12.3 Clock Hands Angular Speed, Angle Between Hands at Given Time, Coincidence & Faulty Clocks

Angular Mechanics of the Clock Dial

A standard analog clock face is a circle of $360^\circ$. The circumference is divided into:

• $12$ equal hour spaces: $\frac{360^\circ}{12} = 30^\circ \text{ per hour space}$.
• $60$ equal minute spaces: $\frac{360^\circ}{60} = 6^\circ \text{ per minute space}$.

                                  12 [0° / 360°]
                             11         1
                        10                   2
                       9 [270°]      +      3 [90°]
                        8                    4
                             7          5
                                   6 [180°]

1. Speed of the Minute Hand

The minute hand completes one full revolution of $360^\circ$ (60 minute spaces) in 60 minutes:

$$\omega_m = \frac{360^\circ}{60 \text{ minutes}} = 6^\circ / \text{minute}$$

2. Speed of the Hour Hand

The hour hand completes one full revolution of $360^\circ$ in 12 hours (720 minutes), or covers one hour space ($30^\circ$) in 60 minutes:

$$\omega_h = \frac{360^\circ}{12 \times 60 \text{ minutes}} = \frac{360^\circ}{720 \text{ minutes}} = 0.5^\circ / \text{minute} = \frac{1}{2}^\circ / \text{minute}$$

3. Relative Speed of the Hands

Because both hands rotate in the same clockwise direction, the minute hand gains on the hour hand at a constant relative angular velocity:

$$\omega_{\text{rel}} = \omega_m - \omega_h = 6^\circ - 0.5^\circ = 5.5^\circ / \text{minute} = \frac{11}{2}^\circ / \text{minute}$$

4. Minute Space Gain Equivalence

In 60 minutes, the minute hand traverses 60 minute spaces while the hour hand traverses 5 minute spaces. Therefore:

• Relative Space Gain: The minute hand gains $60 - 5 = 55 \text{ minute spaces}$ in 60 minutes.
• Rate of Gain: $\frac{55 \text{ spaces}}{60 \text{ minutes}} = \frac{11}{12} \text{ space/minute}$.
• Time to Gain 1 Minute Space: $\frac{60}{55} = \frac{12}{11} \text{ minutes} = 1\frac{1}{11} \text{ minutes}$.

---

The Master Angle Formula

To compute the smaller angle $\theta$ between the minute hand and the hour hand at $H$ hours and $M$ minutes ($0 \le H \le 12$, $0 \le M < 60$):

1. From the 12 o'clock mark, the hour hand rotates through $30^\circ$ for each completed hour, plus $0.5^\circ$ for each elapsed minute:

$$\theta_h = 30H + \frac{1}{2}M$$

1. From the 12 o'clock mark, the minute hand rotates through $6^\circ$ for each elapsed minute:

$$\theta_m = 6M$$

1. First compute the raw absolute separation; reduce H modulo 12:

$$d = |\theta_h - \theta_m| = \left| \left(30H + \frac{1}{2}M\right) - 6M \right| = \left| 30H - \frac{11}{2}M \right|$$ $$d = \left| 30H - \frac{11}{2}M \right| = |30H - 5.5M|$$

+-----------------------------------------------------------------------------+
|                        MASTER ANGLE DECISION RULES                          |
+-----------------------------------------------------------------------------+
| 1. If raw separation d <= 180°:                                                            |
|    - Smaller Angle = d                                             |
|    - Other Angle            = 360° - d                                      |
|                                                                             |
| 2. If raw separation d > 180°:                                                             |
|    - Smaller Angle = 360° - d                                      |
|    - Other Angle            = d                                             |
+-----------------------------------------------------------------------------+

The raw difference $d=\lvert30H-5.5M\rvert$ is not always the smaller angle. Use $\theta=\min(d,360^\circ-d)$, which is between zero and 180 degrees. For example, at 10:10 the raw difference is 245 degrees but the smaller angle is 115 degrees.

Quick-Reference Angular Evaluations

---

Special Alignments & Recurrence Frequencies

RRB CBT examinations frequently ask for the exact number of times clock hands form specific geometric configurations across 12-hour and 24-hour periods.

1. Hands in Coincidence ($0^\circ$ Separation)

The hands point in the exact same direction (overlapping):

• Formula for Coincidence Time: Set $\theta = 0$:

$$30H - \frac{11}{2}M = 0 \implies M = \frac{60}{11}H$$

• Interval Between Successive Coincidences: For the minute hand to gain a full $360^\circ$ over the hour hand at relative speed $\frac{11}{2}^\circ/\text{min}$:

$$T = \frac{360^\circ}{\frac{11}{2}^\circ / \text{min}} = \frac{720}{11} \text{ minutes} = 65\frac{5}{11} \text{ minutes} \approx 65 \text{ min } 27.27 \text{ sec}$$

• Frequency:
  • In a half-open 12-hour interval: 11 times.
  • In 24 hours: 22 times.
  • The Missing Coincidence: Between 11:00 and 1:00 (a 2-hour window), coincidence does not occur twice; it occurs only once, at precisely 12:00:00.

2. Hands in Opposite Directions ($180^\circ$ Separation)

The hands form a straight line pointing in opposite directions:

• Formula: Set $|30H - \frac{11}{2}M| = 180^\circ$.
• Frequency:
  • In a half-open 12-hour interval: 11 times.
  • In 24 hours: 22 times.
  • The Shared Opposite Alignment: Between 5:00 and 7:00 (a 2-hour window), the hands point in opposite directions only once, at precisely 6:00:00.

3. Hands in a Straight Line (Together or Opposite)

A straight line includes both $0^\circ$ (coincidence) and $180^\circ$ (opposite):

$$\text{Total Straight Lines in 12 hours} = 11 + 11 = 22 \text{ times}$$ $$\text{Total Straight Lines in 24 hours} = 22 + 22 = 44 \text{ times}$$

4. Hands at Right Angles ($90^\circ$ Orthogonality)

The hands are perpendicular to each other:

• Formula: With H reduced modulo 12, solve $\lvert30H-5.5M\rvert=90^\circ$ or $270^\circ$, retaining only $0\le M<60$. Both raw separations give a smaller angle of 90 degrees.
• Frequency:
  • In 12 hours: 22 times (not 24!).
  • In 24 hours: 44 times (not 48!).
  • The Shared Right Angles: Under normal hours, hands form a right angle twice. However, in the 2-hour windows between 2:00–4:00 and 8:00–10:00, only 3 right angles occur per window instead of 4, because the 90° alignments at 3:00:00 and 9:00:00 are shared.

Master Clock Alignment Summary

---

Faulty & Incorrect Clocks: Gaining & Losing Time

Assume an ideal analog mechanism with both hands driven by the same uniformly fast or slow clock rate. A clock is described as fast (gaining time) if it registers more than 60 minutes when 60 true solar minutes have elapsed. It is described as slow (losing time) if it registers less than 60 minutes.

Proportional True-Time Correction Formula

If a faulty clock gains or loses $\Delta t$ minutes per hour:

$$\frac{\text{True Elapsed Time}}{\text{Faulty Dial Time}} = \frac{60 \text{ minutes}}{60 \pm \Delta t \text{ minutes}}$$ $$\text{True Time} = \text{Faulty Dial Time} \times \left( \frac{60}{60 \pm \Delta t} \right)$$

Coincidence-Based Faulty Clock Calculations

A standard problem in RRB Section Controller tests states: 'The hands of a clock coincide every $X$ minutes of true time. How much does the clock gain or lose in 24 hours?'

1. In a normal clock, hands coincide every $T_0 = 65\frac{5}{11} = \frac{720}{11} \text{ minutes}$.
2. If $X < 65\frac{5}{11}$, the clock is gaining time (running fast).
3. If $X > 65\frac{5}{11}$, the clock is losing time (running slow).
4. Gain or Loss in $X$ minutes $= \left( 65\frac{5}{11} - X \right) \text{ minutes}$.
5. Total gain or loss in 24 hours ($24 \times 60 = 1440 \text{ minutes}$):

$$\text{Total Gain/Loss in 24 hours} = \left( \frac{65\frac{5}{11} - X}{X} \right) \times 24 \times 60 \text{ minutes}$$

---

Worked Examples

Worked Example 1: Interior and Reflex Angle Determination

Problem: A Section Controller logs a freight train passing a signal cabin at exactly $7:24$ PM. Calculate: (a) The interior angle between the hour and minute hands, and (b) The reflex angle between the hands.

• Step 1: Identify Parameters: $H = 7$, $M = 24$.
• Step 2: Apply Master Angle Formula:

$$\theta = \left| 30H - \frac{11}{2}M \right| = |30(7) - 5.5(24)|$$ $$30 \times 7 = 210^\circ$$ $$5.5 \times 24 = \frac{11}{2} \times 24 = 11 \times 12 = 132^\circ$$ $$\theta = |210^\circ - 132^\circ| = 78^\circ$$

• Step 3: Evaluate Interior vs Reflex: Since $78^\circ \le 180^\circ$, the interior angle is $78^\circ$. The reflex angle is $360^\circ - 78^\circ = \mathbf{282^\circ}$.
• Conclusion: Interior angle is $78^\circ$; reflex angle is $282^\circ$.

---

Worked Example 2: Exact Time of Right Angle Alignment

Problem: At what exact time between 3 o'clock and 4 o'clock do the hands of a clock form a right angle ($90^\circ$) for the second time?

• Step 1: Formulate the Angular Condition: Between 3:00 and 4:00, $H = 3$. The angle formula is:

$$\left| 30(3) - \frac{11}{2}M \right| = 90^\circ \implies \left| 90 - \frac{11}{2}M \right| = 90$$

• Step 2: Solve the Two Algebraic Branches:
  • Branch 1 (First occurrence):

$$90 - \frac{11}{2}M = 90 \implies \frac{11}{2}M = 0 \implies M = 0$$

This represents exactly **3:00:00** (when the hands form their first right angle).

• Branch 2 (Second occurrence):

$$90 - \frac{11}{2}M = -90 \implies \frac{11}{2}M = 90 + 90 = 180$$ $$M = 180 \times \frac{2}{11} = \frac{360}{11} = 32\frac{8}{11} \text{ minutes}$$

• Step 3: State Exact Time: The second right angle occurs at $32\frac{8}{11}$ minutes past 3 (or approximately $3\text{h } 32\text{m } 43.6\text{s}$).
• Conclusion: The hands form a right angle for the second time at $32\frac{8}{11}$ minutes past 3.

---

Worked Example 3: True Time Calculation for a Gaining Clock

Problem: A railway station master's clock was set right at 8:00 AM on Monday. The clock is defective and gains 5 minutes every 24 hours. To the nearest minute, what is the true time when this clock indicates 1:00 PM on the following Wednesday?

• Step 1: Compute Elapsed Faulty Time:
  • Monday 8:00 AM to Wednesday 8:00 AM $= 48 \text{ hours}$.
  • Wednesday 8:00 AM to Wednesday 1:00 PM $= 5 \text{ hours}$.
  • Total elapsed faulty time $= 48 + 5 = 53 \text{ hours}$.
• Step 2: Establish the Proportionality Ratio: In 24 true hours, the faulty clock indicates:

$$24 \text{ hours} + 5 \text{ minutes} = 24 + \frac{5}{60} = 24 + \frac{1}{12} = \frac{289}{12} \text{ hours}$$

Therefore, $\frac{289}{12}$ hours on the faulty dial corresponds to $24$ true hours.

• Step 3: Calculate True Time for 53 Faulty Hours:

$$\text{True Hours} = 53 \times \left( \frac{24}{\frac{289}{12}} \right) = 53 \times \frac{288}{289} = \frac{15264}{289} \approx 52.8166 \text{ hours}$$

Convert to hours and minutes:

$$52 \text{ hours} + (0.8166 \times 60) \text{ minutes} \approx 52 \text{ hours } 49 \text{ minutes}$$

• Step 4: Determine True Clock Reading: 53 faulty hours minus 52 hours 49 minutes $= 11 \text{ minutes}$ ahead. Subtracting 11 minutes from 1:00 PM yields 12:49 PM.
• Conclusion: The true time is approximately 12:49 PM on Wednesday (the clock is about 11 minutes fast).

The master absolute-difference expression gives a raw separation $d$, which can exceed $180^\circ$. Always take $\theta=\min(d,360^\circ-d)$ for the smaller angle; “acute” is not a synonym for “smaller”. Count periodic events on a half-open interval to avoid counting the same endpoint twice.