A and B are two satellites revolving round the earth in circular orbits have time periods 8hr and 1 hr respectively . The ratio of their radius of orbits

Options

A. `8^(3//2) : 1`
B. `8:1`
C. `4:1`
D. `2:1`

Answer

To find the ratio of the radii of the orbits of satellites A and B, we will use Kepler's Third Law of planetary motion, which states that the square of the time period (T) of a satellite is directly proportional to the cube of the radius (r) of its orbit.

### Step-by-step Solution:

1. **Identify the Time Periods**:
- For satellite A, the time period \( T_A = 8 \) hours.
- For satellite B, the time period \( T_B = 1 \) hour.

2. **Apply Kepler's Third Law**:
- According to Kepler's Third Law, we have:
\[ \frac{T_A^2}{T_B^2} = \frac{r_A^3}{r_B^3} \]
- Rearranging gives:
\[ \frac{r_A^3}{r_B^3} = \left(\frac{T_A}{T_B}\right)^2 \]

3. **Substitute the Time Periods**:
- Substitute \( T_A \) and \( T_B \) into the equation:
\[ \frac{r_A^3}{r_B^3} = \left(\frac{8}{1}\right)^2 = 64 \]

4. **Take the Cube Root**:
- To find the ratio of the radii, we take the cube root of both sides:
\[ \frac{r_A}{r_B} = \sqrt[3]{64} = 4 \]

5. **Final Ratio**:
- Therefore, the ratio of the radii of the orbits of satellites A and B is:
\[ \frac{r_A}{r_B} = 4:1 \]

### Conclusion:
The ratio of the radius of the orbits of satellites A and B is \( 4:1 \).

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