The problem asks to find the next time after 12:00 when the minute and hour hands of a clock overlap. The hint suggests using the concept of rotational motion.

1. Problem Description

The problem asks to find the next time after 12:00 when the minute and hour hands of a clock overlap. The hint suggests using the concept of rotational motion.

2. Solution Steps

Let the angular speed of the hour hand be $\omega_h$ and the angular speed of the minute hand be $\omega_m$. We know that the minute hand completes a full rotation ($2\pi$ radians) in 60 minutes, so

$$\omega_m = \frac{2\pi}{60} = \frac{\pi}{30} \text{ rad/min}$$

The hour hand completes a full rotation ($2\pi$ radians) in 12 hours (720 minutes), so

$$\omega_h = \frac{2\pi}{720} = \frac{\pi}{360} \text{ rad/min}$$

Let $t$ be the time in minutes after 12:00 when the minute and hour hands overlap again.

The angle covered by the minute hand is $\theta_m = \omega_m t = \frac{\pi}{30} t$.

The angle covered by the hour hand is $\theta_h = \omega_h t = \frac{\pi}{360} t$.

Since we are looking for the first time after 12:00 when they overlap, the difference in the angles covered by the minute and hour hands must be $2\pi$ radians (one full rotation). So,

$$\theta_m - \theta_h = 2\pi$$

$$\frac{\pi}{30} t - \frac{\pi}{360} t = 2\pi$$

Divide both sides by $\pi$:

$$\frac{t}{30} - \frac{t}{360} = 2$$

Multiply both sides by 360:

$$12t - t = 720$$

$$11t = 720$$

$$t = \frac{720}{11} \approx 65.45 \text{ minutes}$$

The time is approximately 65.45 minutes after 12:

0

0. Convert the decimal part of the minutes to seconds:

$0.45 \text{ minutes} \times 60 \text{ seconds/minute} \approx 27 \text{ seconds}$.

So, the time is approximately 1 hour, 5 minutes, and 27 seconds. More precisely,

$$t = \frac{720}{11} \text{ minutes} = 65\frac{5}{11} \text{ minutes}$$

$$\frac{5}{11} \text{ minutes} = \frac{5}{11} \times 60 \text{ seconds} = \frac{300}{11} \text{ seconds} \approx 27.27 \text{ seconds}$$

So the time when they meet is 1:05:27.27 approximately.

3. Final Answer

The time when the minute and hour hands overlap again is $\frac{720}{11}$ minutes after 12:00, which is approximately 1:05:

2

7. The time is $\frac{720}{11}$ minutes.