The ratio of the angular speed of the hour and the minute hand of a clock is

Options

A. `1:12`
B. `1:6`
C. `1:8`
D. `12:1`

Explanation

To find the ratio of the angular speeds of the hour hand and the minute hand of a clock, we can follow these steps:

### Step 1: Understand the Concept of Angular Speed
Angular speed (ω) is defined as the angle rotated per unit time. For a complete rotation (360 degrees or 2π radians), we need to know how long it takes for each hand to complete one full rotation.

### Step 2: Determine the Time Period for Each Hand
- **Minute Hand**: The minute hand completes one full rotation (2π radians) in 60 minutes.
- **Hour Hand**: The hour hand completes one full rotation (2π radians) in 12 hours.

### Step 3: Convert Time Periods to Seconds
- **Minute Hand**: 60 minutes = 60 × 60 seconds = 3600 seconds.
- **Hour Hand**: 12 hours = 12 × 60 × 60 seconds = 43200 seconds.

### Step 4: Calculate Angular Speed for Each Hand
- **Angular Speed of Minute Hand (ω_m)**:
\[ ω_m = \frac{2π \text{ radians}}{3600 \text{ seconds}} = \frac{2π}{3600} \text{ rad/s} \]

- **Angular Speed of Hour Hand (ω_h)**:
\[ ω_h = \frac{2π \text{ radians}}{43200 \text{ seconds}} = \frac{2π}{43200} \text{ rad/s} \]

### Step 5: Find the Ratio of Angular Speeds
Now, we can find the ratio of the angular speed of the hour hand to that of the minute hand:
\[ \text{Ratio} = \frac{ω_h}{ω_m} = \frac{\frac{2π}{43200}}{\frac{2π}{3600}} \]

### Step 6: Simplify the Ratio
The \(2π\) terms cancel out:
\[ \text{Ratio} = \frac{1}{43200} \times \frac{3600}{1} = \frac{3600}{43200} = \frac{1}{12} \]

### Final Result
Thus, the ratio of the angular speed of the hour hand to the minute hand is:
\[ \text{Ratio} = 1 : 12 \]