A and B are two satellites revolving round the earth in circular orbits have time periods Shr and 1hr 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 can 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 semi-major axis (r) of its orbit. The law can be expressed mathematically as:

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

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Time period of satellite A, \( T_A = 8 \, \text{hr} \)
- Time period of satellite B, \( T_B = 1 \, \text{hr} \)

2. **Express the Time Periods in Terms of a Ratio**:
- The ratio of the time periods is:
\[ \frac{T_A}{T_B} = \frac{8}{1} = 8 \]

3. **Apply Kepler's Third Law**:
- According to Kepler's Third Law:
\[ \frac{T_A^2}{T_B^2} = \frac{r_A^3}{r_B^3} \]
- This can be rearranged to find the ratio of the radii:
\[ \frac{r_A}{r_B} = \left( \frac{T_A}{T_B} \right)^{\frac{2}{3}} \]

4. **Substitute the Values**:
- Substitute the ratio of the time periods into the equation:
\[ \frac{r_A}{r_B} = \left( \frac{8}{1} \right)^{\frac{2}{3}} = 8^{\frac{2}{3}} \]

5. **Simplify the Expression**:
- Calculate \( 8^{\frac{2}{3}} \):
\[ 8 = 2^3 \implies 8^{\frac{2}{3}} = (2^3)^{\frac{2}{3}} = 2^{3 \cdot \frac{2}{3}} = 2^2 = 4 \]

6. **Final Ratio of the Radii**:
- Therefore, the ratio of the radii of the orbits is:
\[ \frac{r_A}{r_B} = 4 \]
- This can be expressed as:
\[ r_A : r_B = 4 : 1 \]

### Final Answer:
The ratio of the radii of the orbits of satellites A and B is \( 4 : 1 \).

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