Set Up the Ratio of the Periods**:
- From the proportionality, we can write:
\[ \frac{T_1^2}{T_2^2} = \frac{R^3}{64R^3} \]
- Simplifying this gives:
\[ \frac{T_1^2}{T_2^2} = \frac{1}{64} \]

4. **Take the Square Root to Find the Ratio of the Periods**:
- Taking the square root of both sides:
\[ \frac{T_1}{T_2} = \frac{1}{8} \]

5. **Express the Ratio**:
- Thus, the ratio of the periods is:
\[ T_1 : T_2 = 1 : 8 \]

### Final Answer:
The ratio of the periods of the two satellites is \( 1 : 8 \).

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