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Clocks

All of us have seen clocks and we also know how to read time on a clock. Solving problems on clocks requires some knowledge about the movement of hands in the clock.
 
Description: 106315.png

The dial of a clock is numbered from 1 to 12 and these numbers are placed at equidistance (angular distance). The angle made by any two consecutive numbers at the centre of the clock is same and it is equal to 30°. The clock has two needles and these needles are unequal in size. The longer needle is known as minute hand and the shorter needle is known as hour hand. Though some of the clocks have seconds hand also, our discussion in this chapter is restricted to the movement of only minute and hour hands.


At twelve o’clock (either at 12:00 noon or at 12:00 midnight), the minute hand and the hour hand overlap. In other words, the angle between the minute hand and the hour hand becomes zero at twelve o’clock. When do these two hands overlap again? To answer this question, we should do some calculations.

The minute hand in a clock makes one revolution in one hour. So, it moves 360° in 1 h. We all know that 60 minutes make 1 hour. Thus, the rotation made by the minute hand in 1 minute
amounts to Description: 106328.png The hour hand takes 12 hour to complete one revolution. The rotation made by the hour hand in 1 minute amounts to Description: 107590.png = 0.5°.

From the above explanation, it follows that the angle between the minute hand and the hour hand changes at the rate of 5.5° minute (6° – 0.5°). Now, we are in a position to find out, as to when the minute and the hour hands overlap again after twelve o’clock.

The angle between the minute and the hour hands changes by 5.5° in 1 minute.
 
So, the time taken to change the angle by 360° is Description: 106357.png minutes, or 1 hour Description: 106372.png minutes. Thus, the minute hand and the hour hand of a clock overlap after every 1 hour and Description: 106384.png minutes.
 
How many times the minute hand and the hour hand of a clock overlap in 12 hours and in 1 day?
 
If we express 1 hour Description: 106392.png minutes in terms of only hours it comes to Description: 106400.png hours. So, in every Description: 106407.png hour, the minute and the hour hands overlap. In a period of 12 hours, these hands overlap Description: 106415.png × 12 = 11 times.

In one day, these two hands overlap 11 × 2 = 22 times.
 
With the above knowledge, now we are in a position to find out the angle between the minute and hour hands at any given point of time. To make this point clear, we shall calculate the angle between the two hands of a clock when the time is 6:10.
 
At six o’clock, the angle between the hands is 180°. This angle goes on decreasing at the rate of 5.5° per minute till the angle between the hands becomes zero and thereafter, the angle between the hands goes on increasing at the same rate.
 
So, in 10 minutes, the change in angle = 5.5 × 10 = 55°.
 
Thus, the angle between two hands at 6:10 = 180° - 55° = 125°.
 
Whenever we refer the angle between two hands, we always mean the interior angle i.e., the angle which is less than 180°.





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