Contents
By learning Physics Topics, we can gain a deeper appreciation for the natural world and our place in it.
What are Two Methods of Measuring Length? What is the Range of Metre Scale?
Measurement of Length
Length measurement methods are of two types
- direct method and
- indirect method.
A physical quantity is measured by comparing it with a stan-dard measurement which defines its unit. For example, when we measure some length, we measure with respect to some standard value like 1 m or 1 cm.
Standard length: Distance travelled by light in 1/299792458 second in vacuum, is taken as 1 metre.
Direct method: In this method, the length to be measured is directly compared with the standard unit of length. A scale or ruler can be used to measure the length, breadth and height of a book. Such scales are already graduated as per the unit of length, and its fractions and multiples. Vernier scale, screw gauge etc. are also used to measure length and other equaivalent quantities like diameter, depth etc.
Indirect method: Very long lengths like the distance of a star from the earth or very short lengths like the diameter of a molecule, cannot be measured by direct methods. In such cases indirect methods like triangulation method, reflection or echo method, parallax method, etc. are used.
Physical quantities like length, height, distance, radius, depth have the same dimension and the units for their measurement are also the same, like cm, m, km, in., ft, mi, etc. But their measured values differ widely in magnitudes. Hence different measuring instruments and techniques are required. The measuring instruments also differ depending on the shape of the body.
Ordinary scale or ruler
It is a thin rectangular metal or wooden strip calibrated in centimetre scale along its length [Fig.]. The smallest scale division is usually 1 mm or 0.1 cm. Its extreme left end is marked 0 cm. However, the marking on the extreme right end reads 15 cm, 30 cm, 50 cm or 100 cm according to the length of the scale. When the marking of scale is from 0 cm to 100 cm it is called a metre-scale.
Ruler of this type is mainly used for measuring the length of a straight line, a straight rod, stretched wires etc. The length of a line can be measured by putting the 0 mark on the scale at one end and taking the reading on the ruler at the other end.
Eye estimation: If the length of a line is 3 cm or 7 cm exactly, the measurement should be written as 3.0 cm or 7.0 cm. It is often seen that although the start of the line coincides with the 0 mark, the end does not coincide with any mark on the scale. It lies between two successive marks. In a measurement, let the extreme right end lie between 7.6 and 7.7 cm. If the end appears to be exactly in the middle of the two marks the length can be estimated as 7.65 cm although the extreme right digit is not reliable. Based on eye estimation only, the reading should not be written as 7.63 cm, 7.66 cm, etc.
Vernier scale
In a ruler or a scale the smallest scale division is usually 1 mm or, sometimes, 0.5 mm. Hence such scales cannot be used to measure accurately any length less than the limits stated. In 1631, Pierre Vernier, a French mathematician, invented the ‘vernier scale’ that can increase the accuracy of measurement.
Description: M, the main scale, is an ordinary scale graduated in cm. Usually the smallest scale division is 1 mm i. e., 0.1 cm [Fig.]. V is a small scale, called the vernier scale, attached to the main scale and can slide along the edge of it. A typical vernier scale is shown in Fig where 10 divisions of this scale equal 9 divisions of the main scale.
Method of Measurement with a vernier scale: The smallest length that a vernier scale can measure is equal to the difference in lengths between 1 smallest main scale division and 1 smallest vernier scale division. The value of this difference is called the vernier constant.
Calculation of the vernier constant:
- Value of 1 smallest main scale division (say, m unit) is noted.
- The 0 marks on the vernier and the main scale are set to coincide.
Starting from zero, the mark on the main scale that coincides with the last mark on the vernier scale is counted.
Now, let y divisions of the vernier scale coincide with x divisions on the main scale.
Hence, length of y divisions of vernier scale = length of x divisions of the main scale.
∴ Length of 1 division of vernier scale = length of \(\frac{x}{y}\) divisions of main scale = \(\frac{x}{y}\) × m unit
Therefore, vernier constant, c = length of 1 main scale division – length of 1 vernier scale division
∴ c = m – \(\frac{x}{y}\) × m = m(1 – \(\frac{x}{y}\)) = m\(\left(\frac{y-x}{y}\right)\) unit
For most of the vernier scales, (y – x) = 1
Hence, vernier constant c = \(\frac{m}{y}\) unit.
As in Fig., m = 1 mm =0.1 cm and y = 10, c = \(\frac{0.1}{10}\) cm or 0.01 cm or 0.1 mm.
Hence, the instrument can measure with an accuracy of 0.1 mm.
Measurement using a vernier scale: To measure the length of a rod R [Fig.], the left end of R is set to coincide with the 0 mark of the main scale. Now, the vernier scale is set in such a way that its 0 mark touches the right end of R . At this stage, the main scale reading that is just on the left of the vernier 0 mark is noted. This reading is denoted as a. In Fig., a = 2.6. Next, the reading on the vernier that coincides best with any one marking on the main scale is noted. Let this reading of the vernier be b. In Fig., b = 5 since the fifth vernier division coincides with a marking of the main scale.
So, the length l of the rod R is given by,
l = main scale reading + vernier scale reading × vernier constant
or, l = a + b × c = (2.6 + 5 × 0.01) cm = 2.65 cm.
Numerical Examples
Example 1.
Find the length of the rod in Fig.
Solution:
From Fig.(a), the value of 1 smallest main scale division is \(\frac{1}{10}\) = 0.1 cm.
10 vernier divisions coincide with 9 main scale divisions (MSD)
∴ Vernier constant, c = (1 – \(\frac{9}{10}\)) × 0.1 cm = 0.01 cm.
From Fig.(b),
0 of vernier scale crossed 2.2 cm mark on the main scale and 5th vernier division coincides with one main scale division.
∴ The length of the rod = (2.2 + 5 × 0.01) cm = 2.25 cm.
Example 2.
Estimate the length of the rod R from Fig.
Solution:
From Fig.(a), the value of 1 smallest main scale division is \(\frac{1}{10}\) = 0.1 cm.
5 vernier divisions coincide with 4 main scale divisions.
∴ Vernier constant, c = (1 – \(\frac{4}{5}\) ) × 0.1 = 0.02 cm.
From Fig.(b),
0 of vernier scale crossed 3.7 cm mark on the main scale and 3rd vernier scale mark coincides with a MSD.
∴ Length = (3.7 + 3 × 0.02) = 3.76 cm.