Which one of the following has the maximum angular velocity

Options

A. hour hand of a clock
B. minute hand of a clock
C. Second hand of a clock
D. A geostationary satellite

Explanation

To determine which of the given options has the maximum angular velocity, we will calculate the angular velocity for each option: the hour hand, minute hand, second hand of a clock, and a geostationary satellite.

### Step-by-Step Solution:

1. **Angular Velocity Definition**:
Angular velocity (ω) is defined as the angle rotated per unit time. The formula is given by:
\[ \omega = \frac{2\pi}{T} \]
where \( T \) is the time period for one complete revolution.

2. **Hour Hand of a Clock**:
- The hour hand takes 12 hours to complete one full revolution (360 degrees).
- Convert 12 hours to seconds:
\[ T_h = 12 \times 60 \times 60 = 43200 \text{ seconds} \]
- Calculate the angular velocity:
\[ \omega_h = \frac{2\pi}{43200} \text{ radians/second} \]

3. **Minute Hand of a Clock**:
- The minute hand takes 60 minutes to complete one full revolution.
- Convert 60 minutes to seconds:
\[ T_m = 60 \times 60 = 3600 \text{ seconds} \]
- Calculate the angular velocity:
\[ \omega_m = \frac{2\pi}{3600} \text{ radians/second} \]

4. **Second Hand of a Clock**:
- The second hand takes 60 seconds to complete one full revolution.
- Calculate the angular velocity:
\[ T_s = 60 \text{ seconds} \]
- Calculate the angular velocity:
\[ \omega_s = \frac{2\pi}{60} \text{ radians/second} \]

5. **Geostationary Satellite**:
- A geostationary satellite has a period equal to the Earth's rotation period, which is 24 hours.
- Convert 24 hours to seconds:
\[ T_g = 24 \times 60 \times 60 = 86400 \text{ seconds} \]
- Calculate the angular velocity:
\[ \omega_g = \frac{2\pi}{86400} \text{ radians/second} \]

6. **Comparison of Angular Velocities**:
- Now we need to compare the angular velocities:
- \( \omega_h = \frac{2\pi}{43200} \)
- \( \omega_m = \frac{2\pi}{3600} \)
- \( \omega_s = \frac{2\pi}{60} \)
- \( \omega_g = \frac{2\pi}{86400} \)

- The denominators indicate the time taken for one complete revolution. The smaller the denominator, the larger the angular velocity.

7. **Conclusion**:
- Comparing the denominators:
- \( 60 < 3600 < 43200 < 86400 \)
- Therefore, the second hand has the maximum angular velocity.

### Final Answer:
The second hand of the clock has the maximum angular velocity.