The ratio of angular speed of a second-had to the hour-hand of a watch is

Options

A. `3600:1`
B. `720:1`
C. `72:1`
D. `60:1`

Explanation

To find the ratio of the angular speed of the second hand to the hour hand of a watch, we can follow these steps:

### Step 1: Understand Angular Speed
Angular speed (ω) is defined as the angle (θ) covered per unit time (t). Mathematically, it is given by:
\[ \omega = \frac{\theta}{t} \]

### Step 2: Determine the Angular Displacement for Each Hand
- The second hand completes a full circle (360 degrees or \(2\pi\) radians) in 60 seconds.
- The hour hand completes a full circle (360 degrees or \(2\pi\) radians) in 12 hours (or 43200 seconds).

### Step 3: Calculate Angular Speed for Each Hand
1. **Angular Speed of the Second Hand (ω_second)**:
\[ \omega_{\text{second}} = \frac{2\pi \text{ radians}}{60 \text{ seconds}} = \frac{\pi}{30} \text{ radians/second} \]

2. **Angular Speed of the Hour Hand (ω_hour)**:
\[ \omega_{\text{hour}} = \frac{2\pi \text{ radians}}{43200 \text{ seconds}} = \frac{\pi}{21600} \text{ radians/second} \]

### Step 4: Find the Ratio of Angular Speeds
Now, we can find the ratio of the angular speed of the second hand to the hour hand:
\[ \text{Ratio} = \frac{\omega_{\text{second}}}{\omega_{\text{hour}}} = \frac{\frac{\pi}{30}}{\frac{\pi}{21600}} \]

### Step 5: Simplify the Ratio
The π cancels out:
\[ \text{Ratio} = \frac{1/30}{1/21600} = \frac{21600}{30} = 720 \]

### Conclusion
Thus, the ratio of the angular speed of the second hand to the hour hand is:
\[ \text{Ratio} = 720:1 \]