The ratio of angular speeds of minutes hand and seconds hand in a watch is

Options

A. `60:1`
B. `30:1`
C. `1:30`
D. `1:60`

Explanation

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

### Step 1: Understand the concept of angular speed
Angular speed (ω) is defined as the angle covered per unit time. It can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
where \(T\) is the time taken to complete one full revolution (in seconds).

### Step 2: Calculate the angular speed of the minute hand
The minute hand completes one full revolution (2π radians) in 60 minutes. We need to convert this time into seconds:
\[ T_m = 60 \text{ minutes} = 60 \times 60 = 3600 \text{ seconds} \]
Now, we can calculate the angular speed of the minute hand (ω_m):
\[ \omega_m = \frac{2\pi}{T_m} = \frac{2\pi}{3600} = \frac{\pi}{1800} \text{ radians per second} \]

### Step 3: Calculate the angular speed of the second hand
The second hand completes one full revolution (2π radians) in 60 seconds:
\[ T_s = 60 \text{ seconds} \]
Now, we can calculate the angular speed of the second hand (ω_s):
\[ \omega_s = \frac{2\pi}{T_s} = \frac{2\pi}{60} = \frac{\pi}{30} \text{ radians per second} \]

### Step 4: Find the ratio of angular speeds
Now, we need to find the ratio of the angular speed of the minute hand to that of the second hand:
\[ \text{Ratio} = \frac{\omega_m}{\omega_s} = \frac{\frac{\pi}{1800}}{\frac{\pi}{30}} \]
We can simplify this:
\[ \text{Ratio} = \frac{\pi}{1800} \times \frac{30}{\pi} = \frac{30}{1800} = \frac{1}{60} \]

### Conclusion
The ratio of the angular speeds of the minute hand to the second hand is:
\[ 1 : 60 \]