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 of a clock, we can follow these steps:

### Step 1: Determine the angular velocities of the hands
- The second hand completes one full revolution (360 degrees or \(2\pi\) radians) in 60 seconds. Therefore, the angular velocity of the second hand (\(\omega_{\text{second}}\)) is:
\[ \omega_{\text{second}} = \frac{2\pi \text{ radians}}{60 \text{ seconds}} = \frac{\pi}{30} \text{ radians/second} \]

- The minute hand completes one full revolution in 60 minutes (or 3600 seconds). Thus, the angular velocity of the minute hand (\(\omega_{\text{minute}}\)) is:
\[ \omega_{\text{minute}} = \frac{2\pi \text{ radians}}{3600 \text{ seconds}} = \frac{\pi}{1800} \text{ radians/second} \]

### Step 2: Calculate the relative angular velocity
- The relative angular velocity (\(\omega\)) of the second hand with respect to the minute hand is given by:
\[ \omega = \omega_{\text{second}} - \omega_{\text{minute}} = \frac{\pi}{30} - \frac{\pi}{1800} \]

- To perform the subtraction, we need a common denominator. The least common multiple of 30 and 1800 is 1800. Thus, we convert:
\[ \frac{\pi}{30} = \frac{60\pi}{1800} \]
\[ \omega = \frac{60\pi}{1800} - \frac{\pi}{1800} = \frac{59\pi}{1800} \text{ radians/second} \]

### Step 3: Calculate the time period of meeting
- The time period (\(T\)) for the hands to meet can be calculated using the formula:
\[ T = \frac{2\pi}{\omega} \]

- Substituting the value of \(\omega\):
\[ T = \frac{2\pi}{\frac{59\pi}{1800}} = \frac{2\pi \times 1800}{59\pi} = \frac{3600}{59} \text{ seconds} \]

### Final Answer
The time period of the meeting of the minute hand and the second hand is:
\[ T \approx 61.02 \text{ seconds} \]

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