Clocks, Circular motion and their application in speed, time and distance

Circular Motion

When two or more bodies are moving around a circular track, the motion around this circular track is known as circular motion.
The relative motion of two bodies moving around a circle in the same direction is taken as
= S₁ - S₂
Also, when bodies are moving in opposite direction, then relative speed
= S₁ + S₂

Clocks

Problems on clocks are based on the movement of the minute hand and that of the hour hand as well as on the relative movement between two. For the simplicity, we don't take the second hand. It is best to solve problems of clocks by considering a clock to be in a circular track having a circumference of 60 km and each kilometer being represented by one minute on the dial of the clock.

Then, we can look at the minute hand as runner running at the speed of 60kmph while we can also look at the hour hand as a runner running at an average 5kmph $$\left( {\frac{{60}}{{12}} = 5\,{\text{kmph}}} \right).$$   Since, minute hand and hour hand are running in same direction, the relative speed of minute hand with respect to hour hand is 55 kmph (60 - 5 = 55 kmph). This means that for every hour elapsed, the minute hand goes 55 km (minute) more than the hour hand.

Key facts:

==> A clock makes two right angles between any 2 hours.
==> Right angles are formed when the distance between the minute hand and hour hand is equal to 15 minute.
==> A clock makes two straight lines between any 2 hours. Straight lines are formed when the distance between the minute hand and the hour hand is equal to either 0 minutes or 30 minutes.

Illustration: At what time between 2-3 pm is first right angle formed by hands of clocks?

Solution:

At 2 pm minute hand can be visualized as being 10 kilometers behind the hour hand as the minute hand is 10 minute behind the hour hand.
The first right angle between 2-3 is formed when the minute hand is 15 kilometers ahead of the hour hand. Thus, the minute hand has to cover 25 kilometer over hour hand.
Distance covered by minute hand over hour hand; 55 kilometer _________ in 1 hour.
Then, 1 km ______________ in $$\frac{1}{{55}}$$ hour.
Hence, 25 km ___________ in $$\frac{{25}}{{55}} = \frac{5}{{11}}$$  hour.
Thus, the first right angle between 2-3 is formed at $$\frac{5}{{11}}$$ hours past 2 o'clock.
$$\eqalign{ & \frac{5}{{11}} = \frac{{5 \times 60}}{{11}} = 27\left( {\frac{3}{{11}}} \right)\,{\text{minutes}} \cr & \frac{3}{{11}}\,{\text{minute}} = \frac{{3 \times 60}}{{11}} = 16.3636\,{\text{second}} \cr} $$

Hence, the required answer is; 2:27:16.36 pm.

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