Two planets `S_1` and `S_2` rotates about own axis `S_1` completes one rev in 1hr and `S_2` completes one rev in 8hr then find ratio of angular velocity of `S_1 and S_2` (`omega_1/omega_2`)

Options

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

Answer

To find the ratio of the angular velocities of two planets \( S_1 \) and \( S_2 \), we can follow these steps:

### Step 1: Understand the Time Periods
- The time period \( T_1 \) for planet \( S_1 \) is given as 1 hour.
- The time period \( T_2 \) for planet \( S_2 \) is given as 8 hours.

### Step 2: Recall the Relationship Between Time Period and Angular Velocity
The relationship between the time period \( T \) and angular velocity \( \omega \) is given by the formula:
\[ T = \frac{2\pi}{\omega} \]
From this, we can express angular velocity in terms of the time period:
\[ \omega = \frac{2\pi}{T} \]

### Step 3: Calculate Angular Velocities
Now, we can find the angular velocities for both planets.

For planet \( S_1 \):
\[ \omega_1 = \frac{2\pi}{T_1} = \frac{2\pi}{1 \text{ hour}} \]

For planet \( S_2 \):
\[ \omega_2 = \frac{2\pi}{T_2} = \frac{2\pi}{8 \text{ hours}} \]

### Step 4: Find the Ratio of Angular Velocities
Now, we need to find the ratio \( \frac{\omega_1}{\omega_2} \):
\[ \frac{\omega_1}{\omega_2} = \frac{\frac{2\pi}{1}}{\frac{2\pi}{8}} = \frac{2\pi}{1} \times \frac{8}{2\pi} \]
The \( 2\pi \) cancels out:
\[ \frac{\omega_1}{\omega_2} = \frac{8}{1} = 8 \]

### Final Answer
Thus, the ratio of the angular velocities of planets \( S_1 \) and \( S_2 \) is:
\[ \frac{\omega_1}{\omega_2} = 8 \]

---