Consider two satellites `S_1` and `S_2` with periods of revolution 1 hr. and 8hr. respectively revolving around a planet in circular orbits. The ratio of angular velocity of satellite `S_1` to the angular velocity of satellites `S_2` is:

Options

A. `8:1`
B. `1:4`
C. `2:1`
D. `1:8`

Answer

To solve the problem, we need to find the ratio of the angular velocities of two satellites, \( S_1 \) and \( S_2 \), which have periods of revolution \( T_1 = 1 \) hour and \( T_2 = 8 \) hours, respectively.

### Step-by-Step Solution:

1. **Understanding Angular Velocity**:
The angular velocity \( \omega \) of a satellite is related to its time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
where \( \omega \) is measured in radians per second and \( T \) is the time period in seconds.

2. **Calculating Angular Velocity for \( S_1 \)**:
For satellite \( S_1 \):
\[ T_1 = 1 \text{ hour} = 3600 \text{ seconds} \]
Therefore, the angular velocity \( \omega_1 \) is:
\[ \omega_1 = \frac{2\pi}{T_1} = \frac{2\pi}{3600} \]

3. **Calculating Angular Velocity for \( S_2 \)**:
For satellite \( S_2 \):
\[ T_2 = 8 \text{ hours} = 8 \times 3600 \text{ seconds} = 28800 \text{ seconds} \]
Therefore, the angular velocity \( \omega_2 \) is:
\[ \omega_2 = \frac{2\pi}{T_2} = \frac{2\pi}{28800} \]

4. **Finding the Ratio of Angular Velocities**:
We need to find the ratio \( \frac{\omega_1}{\omega_2} \):
\[ \frac{\omega_1}{\omega_2} = \frac{\frac{2\pi}{3600}}{\frac{2\pi}{28800}} = \frac{28800}{3600} \]
Simplifying this gives:
\[ \frac{\omega_1}{\omega_2} = \frac{28800 \div 3600}{1} = 8 \]

5. **Final Result**:
The ratio of the angular velocities of satellite \( S_1 \) to satellite \( S_2 \) is:
\[ \frac{\omega_1}{\omega_2} = 8:1 \]

### Conclusion:
The ratio of the angular velocity of satellite \( S_1 \) to the angular velocity of satellite \( S_2 \) is \( 8:1 \).

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