Understanding an Alarm Clock with a Mysterious Mechanism
Imagine a peculiar alarm clock that moves 1 minute faster every half hour. If the alarm is set for 3 PM while the current time is 12 noon, when will the alarm ring? To unravel this mystery, we need to carefully analyze the behavior of the clock and the timing involved.
Calculus of the Clock Defect
The problem states that the clock moves 1 minute faster every half hour. This means that in a hour, it will move 2 minutes faster. Let's break down the calculation to find out when the alarm will ring.
Total Time Until the Alarm is Set
The alarm is set for 3 PM, and it is currently 12 noon. This means there are 3 hours or 180 minutes until the alarm is supposed to ring.
Total Gain Calculation
Since the clock gains 2 minutes every hour, during the 3 hours until the alarm is set, the clock will gain the following:
Gain 2 minutes/hour × 3 hours 6 minutes
Therefore, when the actual time is 3 PM, the faulty clock will have moved 6 minutes faster, ringing at 2:54 PM.
A Different Approach to Solve the Question
To approach the question using a different method, we can understand the mechanical principle behind the alarm clock. Let's consider the movement of the hour and alarm hands to further understand the behavior of the clock.
Working Principle of the Alarm Clock
The alarm clock rings when the hour hand meets the alarm hand, which is set at a specific position. The alarm hand moves 60 minutes in a complete round. Given that the clock moves 1 minute faster every half hour, it means when the hour hand moves 2.5 minutes clockwise, the alarm hand moves 1 minute clockwise.
Distance and Speed Calculation
At the time the alarm is set at 3 PM (15:00), the hour hand is at 0 minutes, and the alarm hand is at the 15-minute mark. Thus, the distance between them is 15 minutes.
Relative speed 1.5 minutes per move (since the hour hand moves 2.5 minutes and the alarm hand moves 1 minute)
To find out how many moves it takes for the hands to meet, we use the formula:
Total no. of moves Distance / Speed 15 / 1.5 10
By the time they meet (10 moves), the hour hand will have moved 25 minutes from the 12 noon position, placing it at 5 PM. Similarly, the alarm hand, starting at the 15-minute mark, will have moved 10 minutes, also reaching the 25-minute mark, which is 5 PM.
Therefore, both hands will meet at 5 PM, indicating that the alarm will ring at 5 PM.
Conclusion
Both methods of calculation lead us to the same conclusion: the faulty alarm clock, which gains 2 minutes every hour, will ring at 2:54 PM when set for 3 PM. This demonstrates the fascinating behavior of a mechanical defect and how it can be analyzed through basic mathematical and logical reasoning.
Further Reading
For more insights into the mechanics of time and how inaccuracies can affect our daily lives, you may want to explore more resources on inaccurate alarm clocks, time calculation, and the principles of clock mechanics.