Chapter 3.3 - Direct proportions in Angular measurements

In the previous section we saw solved examples on direct proportional changes of various quantities. In this section we will see direct proportional changes in 'angular measurements'. 

For this, we do an experiment with clock readings. Consider the fig.3.3(a). It shows a clock. The time shown is exact 2.00.

Fig.3.3(a) First stage

The minute hand is at 12 and the hour hand is at 2. [For our experiment we must completely ignore the hour hand. It's position or angle does not have any relevance in this experiment]. After 15 minutes, that is at 2.15, the position of the minute hand will be at 3. This is shown in fig.3.3(b) below. [The hour hand has turned a little bit away from 2, towards 3. But as mentioned earlier, we simply ignore it's position]. The minute hand has turned through an angle of 90^o during this 15 minutes.

Fig.3.3(b) Second stage

We will note down the stages that the minute hand has gone through so far:

• Initial stage → Time is zero, and the angle turned is also zero
• Second stage → Time passed is 15 minutes, and angle turned is 90^o

Let us continue: After 15 more minutes, the time is 2.30. This is shown in fig.3.3(c). The minute hand is at 6. Time passed is 30 minutes from the initial stage. 

Fig.3.3(c) Third stage

During these 30 minutes, the minute hand has turned through an angle of 180^o. We will add this also to the above list of stages:
• Initial stage → Time is zero, and the angle turned is also zero
• Second stage → Time passed is 15 minutes, and angle turned is 90^o
• Third stage → Time passed is 30 minutes, and angle turned is 180^o

Let us continue. After 15 more minutes, the time is 2.45. This is shown in fig.3.3(d). The minute hand is at 9. Time passed is 45 minutes from the initial stage. 

Fig.3.3(d) Fourth stage

During these 45 minutes, the minute hand has turned through an angle of 270^o. We will add this also to the above list of stages:


• Initial stage → Time is zero, and the angle turned is also zero
• Second stage → Time passed is 15 minutes, and angle turned is 90^o
• Third stage → Time passed is 30 minutes, and angle turned is 180^o
• Fourth stage → Time passed is 45 minutes, and angle turned is 270^o

Let us continue. This is the final reading in our experiment. After 15 more minutes, the time is 3.00. This is shown in fig.3.3(e). The minute hand is back at 12. Time passed is 60 minutes from the initial stage. 

Fig.3.3(e) Fifth stage

During these 60 minutes, the minute hand has turned through an angle of 360^o. We will add this also to the above list of stages:

• Initial stage → Time is zero, and the angle turned is also zero
• Second stage → Time passed is 15 minutes, and angle turned is 90^o
• Third stage → Time passed is 30 minutes, and angle turned is 180^o
• Fourth stage → Time passed is 45 minutes, and angle turned is 270^o
• Fifth stage → Time passed is 60 minutes, and angle turned is 360^o

Written above is the final list. We have taken all the required readings. We will tabulate them. There are two quantities: (1) Time passed (let us take it as x) and (2) Angle turned by the minute hand (let us take it as y). The table is shown below:

The y/x ratio for each column is also computed in the same table. We find that y/x is a constant. So the two quantities are in direct proportion. They satisfy the relation y = kx. And the value of k is 6.00. This gives us an easy method to calculate the angle turned during any 'duration of time'. This is explained below:

In the table, The 'time passed' is a 'duration of time'. In a duration of 15 minutes, the minute hand turns through 90^o. For a duration of 45 minutes, the minute hand turns through 90^o and so on. Suppose we want to find the angle turned in a duration of 25 minutes. This 25 minutes, and it's corresponding angle y1 will also have a place in the above table. And they must give the same constant 6.00. So the above table can be modified as:

So we can write:
y1/25 = 6 ⇒ y1 = 6  × 25 = 150^o

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Let us see some solved examples based on the above discussion:

Solved example 3.7

The first reading is taken when the minute hand is at 7, and the second reading when it is at 11. What is the angle through which the minute hand turned during this time interval?

Solution:

• First reading is at 7. That is 35 minutes.

• Second reading is at 11. That is 55 minutes.

• So the time interval is 55 – 35 = 20 minutes.

As the two quantities are in direct proportion, this 20 minutes and the corresponding angle y1 have a place in the table. So the table can be formed as:

This y1 and 20 should give the same constant 6.00. So we can write:
y1/20 = 6 ⇒ y1 = 6  × 20 = 120^o

Solved example 3.8
The first reading was taken when the minute hand was at the 2^nd small division after 4. The angle turned during a certain time interval from this first reading was found to be 84^o. 
(i) What was the time interval? 
(ii) Where was the minute hand after this time interval ? 
Solution:
• Minute hand at 4 indicates 20 minutes.
• But 2 small divisions were also passed. Each small division is one minute.
• So the time at first reading = 20 + 2 = 22 minutes.

The angle is given as 84 deg. This 84 and the corresponding time x1 duration have a place in the table. So the table can be formed as:

84 and x1 will give the same constant 6.00. So we can write:
84/x1 = 6 ⇒ x1 = 84/6 = 14 minutes. This is the answer to the first part.
(ii) • The total time passed after the first reading = 22 + 14 = 36 minutes
• When the minute hand is at 7 it indicates 35 minutes
• One small division is one minute.
• So to indicate 36, the minute hand is at the 1^st division after 7

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In the next section we will see the relation between direct proportion and the 'scales' in maps and drawings.

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