Question

The ratio of angular speeds of the minute hand and second hand of a watch is

• A. \(1:12 \)
• B. \(1:6 \)
• C. \(1:60 \)
• D. \(12:1 \)
• E. \(60:1 \)

Hint:

Smaller time period → larger angular speed.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Angular speed (\(\omega\)) is defined as the rate of change of angular displacement (\(\theta\)) with respect to time (\(t\)), i.e., \(\omega = \frac{\Delta\theta}{\Delta t}\). To find the ratio of angular speeds, we can calculate the angular speed for each hand by considering the angle it covers in a specific time period.
Step 2: Key Formula or Approach:
1. Calculate the angular speed of the minute hand (\(\omega_{min}\)). 2. Calculate the angular speed of the second hand (\(\omega_{sec}\)). 3. Find the ratio \(\omega_{min} : \omega_{sec}\). A full circle corresponds to an angular displacement of \(2\pi\) radians.
Step 3: Detailed Explanation:
Angular speed of the minute hand (\(\omega_{min}\)): The minute hand completes one full revolution (\(2\pi\) radians) in 60 minutes. Time period, \(T_{min} = 60 \text{ minutes} = 60 \times 60 \text{ seconds} = 3600 \text{ s}\). \[ \omega_{min} = \frac{2\pi}{T_{min}} = \frac{2\pi}{3600} \text{ rad/s} \] Angular speed of the second hand (\(\omega_{sec}\)): The second hand completes one full revolution (\(2\pi\) radians) in 60 seconds. Time period, \(T_{sec} = 60 \text{ s}\). \[ \omega_{sec} = \frac{2\pi}{T_{sec}} = \frac{2\pi}{60} \text{ rad/s} \] Ratio of the angular speeds: We need to find the ratio \(\omega_{min} : \omega_{sec}\). \[ \frac{\omega_{min}}{\omega_{sec}} = \frac{2\pi/3600}{2\pi/60} = \frac{60}{3600} = \frac{1}{60} \] So, the ratio is 1:60.
Step 4: Final Answer:
The ratio of the angular speeds of the minute hand to the second hand is 1:60.