The difference between angular speed of minute hand and second hand of a clock is

Options

A. `(59pi)/900rad//s`
B. `(59pi)/1800rad//s`
C. `(59pi)/2400rad//s`
D. `(59pi)/3600rad//s`

Answer

To find the difference between the angular speeds of the minute hand and the second hand of a clock, we can follow these steps:

### Step-by-Step Solution

1. **Understand Angular Speed**:
Angular speed (ω) is defined as the angle covered per unit time. For a complete revolution, the angle covered is \(2\pi\) radians.

2. **Determine the Time Period of Each Hand**:
- **Minute Hand (Tm)**: The minute hand takes 60 minutes to complete one full revolution.
\[ T_m = 60 \text{ minutes} = 60 \times 60 \text{ seconds} = 3600 \text{ seconds} \]
- **Second Hand (Ts)**: The second hand takes 60 seconds to complete one full revolution.
\[ T_s = 60 \text{ seconds} \]

3. **Calculate Angular Speeds**:
- **Angular Speed of Minute Hand (ωm)**:
\[ \omega_m = \frac{2\pi}{T_m} = \frac{2\pi}{3600} \text{ rad/s} \]
- **Angular Speed of Second Hand (ωs)**:
\[ \omega_s = \frac{2\pi}{T_s} = \frac{2\pi}{60} \text{ rad/s} \]

4. **Find the Difference in Angular Speeds**:
The difference between the angular speeds of the second hand and the minute hand is given by:
\[ \Delta \omega = \omega_s - \omega_m \]
Substituting the values we calculated:
\[ \Delta \omega = \frac{2\pi}{60} - \frac{2\pi}{3600} \]

5. **Simplify the Expression**:
To simplify, we can find a common denominator:
\[ \Delta \omega = \frac{2\pi \cdot 60}{3600} - \frac{2\pi}{3600} = \frac{2\pi(60 - 1)}{3600} = \frac{2\pi \cdot 59}{3600} \]

6. **Final Result**:
Thus, the difference in angular speeds of the minute hand and the second hand is:
\[ \Delta \omega = \frac{59\pi}{1800} \text{ rad/s} \]

### Conclusion:
The difference between the angular speed of the minute hand and the second hand of a clock is \(\frac{59\pi}{1800}\) rad/s.

---