In a clock, what is the time period of meeting of the minute hand and the second hand ?

Options

A. `59s`
B. `(60)/(59)s`
C. `(59)/(60)s`
D. `(3600)/(59)s`

Explanation

To find the time period of the meeting of the minute hand and the second hand on a clock, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Movement of the Hands**:
- The minute hand completes one full revolution (360 degrees or \(2\pi\) radians) in 60 minutes (3600 seconds).
- The second hand completes one full revolution in 60 seconds.

2. **Calculate Angular Velocities**:
- The angular velocity of the minute hand (\(\omega_m\)) is:
\[ \omega_m = \frac{2\pi \text{ radians}}{3600 \text{ seconds}} = \frac{\pi}{1800} \text{ radians/second} \]
- The angular velocity of the second hand (\(\omega_s\)) is:
\[ \omega_s = \frac{2\pi \text{ radians}}{60 \text{ seconds}} = \frac{\pi}{30} \text{ radians/second} \]

3. **Relative Angular Velocity**:
- The relative angular velocity (\(\omega_{rel}\)) of the second hand with respect to the minute hand is:
\[ \omega_{rel} = \omega_s - \omega_m = \frac{\pi}{30} - \frac{\pi}{1800} \]
- To subtract these, we need a common denominator. The least common multiple of 30 and 1800 is 1800:
\[ \omega_{rel} = \frac{60\pi}{1800} - \frac{\pi}{1800} = \frac{59\pi}{1800} \text{ radians/second} \]

4. **Time to Meet**:
- The second hand and minute hand start at the same position at 12:00. To find the time taken for them to meet again, we need to find how long it takes for the second hand to gain a full \(2\pi\) radians over the minute hand:
\[ \text{Time} (t) = \frac{\text{Angle to gain}}{\text{Relative Angular Velocity}} = \frac{2\pi}{\frac{59\pi}{1800}} = \frac{2\pi \cdot 1800}{59\pi} = \frac{3600}{59} \text{ seconds} \]

5. **Conclusion**:
- The time period of the meeting of the minute hand and the second hand is:
\[ t = \frac{3600}{59} \text{ seconds} \]
- This corresponds to option 2 in the given choices.

### Final Answer:
The time period of meeting of the minute hand and the second hand is \(\frac{3600}{59}\) seconds.

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