Understanding Clock Mechanics and Angular Speed
To master clock problems, we must first view the clock as a 360-degree circular path. A standard clock is divided into 12 hour-marks, representing 360 degrees. Therefore, each hour space represents 30 degrees (360°/12 = 30°). When we look at the movement of the hands, we realize that they move at constant, distinct speeds.
The minute hand completes a full circle (360°) in 60 minutes. This means its speed is 6° per minute. In contrast, the hour hand completes a full circle in 12 hours (720 minutes). Consequently, its speed is 0.5° per minute. This difference in speed is the fundamental logic behind every calculation in this chapter.
The relative speed of the minute hand with respect to the hour hand is 5.5° per minute (6° – 0.5° = 5.5°). This value is the “magic number” used to solve almost all problems regarding the coincidence or separation of hands.
Calculating the Angle Between Hands
In competitive exams, you are frequently asked to calculate the angle between the hands at a specific time, such as 4:20 or 8:15. While you could draw a diagram, the most efficient approach is to use the standard angular formula. This formula accounts for the displacement of the hour hand due to the minutes passed.
The formula for the angle (θ) between the hands at H hours and M minutes is: θ = |30H – 5.5M|. Here, H represents the hour (1 to 12) and M represents the minutes. If the result is greater than 180°, subtract it from 360° to get the reflex angle, as the question usually asks for the smaller angle.
- Example: Find the angle at 3:30.
- H = 3, M = 30.
- θ = |30(3) – 5.5(30)| = |90 – 165| = 75°.
Coincidence, Right Angles, and Straight Lines
Hands of a clock follow specific patterns of alignment throughout a 12-hour cycle. The minute hand gains 360° over the hour hand every 65 5/11 minutes. This leads to the phenomenon where hands overlap, form right angles, or point in opposite directions.
Hands coincide (overlap) 11 times in every 12 hours. They do not coincide 12 times because the overlap between 11:00 and 1:00 occurs exactly at 12:00, covering both hours. Similarly, they form a straight line (180°) 11 times in 12 hours. Understanding these frequency constraints prevents common errors in time-based logic questions.
Faulty Clocks: Gain and Loss
Problems involving faulty clocks test your ability to work with ratios. If a clock gains or loses time, it is essentially moving at a speed different from the standard reference time. To solve these, compare the “incorrect time” elapsed with the “correct time” elapsed using a proportion.
If a clock gains 5 minutes every hour, for every 60 minutes of correct time, the clock shows 65 minutes. You can set up an equation: (Correct Time) / (Faulty Time) = (Standard Rate) / (Faulty Rate). This ratio method is far more reliable than guessing or manual counting for long durations like 24 or 48 hours.
Important Facts and Formulas
| Condition | Frequency (12 Hours) | Frequency (24 Hours) |
|---|---|---|
| Coincidence (0°) | 11 times | 22 times |
| Opposite (180°) | 11 times | 22 times |
| Right Angle (90°) | 22 times | 44 times |
Previous Year Question Hints
- Type 1: “At what time between 2 and 3 o’clock will the hands be at 90 degrees?” Use the formula |30H – 5.5M| = 90. Solve for M.
- Type 2: “A clock loses 10 minutes every 24 hours. When will it show the correct time again?” Calculate the total loss needed to reach 12 hours (720 minutes) and multiply by the rate.
Quick Revision Summary
- Minute hand moves at 6°/min; Hour hand moves at 0.5°/min.
- Relative speed is 5.5°/min.
- Angle formula: θ = |30H – 5.5M|.
- Hands overlap 11 times in 12 hours.
- Right angles occur 22 times in 12 hours.
- Always check if the answer requires a reflex angle (>180°).
- For faulty clocks, use the ratio of (Correct Time / Faulty Time).
- The hour hand moves only 30° in one hour.
