Problems on
Clocks - Concept
1. The
face of the dial of a watch is a circle whose circumference is divided into 60
equal parts called minute spaces.
2. A
clock has two hands; the smaller one is called the hour hand or short hand
while the larger one is called the minute hand or long hand.
3. In
60 minutes, the minute hand covers 60 minute spaces and the hour hand covers 5
minute spaces. So in 1 hour, the minute hand gains 55 minute spaces on the hour
hand.
4. Minute
hand covers 360° in 1 hour, i.e., in 60 minutes. Hence, the minute hand covers
6° per minute.
1-minute space = 3600/60 =60
(3600 of the circle is divided into 60 minutes).
5. Hour
hand covers 360° in 12 hours. Hence, hour hand moves 30° in an hour i.e., 1/2° per minute.
In one hour, the hour hand moves 5-minute
spaces = 5 x 60 = 300.
In one minute, hour hand movies 300/
60 = ½0
6. Thus
in one minute, the minute hand gains 5 ½0 over the hour hand.
7. The hour hand and the minute hand
of the clock move in relation to each other continuously, and at any given
point in time, they form an angle between 0° and 180° with each other.
8. If
the time shown by the clock is known, the angle between the hands can be
calculated. Similarly, if the angle between two hands is known, the time shown
by the clock can be found.
When
we say angle between the hands, we normally refer to the acute/obtuse angles
(up to 180°) between the two hands and not the reflex angle (> 180°).
9. Hands make an angle of
0° with each other (i.e., they coincide with each other)
In
every hour the two hands coincide once. When they coincide the two hands are 00
apart. In 12 hours. There are 11 coincidences of the two hands and 22 coincidences
in 24 hours or a day.
10.
The time gap between any two
coincidences of the hour hand and the minute hand is 12/11 hours or 65 5/11
minutes.
11.
Hands make an angle of 180° with each other
When
the two hands make an angle of 1800 each other, they point from a
straight line in exactly opposite directions or the two hands are 30-minute
spaces apart.
It
happens once in an hour 11 times in 12 hours and 22 times and 24 hours (a day)
12.
Hands
make an angle of 900 each other.
When
the two hands make an angle of 900 each other, they are at right
angles or perpendicular to each other or the two hands are 15-minute spaces
apart.
It
happens twice in an hour, 22 times in 12 hours, and 44 times in 24 hours ( a
day)
13.
Relative Velocity
In
1 minute, the Minute hand covers 60 and the Hour hand covers 1/2°.
Therefore,
Relative Velocity =6 – ½ = 5 1/2° per minute.
In
1 hour, the minute hand covers 60-minute divisions whereas the hour hand covers
5-minute divisions.
Therefore,
Relative Speed =60 — 5 =55 minutes per hour.
14.
Incorrect Clock: An incorrect clock can
either be a fast clock which gains time or a slow clock which loses time.
If
a clock indicates 10.30, when the correct time is 10.15, it is said to be too
fast and running 15 minutes fast.
On
the other hand, if the clock indicates the time 10.30 when the correct time is
10.45, it is said to be too slow and running 15 minutes slow.
16. Assume
that the hands of the clock of the incorrect clock (which does not show the
correct time) coincide every n minutes.
If
n > 65 5/11, then the watch is going slow or losing time.
If
n < 65 5/11, then the watch is going fast or gaining time.
17.
To calculate the angle between the
hands of a clock, we use the following formula
( where m =
minutes and h = hours)
i)Angle = 11/2m – 30h(when 11/2 m>30h)
ii)Angle = 30h- 11/m(when 30h> 11/2 m)
