The Overlapping of Clock Hands

One of the fascinating aspects of analog clocks is the continuous movement of their hands. A common question often asked is: When will the hour and minute hands overlap after a certain time? Let's explore this problem with a detailed breakdown and solve a related question.

The First Overlapping After 12 Noon

The hour hand has an angular speed of 6° per minute.

The minute hand has an angular speed of 360° per minute.

To find the time when the hour and minute hands overlap after 12 noon, we can use the following equation:

360t - 360 6t

Solving for t, we get:

360t - 6t 360

354t 360

t 360 / 354

t ≈ 6.0674 minutes

Therefore, after 12 noon, the next overlapping of the hour and minute hands is at approximately 12:06:04.

The Next Overlap at 1:05:27 PM

Calculating the exact time when the minute and hour hands overlap after 1:00 PM, we can use the equation:

130 - 5.5x 0

5.5x 30

x 30 / 5.5 ≈ 5.4545 minutes

This gives us a time of 1:05:27.27 PM.

Starting Point and Context

The question itself is incomplete as it does not specify the starting point. If it is 1 minute before 12, then the next overlap would indeed be at 12:06:04. However, during practical use, such as with a watch, the context and start time play crucial roles.

Curious Fact About Overlapping

Interestingly, there are eleven times when the hands of an analog clock align from 12 o'clock on a minute until noon. This might seem counterintuitive as one might initially think it is twelve times. However, the angles and relative speeds make the first overlap at 01:05:33 and the tenth overlap at 10:54:27.

The time between these overlaps can be calculated using the formula derived earlier. For instance, the time taken between the first and second overlaps is approximately 65 minutes and 27.27 seconds after 12 o'clock.

Thus, the question might have been asking for the time taken between 12 o'clock and the next overlap, which would be approximately 1 hour 5 minutes and 33 seconds.

Conclusion

Continuous movement and relative circular motion play key roles in understanding how and when the hands of a clock align. Here, we have explored the overlap problem from different perspectives and provided a detailed solution using relative speeds.