Match list I with list II
`{:("List-I", "List-II"),((A)"Ratio of angular velocities of hours hand of a clock and self rotation of the earth", (e) 12:1),((b)"Ratio of angular velocities of seconds hand to minutes hand of a clock",(f) 60:1),((c)"Ratio of angular velocities of seconds hand to hours hand of a clock",(g)2:1),((d) "Ratio of angular velocities of minutes hand to hours hand of a clock",(h) 720:1):}`

Options

A. a-h,b-f,c-e,d-h
B. a-f,b-g,c-h,d-e
C. a-g,b-f,c-h,d-e
D. a-g,b-e,c-f,d-h

Answer

To solve the problem of matching List I with List II, we will calculate the ratios of angular velocities for each of the cases mentioned in List I.

### Step-by-Step Solution:

1. **Understanding Angular Velocity**:
- Angular velocity (ω) is defined as the angle rotated per unit time. It can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
where \( T \) is the time period for one complete rotation.

2. **Calculate Angular Velocity of Hour Hand**:
- The hour hand of a clock completes one rotation in 12 hours.
- Therefore, the angular velocity of the hour hand (ω_h) is:
\[ \omega_h = \frac{2\pi}{12 \text{ hours}} = \frac{2\pi}{12 \times 3600 \text{ seconds}} = \frac{2\pi}{43200} \text{ rad/s} \]

3. **Calculate Angular Velocity of Earth**:
- The Earth completes one rotation in 24 hours.
- Therefore, the angular velocity of the Earth (ω_e) is:
\[ \omega_e = \frac{2\pi}{24 \text{ hours}} = \frac{2\pi}{24 \times 3600 \text{ seconds}} = \frac{2\pi}{86400} \text{ rad/s} \]

4. **Ratio of Angular Velocities of Hour Hand to Earth**:
- The ratio \( \frac{\omega_h}{\omega_e} \) is:
\[ \frac{\omega_h}{\omega_e} = \frac{\frac{2\pi}{43200}}{\frac{2\pi}{86400}} = \frac{86400}{43200} = 2:1 \]
- Thus, **A matches with (g) 2:1**.

5. **Calculate Angular Velocity of Second Hand**:
- The second hand completes one rotation in 60 seconds.
- Therefore, the angular velocity of the second hand (ω_s) is:
\[ \omega_s = \frac{2\pi}{60} \text{ rad/s} \]

6. **Calculate Angular Velocity of Minute Hand**:
- The minute hand completes one rotation in 3600 seconds (60 minutes).
- Therefore, the angular velocity of the minute hand (ω_m) is:
\[ \omega_m = \frac{2\pi}{3600} \text{ rad/s} \]

7. **Ratio of Angular Velocities of Second Hand to Minute Hand**:
- The ratio \( \frac{\omega_s}{\omega_m} \) is:
\[ \frac{\omega_s}{\omega_m} = \frac{\frac{2\pi}{60}}{\frac{2\pi}{3600}} = \frac{3600}{60} = 60:1 \]
- Thus, **B matches with (f) 60:1**.

8. **Ratio of Angular Velocities of Second Hand to Hour Hand**:
- The ratio \( \frac{\omega_s}{\omega_h} \) is:
\[ \frac{\omega_s}{\omega_h} = \frac{\frac{2\pi}{60}}{\frac{2\pi}{43200}} = \frac{43200}{60} = 720:1 \]
- Thus, **C matches with (h) 720:1**.

9. **Ratio of Angular Velocities of Minute Hand to Hour Hand**:
- The ratio \( \frac{\omega_m}{\omega_h} \) is:
\[ \frac{\omega_m}{\omega_h} = \frac{\frac{2\pi}{3600}}{\frac{2\pi}{43200}} = \frac{43200}{3600} = 12:1 \]
- Thus, **D matches with (e) 12:1**.

### Final Matching:
- A → g (2:1)
- B → f (60:1)
- C → h (720:1)
- D → e (12:1)