Clocks Concept
The dial of the clock is circular in shape and was divided into 60 equal minute spaces
60 minute spaces traces an angle of 360°
∴ 1minute space traverses an angle of 6°
In 1 hour, Minute hand traverses 60 minute space or 360° ,Hour hand traverses 5 minute space or 30°
The hands of the clock are perpendicular in 15 minute spaces apartMinute Spaces
The face or dial of watch is a circle whose circumference is divided into 60 equal parts, called minute spaces.
Hour Hand and Minute Hand:
A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called minute hand or long hand .
i. In 60 minutes, the minute hand gains 55 minutes on the hour on the hour hand.
ii. In every hour, both the hands coincide once.
iii. The hands are in the same straight line when they are coincident or opposite to each other.
iv. When the two hands are at right angles, they are 15 minute spaces apart.
v. When the hands are in opposite directions, they are 30 minute spaces apart.
vi. Angle traced by hour hand in 12 hrs = 360°
vii. Angle traced by minute hand in 60 min. = 360°.
viii. If a watch or a clock indicates 8.15, when the correct time is 8, it is said to be 15 minutes too fast .
ix. On the other hand, if it indicates 7.45, when the correct time is 8, it is said to be 15 minutes too slow
Example
Qn.
My watch, which gains uniformly, is 2 min, & show at noon on Sunday, and is 4 min
48 seconds fast at 2 p.m on the following Sunday when was it correct ?
Sol:
From Sunday noon to the following Sunday at 2 p.m there are 7 days 2 hours or 170hours.
The watch gains (2+4 $\dfrac{4}{5}$) min in 170 hrs.
∴ the watch gains 2 min in ($\dfrac{2}{6 \dfrac{4}{5}} \times 170$) hrs i.e., 50 hours
Now 50 hours = 2 days 2 hrs.
∴ 2 days 2 hours from Sunday noon = 2 p.m on Tuesday.
Key Notes
The hands of the clock are in straight line and apposite to each other in 30 minute spaces apart.
The hands of the clock are in straight line when they coincide or opposite to each other.
The hands of the clock are perpendicular to each other for 22 times in 12 hours and for 44 times in a day.
The hands of the clock are opposite to each other for 11 times in 12 hours and 22 times in a day.
The hands of the clock coincides with each other for 11 times in 12 hours and 22 times per day
The hands of the clock are 44 times in a straight line per day
The minute hand gain 55 minutes over hour hand per hour.
Hence x minute space to be gained by minute hand over hour hand can be x.(60/55) or x.(12/11)
Example
Qn.
At what time between 2'O clock and 3'O clock the hands of the clock are opposite to each other.Sol:
At 2'O clock the minute hand will be at 12 as shown below
The minutes hand to coincide with the hour hand it should trace at first 10 minute spaces
And then the hands of the clocks to be opposite to each other minute hand
should trace 30 minute spaces i.e. totally it should gain 10+30=40 minute spaces to be opposite to that of hour hand
We know that,
Minute hand gains 55 minute spaces over hour hand in 1 hour
∴ Minute hand gain 40 minute spaces over hour hand in 40 × (60/55) = 43(7/11)
Hence the hand of the clock will minutes be opposite to each at 43( 7/11 ) past 2'Oclock
∴ Answer is 43( 7/11 ) past 2'Oclock