CHAPTER 1: Units and Measurements
 INTRODUCTION
Measurement of any physical quantity involves comparison with a certain basic, arbitrarily
chosen, internationally accepted reference standard called unit.
The result of a measurement of a physical quantity is expressed by a number (or numerical
measure) accompanied by a unit. Although the number of physical quantities appears to be
very large, we need only a limited number of units for expressing all the physical quantities,
since they are interrelated with one another. The units for the fundamental or base
quantities are called fundamental or base units.
The units of all other physical quantities can be expressed as combinations of the base
units. Such units obtained for the derived quantities are called derived units.
A complete set of these units, both the base units and derived units, is known as the system
of units.
 THE INTERNATIONAL SYSTEM OF UNITS
In earlier time scientists of different countries were using different systems of units for
measurement. Three such systems, the CGS, the FPS (or British) system and the MKS system
were in use extensively till recently.
The base units for length, mass and time in these systems were as follows :
• In CGS system they were centimetre, gram and second respectively.
• In FPS system they were foot, pound and second respectively.
• In MKS system they were metre, kilogram and second respectively.
The system of units which is at present internationally accepted for measurement is the
Système Internationale d(SI)’ Unites (French for International System of Units), abbreviated
as SI. The SI, with standard scheme of symbols, units and abbreviations, was developed and
recommended by General Conference on Weights and Measures in 1971 for international
usage in scientific, technical, industrial and commercial work. Because SI units used decimal
system, conversions within the system are quite simple and convenient. We shall follow the
SI units.
 Fundamental Units and Derived Units
In SI, there are seven base units as given in Table Besides the seven base units, there are two
more units that are defined for (a) plane angle dθ as the ratio of length of arc ds to the
radius r and (b) solid angle dΩ as the ratio of the intercepted area dA of the spherical
surface, described about the apex O as the centre, to the square of its radius r, as shown in
Fig. (a) and (b) respectively. The unit for plane angle is radian with the symbol rad and the
unit for the solid angle is steradian with the symbol sr. Both these are dimensionless
quantities
Other physical quantities derived from the base quantities can be expressed as a
combination of the base units and are called derived units.A complete set of units both
fundamental and derived units are called a system of units. Example :- volume, density etc.
 Scalar and Vector Quantity
A quantity which does not depend on direction is called a scalar quantity. Vector quantities
have two characteristics, a magnitude and a direction. Scalar quantities have only a
magnitude. When comparing two vector quantities of the same type, you have to compare
both the magnitude and the direction. For scalars, you only have to compare the magnitude.
When doing any mathematical operation on a vector quantity (like adding, subtracting,
multiplying ..) you have to consider both the magnitude and the direction. This makes
dealing with vector quantities a little more complicated than scalars.
Scalar quantities that have the same physical units can be added or subtracted according to
the usual rules of algebra for numbers. For example, a class ending 10 min earlier than 50
min lasts 50min−10min=40min50min−10min=40min. Similarly, a 60-cal serving of corn
followed by a 200-cal serving of donuts gives 60cal+200cal=260cal60cal+200cal=260cal of
energy. When we multiply a scalar quantity by a number, we obtain the same scalar quantity
but with a larger (or smaller) value.
Physical quantities specified completely by giving a number of units (magnitude) and a
direction are called vector quantities. Examples of vector quantities include displacement,
velocity, position, force, and torque. In the language of mathematics, physical vector
quantities are represented by mathematical objects called vectors. We can add or subtract
two vectors, and we can multiply a vector by a scalar or by another vector, but we cannot
divide by a vector. The operation of division by a vector is not defined.
When vectors lie in a plane—that is, when they are in two dimensions—they can be
multiplied by scalars, added to other vectors, or subtracted from other vectors in
accordance with the general laws expressed. However, the addition rule for two vectors in a
plane becomes more complicated than the rule for vector addition in one dimension. We
have to use the laws of geometry to construct resultant vectors, followed by trigonometry to
find vector magnitudes and directions. This geometric approach is commonly used in
navigation. In this section, we need to have at hand two rulers, a triangle, a protractor, a
pencil, and an eraser for drawing vectors to scale by geometric constructions.
 Parallelogram rule of vector addition.
For a geometric construction of the sum of two vectors in a plane, we follow the
parallelogram rule. Suppose two vectors are at the arbitrary positions shown in Figure.
Translate either one of them in parallel to the beginning of the other vector, so that after the
translation, both vectors have their origins at the same point. Now, at the end of vector we
draw a line parallel to vector and at the end of vector we draw a line parallel to vector (the
dashed lines in (Figure)). In this way, we obtain a parallelogram. From the origin of the two
vectors we draw a diagonal that is the resultant of the two vectors: . The other diagonal of
this parallelogram is the vector difference of the two vectors , as shown in (Figure) (b).
Notice that the end of the difference vector is placed at the end of vector .
 Vernier Calliper
A vernier calliper is a device used to measure the distance between two symmetrically
opposing sides.A vernier calliper can be as simple as a compass with inward or outward
facing points. The tip of thecalliper is adjusted across the points to be measured, the calliper
is then removed and the distance is read by measuring between the tips with a measuring
tool such as a ruler. They areused in many fields such as metal working, mechanical
engineering, gunsmithing, woodworking and wood turning.
Parts of Vernier Calliper:
1. Outside Jaws: Used to measure external lengths
2. Inside Jaws: Used to measure internal lengths
3. Depth Probe: Used to measure depths
4. Main Scale (cm)
5. Main Scale (Inch)
6. Vernier (cm)
7. Vernier (Inch)
8. Retainer: used to block movable part to allow easy transferring a
measurement
The least count is an important criterion for the selection of
instruments. It is the minimum measurement that can be
used to measure through the instrument. The least count of
Vernier Calliper is given as
Least Count(L.C) =
 Micrometer Screw Gauge:
The micrometer Screw gauge is used to measure even smaller
dimensions than the Vernier Caliper. The micrometer screw gauge also
uses an auxiliary scale (measuring hundreds of millimeter) which is
marked on a rotary thimble. Basically, it is a screw with an accurately
constant pitch (the amount by which the thimble moves forward and
backward for one complete revolution). Following is the block diagram
of the micrometer Screw Gauge.:
Parts of Screw gauge:
1. Anvil: The spindle is moved towards the object to
be measured that is put onthe anvil.
2. Thimble lock: This element can be tightened to
keep the spindle in positionwhile measuring. It's
sometimes referred to as a lock nut or a lock ring.
3. Spindle: The spindle will travel towards the anvil as the thimble
is rotated.
4. Thimble: This is the rotating portion of the screw gauge.
5. Screw: This is the screw gauge's key component that aids in
measurement.
6. Ratchet stop: It's utilized to keep the thimble from
rotating any further than it needs to.
7. Barrel/Main Scale: It's also known as a sleeve or
stock, and it's a stationary component with linear or
vernier markings on it. The barrel of some screw
gauge versions is a rotatable cylinder, which allows
for zero value by adjustingthe cylindrical barrel.
The least count of Micrometer Screw is given by,
Least Count(L.C) =
 ACCURACY, PRECISION OF INSTRUMENTS
We shall distinguish between two terms: accuracy and precision. The accuracy of a
measurement is a measure of how close the measured value is to the true value of the
quantity. Precision tells us to what resolution or limit the quantity is measured.
The accuracy in measurement may depend on several factors, including the limit or the
resolution of the measuring instrument. For example, suppose the true value of a certain
length is near 3.678 cm. In one experiment, using a measuring instrument of resolution 0.1
cm, the measured value is found to be 3.5 cm, while in another experiment using a
measuring device of greater resolution, say 0.01 cm, the length is determined to be 3.38 cm.
The first measurement has more accuracy (because it is closer to the true value) but less
precision (its resolution is only 0.1 cm), while the second measurement is less accurate but
more precise.
 ERRORS IN MEASUREMENT
Measurement is the foundation of all experimental science and technology. The result of
every measurement by any measuring instrument contains some uncertainty. This
uncertainty is called error.
Thus every measurement is approximate due to errors in measurement. In general, the
errors in measurement can be broadly classified as (a) systematic errors and (b) random
errors.
(a) Systematic errors :
The systematic errors are those errors that tend to be in one direction, either positive or
negative. Some of the sources of systematic errors are :
(1) Instrumental errors that arise from the errors due to imperfect design or calibration of
the measuring instrument, zero error in the instrument, etc. For example, the temperature
graduations of a thermometer may be inadequately calibrated (it may read 104 °C at the
boiling point of water at STP whereas it should read 100 °C); in a vernier callipers the zero
mark of vernier scale may not coincide with the zero mark of the main scale, or simply an
ordinary metre scale may be worn off at one end.
(2) Imperfection in experimental technique or procedure To determine the temperature of
a human body, a thermometer placed under the armpit will always give a temperature lower
than the actual value of the body temperature. Other external conditions (such as changes
in temperature, humidity, wind velocity, etc.) during the experiment may systematically
affect the measurement.
(3) Personal errors that arise due to an individual’s bias, lack of proper setting of the
apparatus or individual’s carelessness in taking observations without observing proper
precautions, etc. For example, if you, by habit, always hold your head a bit too far to the
right while reading the position of a needle on the scale, you will introduce an error due to
parallax.
(b) Random errors:
The random errors are those errors, which occur irregularly and hence are random with
respect to sign and size. These can arise due to random and unpredictable fluctuations in
experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply,
mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the
observer taking readings, etc. For example, when the same person repeats the same
observation, it is very likely that he may get different readings everytime.
 Absolute Error, Relative Error and Percentage Error
(a) Suppose the values obtained in several measurements are a1 , a2 , a3 ...., an . The
arithmetic mean of these values is taken as the best possible value of the quantity under the
given conditions of measurement as :
amean = (a1 +a2 +a3 +...+an ) / n
This is because, as explained earlier, it is reasonable to suppose that individual
measurements are as likely to overestimate as to underestimate the true value of the
quantity.
The magnitude of the difference between the individual measurement and the true value
of the quantity is called the absolute error of the measurement.
This is denoted by |∆a |. In absence of any other method of knowing true value, we
considered arithmatic mean as the true value. Then the errors in the individual
measurement values from the true value, are
∆a1 = a1 – amean ,
∆a2 = a2 – amean ,
.... .... .... .... .... ....
∆an = an – amean
The ∆a calculated above may be positive in certain cases and negative in some other cases.
But absolute error |∆a| will always be positive.
(b) The arithmetic mean of all the absolute errors is taken as the final or mean absolute error
of the value of the physical quantity a. It is represented by ∆amean . Thus,
∆amean = (|∆a1 |+|∆a2 |+|∆a3 |+...+ |∆an |)/n
(c) Instead of the absolute error, we often use the relative error or the percentage error
(δa). The relative error is the ratio of the mean absolute error ∆amean to the mean value
amean of the quantity measured.
Relative error = ∆amean /amean
When the relative error is expressed in per cent, it is called the percentage error (δa).
Thus, Percentage error
δa = (∆amean /amean ) × 100%
Example :1 We measure the period of oscillation of a simple pendulum. In successive
measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s and 2.80 s. Calculate
the absolute errors, relative error or percentage error.
The mean period of oscillation of the pendulum
Tmean = (2.63 2.56 2.42 2.71 2.80) / 5
= 13.12 / 5
= 2.624 s
= 2.62 s
As the periods are measured to a resolution of 0.01 s, all times are to the second decimal; it
is proper to put this mean period also to the second decimal.
The errors in the measurements are
2.63 s – 2.62 s = 0.01 s
2.56 s – 2.62 s = – 0.06 s
2.42 s – 2.62 s = – 0.20 s
2.71 s – 2.62 s = 0.09 s
2.80 s – 2.62 s = 0.18 s
∆Τmean = [(0.01+ 0.06+0.20+0.09+0.18)]/5
= 0.54 /5
= 0.11 s
the relative error
δa = 0.1/2.6 = 0.04
the percentage error is
δa = 0.1/2.6 x 100 = 4%