If `omega_(1),omega_(2)omega_(3)omega_(4)` are angular velocities of rotation of earth, seconds hand, minutes hand. hours hand of a watch respectively these values in decreasing order is

Options

A. `omega_(1),omega_(4)omega_(3)omega_(2)`
B. `omega_(2),omega_(3)omega_(4)omega_(1)`
C. `omega_(1),omega_(2)omega_(3)omega_(4)`
D. `omega_(4),omega_(3)omega_(2)omega_(1)`

Answer

To solve the question regarding the angular velocities of the Earth, seconds hand, minutes hand, and hour hand, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Time Periods**:
- The time period \( T \) is the time taken for one complete rotation.
- For the Earth, \( T_1 = 24 \) hours.
- For the seconds hand, \( T_2 = 60 \) seconds.
- For the minutes hand, \( T_3 = 60 \) minutes = \( 3600 \) seconds.
- For the hour hand, \( T_4 = 12 \) hours.

2. **Convert All Time Periods to the Same Unit**:
- Convert hours to seconds for consistency:
- \( T_1 = 24 \times 3600 = 86400 \) seconds.
- \( T_4 = 12 \times 3600 = 43200 \) seconds.
- Now we have:
- \( T_1 = 86400 \) seconds (Earth)
- \( T_2 = 60 \) seconds (Seconds hand)
- \( T_3 = 3600 \) seconds (Minutes hand)
- \( T_4 = 43200 \) seconds (Hour hand)

3. **Determine the Angular Velocities**:
- Angular velocity \( \omega \) is inversely proportional to the time period \( T \):
\[ \omega = \frac{2\pi}{T} \]
- Therefore, the relationship between angular velocities and time periods is:
- \( \omega_1 \propto \frac{1}{T_1} \)
- \( \omega_2 \propto \frac{1}{T_2} \)
- \( \omega_3 \propto \frac{1}{T_3} \)
- \( \omega_4 \propto \frac{1}{T_4} \)

4. **Rank the Time Periods**:
- From the calculated time periods:
- \( T_1 = 86400 \) seconds (Earth) - longest
- \( T_4 = 43200 \) seconds (Hour hand)
- \( T_3 = 3600 \) seconds (Minutes hand)
- \( T_2 = 60 \) seconds (Seconds hand) - shortest
- Thus, in terms of time period: \( T_1 > T_4 > T_3 > T_2 \)

5. **Rank the Angular Velocities**:
- Since angular velocity is inversely proportional to the time period, we have:
- \( \omega_1 < \omega_4 < \omega_3 < \omega_2 \)
- Therefore, in decreasing order of angular velocities:
- \( \omega_2 > \omega_3 > \omega_4 > \omega_1 \)

### Final Answer:
The angular velocities in decreasing order are:
\[ \omega_2, \omega_3, \omega_4, \omega_1 \]