The ratio of angular speeds of minute hand and hour hand of a watch is

Options

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

Explanation

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

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

### Step 2: Calculate Angular Speed of the Minute Hand
The minute hand completes one full revolution (2π radians) in 60 minutes. Therefore, the angular speed of the minute hand (ω_m) can be calculated as:
\[ \omega_m = \frac{2\pi \text{ radians}}{60 \text{ minutes}} = \frac{2\pi}{60} \text{ radians/minute} \]

### Step 3: Calculate Angular Speed of the Hour Hand
The hour hand completes one full revolution (2π radians) in 12 hours (which is 720 minutes). Therefore, the angular speed of the hour hand (ω_h) can be calculated as:
\[ \omega_h = \frac{2\pi \text{ radians}}{720 \text{ minutes}} = \frac{2\pi}{720} \text{ radians/minute} \]

### Step 4: Find the Ratio of Angular Speeds
Now, we can find the ratio of the angular speeds of the minute hand to the hour hand:
\[ \text{Ratio} = \frac{\omega_m}{\omega_h} = \frac{\frac{2\pi}{60}}{\frac{2\pi}{720}} \]

### Step 5: Simplify the Ratio
The \(2\pi\) cancels out:
\[ \text{Ratio} = \frac{1/60}{1/720} = \frac{720}{60} = 12 \]

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