A close look at the hands of a clock

 By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

03 November 2013

In Kenya, the term overlap means the act of overtaking a long queue of traffic. I don’t know how this terminology came about but it is technically inappropriate. There is no lap on the road so one car cannot overlap another! Perhaps a more apt description for the action would be “queue-jumping”.

Nevertheless, Lillian Otieno has been wondering about another overlapping process. She writes: “…the digital revolution will soon kill the traditional clock before I can get an answer to a question that has nagged me for many years. You know the way the hands move at different speeds, if we start at 12:00 o’clock, at what times will the minute had overlap the hour hand?”

Lillian, I cannot understand why anyone would be wondering about the overtaking process of the hands of a clock, but all the same I shall try to explain how you can evaluate it. Obviously the two hands move at different speeds, but how do we express those speeds?

I think the least confusing way is to use the number of degrees covered per minute. Thus the minute hand completes one cycle or 360^O in 60 minutes; therefore its speed is 6^O per minute. The hour hand does 360^O in 12 hours or 720 minutes. Its speed comes to 0.5^O per minute.

Now at 1:00 o’clock, the minute hand is on 12 and the hour hand at 1. Thus the latter is 30^O ahead of the former. The question then comes down to this: how long will it take the fast hand to catch up with the slow one?

The “distance” (number of degrees) that each hand covers is simply the speed (degrees per minute) multiplied by the time taken (in minutes). But we must bear in mind that the minute hand is starting this race 30^O ahead. At the point of overlapping, the two hands will be the same “distance” from the 12 o’clock mark.

Therefore, we may form a mathematical equation with the two “distances” on either side. The only unknown quantity in that equation is the time. And the calculation yields that it takes 5.4545 minutes for the overlapping to occur thus it happens at 1:05:27 o’clock

Obviously, at 2:00 o’clock, the hour hand is now 60^O ahead, thus the minute hand will need twice as much time to catch up; thus the second overlap will happen after 10.0909 minutes, that is, at 2:10:55 o’clock. Subsequent overlaps can be calculated the same way.

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There is a total solar eclipse expected this afternoon. The event will be observable in countries lying along the central belt of Arica.

In Kenya, the total eclipse will pass through the northwestern corner, running over Lake Turkana late in the afternoon – just before sunset. But the rest of Kenyans will only have a partial eclipse. Viewed from Nairobi, the moon will block about 75 per cent of the Sun thereby creating a crescent.

Whatever you do, do not look at this spectacle with naked eyes – you might go blind!