We've covered this topic in our YouTube class. In this post you'll find the class recording, downloadable PDFs, and practice questions.
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Clock (घड़ी - तर्कशक्ति)
1. Fundamental Hand Speeds & Relative Motion Theorems
Clock dial angles are governed by constant angular velocities of the hands traversing the circular dial of \(360^\circ\)[cite: 15]:
- Minute Hand Speed (मिनट की सुई की गति): Moves \(360^\circ\) in \(60\text{ minutes} = 6^\circ\text{ per minute}\)[cite: 15].
- Hour Hand Speed (घंटे की सुई की गति): Moves \(360^\circ\) in \(12\text{ hours} (720\text{ min}) = 30^\circ\text{ per hour} = 0.5^\circ\text{ per minute}\)[cite: 15].
- Relative Speed Advantage (सापेक्ष गति): The minute hand gains over the hour hand at: $$\text{Relative Speed} = 6^\circ - 0.5^\circ = 5.5^\circ\text{ per min} = \frac{11}{2}^\circ\text{ per min}$$[cite: 15] In \(60\text{ minutes}\), the minute hand gains \(55\text{ minute spaces}\), establishing the fundamental scaling factor \(\frac{60}{55} = \frac{12}{11}\)[cite: 15].
- Angle Master Formula: Between hour \(H\) and minute \(M\): $$\theta = \left|30H - \frac{11}{2}M\right|$$[cite: 15] The reflex angle is \(360^\circ - \theta\)[cite: 15].
- Coincidence (\(0^\circ\) / संपाती): Occurs \(11\text{ times in } 12\text{ hours}\) and \(22\text{ times in } 24\text{ hours}\) (the overlap between 11:00 and 1:00 merges at exactly 12:00)[cite: 15]: $$M = \frac{2}{11} \times (30H)$$[cite: 15] Successive overlaps repeat precisely every \(\frac{720}{11} = 65\frac{5}{11}\text{ minutes}\)[cite: 15].
- Straight Line Opposite (\(180^\circ\) / विपरीत): Occurs \(11\text{ times in } 12\text{ hours}\) and \(22\text{ times in } 24\text{ hours}\) (single occurrence between 5:00 and 7:00 at 6:00)[cite: 15]: $$M = \frac{2}{11} \times |30H - 180|$$[cite: 15]
- Right Angle (\(90^\circ\) / समकोण): Occurs \(2\text{ times per hour}\), \(22\text{ times in } 12\text{ hours}\), and \(44\text{ times in a full day (24 hours)}\)[cite: 15]: $$M = \frac{2}{11} \times (30H \pm 90)$$[cite: 15]
- Reflection Invariants (दर्पण व जल प्रतिबिम्ब):
- Mirror Reflection: Subtract given time from \(11:60\) (or \(23:60\))[cite: 15].
- Water Reflection: Subtract from \(18:30\) (or borrow 60 minutes to use \(17:90\) when minutes exceed 30)[cite: 15].
2. Faulty Clock & Gaining/Losing Rates
- Repetition Dial Cycle: For a slow or fast clock to indicate the exact correct time again, it must accumulate a net error of \(12\text{ hours} = 720\text{ minutes}\)[cite: 15].
- Relative Divergence Between Clocks: If Clock A gains \(x\text{ min/hr}\) and Clock B loses \(y\text{ min/hr}\), relative separation is \(x + y\text{ min/hr}\)[cite: 15]. They will display identical times after \(\frac{720}{x+y}\text{ hours}\)[cite: 15].
3. Worked Examples
- Angle at 5:40: $$\theta = \left|30(5) - \frac{11}{2}(40)\right| = |150 - 220| = 70^\circ$$[cite: 15]
- Coincidence Between 2 and 3 O'Clock: $$M = \frac{2}{11} \times (2 \times 30) = \frac{120}{11} = 10\frac{10}{11}\text{ min} \implies 2:10\frac{10}{11}$$[cite: 15]
- Water Image of 1:50: Since \(50 > 30\), use base \(17:90\)[cite: 15]: $$17:90 - 01:50 = 16:40 \equiv 4:40$$[cite: 15]
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