Understanding the First 120° Angle Between Hour and Minute Hands on a Clock

The interesting mechanics of a clock's hands reveal fascinating insights into the symmetries and ratios involved. A common question often posed is: After how many minutes from 12:00 o'clock noon will the minute hand and the hour hand of a clock form an angle of 120° for the first time?

Mathematical Approach to the Problem

To find out when the minute hand and the hour hand of a clock make an angle of 120° for the first time after 12:00 noon, we can use the following steps:

Position of the Hour Hand and the Minute Hand

The hour hand moves at a rate of 0.5 degrees per minute. This rate is derived from the fact that the hour hand completes a full 360-degree circle in 12 hours, or 30 degrees per hour, which translates to 30/60 0.5 degrees per minute. At t minutes after 12:00, the angle of the hour hand from the 12 o'clock position is:

Angle of the hour hand: 0.5t degrees

The minute hand moves at a rate of 6 degrees per minute. This is because it completes a full 360-degree circle in 60 minutes. At t minutes after 12:00, the angle of the minute hand from the 12 o'clock position is:

Angle of the minute hand: 6t degrees

Calculation of the Angle Between the Hands

The angle between the hour hand and the minute hand can be calculated by subtracting the angle of the hour hand from the angle of the minute hand:

Angle between the hand: 6t - 0.5t 5.5t degrees

Solving for the First 120° Angle

We want this angle to be 120°, so we set up the following equation:

5.5t 120

Solving for t:

t 120 / 5.5 ≈ 21.8181 minutes

Converting Minutes and Seconds

- The integer part of 21.8181 minutes is 21 minutes.
- The decimal part, 0.8181 minutes, can be converted to seconds by multiplying by 60:

0.8181 × 60 ≈ 49.09 seconds

Thus, the minute hand and the hour hand will for the first time form an angle of 120° approximately 21 minutes and 49 seconds after 12:00 noon.

In summary, the answer is 21 minutes and 49 seconds.

Alternative Solution Using Ratios

Another approach involves using ratios. Since 120° is equivalent to 1/3 of a revolution, and the hour hand makes 1 revolution every 12 hours or 1/12 of a revolution every hour, while the minute hand completes 1 full revolution per hour, we can determine the time when the hands form a 120° angle:

11/12 hours / 1 hour 1/3 hours / n hours

11/12n 1/3

n 4/11 of an hour

4/11 of an hour is approximately 1309 seconds or 21 minutes 49 seconds.

Key Takeaways

The angle between the hour and minute hands of a clock can be calculated using the rates of angles moved by each hand. Ratios can be used to solve similar problems, providing an alternative method. Understanding these concepts enhances one's appreciation for the symmetry and precision in mechanical clocks.

By exploring such problems, we not only deepen our mathematical understanding but also gain insight into the fascinating world of timekeeping and mechanics.