Clock

QUIZ

Clock

A clock is a device used to measure and indicate time. An ordinary analog clock has two hands:

  • Hour hand (H.H.) — the shorter hand, which shows the hour
  • Minute hand (M.H.) — the longer hand, which shows the minutes

The clock face is divided into:

  • 12 equal hour divisions, numbered from 1 to 12
  • 60 equal minute divisions around the dial

Since the clock has 60 minute divisions and only 12 hour numbers, each hour number is separated by:

\[\frac{60}{12} = 5 \text{ minute divisions}\]

Angular Movement

A complete rotation around the clock dial corresponds to a full circle:

\[1 \text{ complete circle} = 360^\circ\]

Hence:

  • One minute division corresponds to

    \[\frac{360^\circ}{60} = 6^\circ\]

  • Five minute divisions (distance between two consecutive hour numbers) correspond to

    \[6^\circ \times 5 = 30^\circ\]

Therefore:

  • The hour hand moves $30^\circ$ in one hour

In one minute:

  • Hour hand moves

    \[\frac{30^\circ}{60} = \frac{1}{2}^\circ\]

  • Minute hand moves

    \[6^\circ\]

Thus, in one minute, the minute hand gains over the hour hand:

\[6^\circ - \frac{1}{2}^\circ = 5\frac{1}{2}^\circ\]

This relative gain is the key idea used in solving all clock problems.

Relative Motion of Hands

Since the minute hand gains $5\frac{1}{2}^\circ$ per minute over the hour hand, in one hour (60 minutes) it gains:

\[5\frac{1}{2}^\circ \times 60 = 330^\circ\]

As a full circle is $360^\circ$, the remaining angle is:

\[360^\circ - 330^\circ = 30^\circ\]

This shows that the relative position of the two hands repeats after the minute hand gains $360^\circ$ over the hour hand.

Hence:

  • Any particular relative position of the two hands occurs 11 times in 12 hours

Special Positions of Clock Hands

Certain positions of the hands are of special importance:

  • Coincidence: both hands overlap and are in the same direction
  • Opposite direction: hands are $180^\circ$ apart, corresponding to 30 minute divisions
  • Right angle: hands are $90^\circ$ apart, corresponding to 15 minute divisions

From the relative motion analysis, the following standard results are obtained:

  • In every hour, the hands coincide once
  • In 12 hours, the hands coincide 11 times
  • In a day (24 hours), the hands coincide 22 times
  • In 12 hours, the hands are in opposite directions 11 times
  • In 12 hours, the hands are at right angles 22 times
  • In one hour, the hands are at right angles 2 times
  • In one hour, the hands are in opposite directions once
  • In a day, the hands are at right angles 44 times

Time Between Coincidences

Since the hands coincide 11 times in 12 hours, the time between two successive coincidences is:

\[\frac{12 \times 60}{11} = 65\frac{5}{11}\ \text{minutes}\]

This value is taken as the standard interval between coincidences for a correct clock.

  • If the hands coincide in less than $65\frac{5}{11}$ minutes, the clock is said to be fast
  • If the hands coincide in more than $65\frac{5}{11}$ minutes, the clock is said to be slow

Fast and Slow Clock

A clock is said to be:

  • Fast, if it shows a time ahead of the correct time
    Example: Showing 6:10 when the correct time is 6:00
  • Slow, if it shows a time behind the correct time
    Example: Showing 5:50 when the correct time is 6:00

If the hands of a clock coincide at an interval of $x$ minutes:

  • When $x < 65\frac{5}{11}$, the clock is fast and the time gained per hour is:

    \[\frac{65\frac{5}{11} - x}{x}\ \text{minutes}\]

  • When $x > 65\frac{5}{11}$, the clock is slow and the time lost per hour is:

    \[\frac{x - 65\frac{5}{11}}{x}\ \text{minutes}\]

These relations are frequently used in numerical problems involving gain or loss of time in clocks.