Physics Notes I Sem.

(NEP) Unit – 01, Chapter – 01

01 Units and Measurements

Syllabus: Units and measurements: System of units (CGS and SI), measurement of
length, mass and time, dimensions of physical quantities, dimensional formulae.
Minimum deviation, errors. [05 hours]

Introduction:
In measurement of a physical quantity (A physical quantity is that which can be
measured), we require some ‘reference standard’. The reference standard of
measurement is called a unit.
“The unit of a physical quantity is defined as the reference standard used to
measure it”.
“The magnitude of the physical quantity is the product of unit (𝒖) in which the
quantity is measured and the number (𝒏) of times that unit is contained in the given
quantity”.
The unit selected for measuring a physical quantity must have the following
characteristics:
(i) The unit should be well defined.
(ii) The unit should be neither too small nor too large in comparison with the
physical quantity to be measured. In simple words, the unit should be of some
suitable size.
(iii) The unit should be imperishable.
(iv) The unit should neither change with time nor with physical conditions like
pressure, temperature etc.
(v) The unit should be easily reproducible.

Need of a System of Units:

There are as many units as there are independent quantities. Let us consider
three physical quantities mass, length and time. These quantities are independent of
each other. So, three separate units are required for the measurements of these
quantities. Thus, it becomes important to establish a system of units. Historically, the
choice of every unit has been changing. As an example, the metre was originally
defined in terms of the distance from north pole to the equator. This distance is very
nearly equal to 107 𝑚. Until recently, the standard metre of the world was distance
between two scratches on a platinum-iridium alloy bar that has been kept at the
International Bureau of weights and Measurements in France. Presently, the standard
metre in France has been specified in terms of number of wavelengths of light of a
specific spectral line of the isotope of 𝐾𝑟 86.

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 Physics Notes I Sem. (NEP) Unit – 01, Chapter – 01

Fundamental and Derived Physical Quantities:

“Fundamental physical quantities are those which cannot be defined in terms
of other quantities”.
“Derived physical quantities are those which can be defined in terms of
fundamental physical quantities”.

Fundamental and Derived Units:

Units are classified into two categories – basic or fundamental and derived units.
“The basic or fundamental units are the units of fundamental quantities”. These
units are so named because they can neither be derived from one another nor can be
further resolved into other simpler units.
In physics, we deal with a very large number of physical quantities. But the
minimum convenient number of units is only seven. In other words, the number of
basic units is only seven. These are the units of length, mass, time, electric current,
temperature, intensity of light and amount of substance.
“Derived units are those units which are derived from basic or fundamental
units”. As an example, the unit of speed is 𝑚𝑠 −1 and the unit of acceleration is 𝑚𝑠 −2.
These are the derived units.

Systems of Units:
Following are the commonly used systems of measurements:
a) cgs system: It is the French system in which ‘c’ stands for centimetre, ‘g’ stands
for gram and ‘s’ stands for second. These systems deals with only three
mechanical quantities length, mass and time. In this metric system, the basic
units of length, mass and time are centimetre, gram and second respectively.
b) fps system: It is British system in which ‘f’ stands for foot, ‘p’ stands for pound
and ‘s’ stands for second. In this system, the basic unit of length, mass and time
are foot, pound and second respectively. It is not a metric system. This system
is not in much use now a days.
c) mks system: In this system, which is originated in France, ‘m’ stands for metre,
‘k’ stands for kilogram and ‘s’ stands for second. In this system, the basic units
of length, mass and time are metre, kilogram and second respectively. This
system is closely related to cgs system and is a metric system.
d) International System (S.I.) of Units: The French name for this system is “System
International d’ Unites”. It is abbreviated as S.I. This system is in fact the
improved and extended version of M.K.S. system of units.
In the year 1960, the Eleventh General Conference of Weights and
Measurements made an attempt to improve the two metric systems – C.G.S. system
and M.K.S. system. This attempt was made to meet the varied needs to scientists,
technologists and engineers. This conference introduced the International System of
Units. The recommendation of this conference was endorsed by both International

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 Physics Notes I Sem. (NEP) Unit – 01, Chapter – 01

Standards Organization (I.S.O.) and International Electromechanical Commission

(I.E.C.) in 1962.
In the present study we shall use S.I. units.
Following are the seven fundamental units of ‘International System of Units’.
1) metre (m): “It is defined as the distance occupied by 1,650,763.73 wavelengths
in vacuum of the radiation emitted by the krypton-86 atom in its transition
between 2𝑝10 and 5𝑑5 ”.
2) kilogram (kg): “It is defined as the mass of a platinum-indium cylinder of
diameter equal to its height which is preserved in a vault at International Bureau
of Weights and Measurements at Sevres near Paris”.
[Why the cylinder used is defined kilogram made of platinum-iridium alloy?
“This is because platinum-iridium alloy is least affected by environment and
time”.]
3) second (s): “It is the duration of 9,192,631,770.0 periods of radiation
corresponding to the transition between the two hyperfine levels of the ground
state of cesium-133 atom”.
“It is also defined as the time taken by cesium-133 atom to make
9,192,631,770 vibrations”.
4) ampere (A); “It is the constant current which when flowing in two straight
conductors of infinite length and of negligible area of cross-section and placed
one metre apart in vacuum, would produce between the conductors a force
equal to 2 × 10−7newton per metre of length”.
1
5) kelvin (K): “It is equal to of the thermodynamic temperature of triple point
273.16
of water”.
[What is triple point of water?
“It is the temperature at which ice, water and water vapour co-exist”.]
6) candela (cd): “It is the luminous intensity in the direction at right angles to a
1
surface of square metre area of a black body at a temperature of freezing
600000
platinum under a pressure of 101,325 newton per square metre”.
7) mole (mol): “It is defined as the amount of substance which contains as many
elementary units as there are carbon atoms in exactly 0.012 kg of 𝐶 12”.

In addition to seven fundamental units, there are following two supplementary units:
1) radian (rad): “It is defined as the plane angle between two radii of a circle which
cut off on the circumference an arc of length equal to the radius of that circle”.
“It is the angle subtended at the centre of a circle of radius 𝑟 by an arc of length
𝑙
𝑙 ”. It is given by 𝜃 = 𝑟𝑎𝑑𝑖𝑎𝑛
𝑟
2) steradian (sr): “It is defined as the solid angle subtended at the centre of a sphere
by an area of its surface equal to the radius of the sphere”.

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 Physics Notes I Sem. (NEP) Unit – 01, Chapter – 01

The solid angle subtended at the centre of a sphere of radius 𝑟 by an area Δ𝑆 on

Δ𝑆
the surface of the sphere is given by Ω = 2 𝑠𝑡𝑒𝑟𝑎𝑑𝑖𝑎𝑛
𝑟
The international system of units has the following advantages over other
system of units:
i. This system makes use of only one unit for one physical quantity, i.e., it is a
rational system of units.
ii. In this system, all the derived units can be easily obtained from basic and
supplementary units, i.e., it is a coherent system of units.
[Coherent system of units is that system of units which is based upon a set of
fundamental units from which all other units can be derived by either
multiplying them without introducing numerical factors.]

iii. It is metric system, i.e., multiples and submultiples can be expressed as powers
of 10.

Length:
The concept of length involves the comparison of two bodies. Suppose we have
a rubber cord placed against a standard metre scale. Let its length be 50 cm. Now pull
the cord from both its ends and measure the length. Let the new length be 60 cm.
Since the final length reading is more than the initial reading therefore we say that the
cord has increased in length. When we say that the length of a wire is 10 m, it implies
that the wire is being compared with a standard metre and its length is 10 times that
of the standard metre.
Originally, metre was defined in terms of the distance of north pole from the
equator of Earth. This distance is 1/4 th of the circumference of Earth. The
circumference of the Earth is nearly 4 × 107 𝑚. So, the distance of north pole from
1
the equator of Earth is × 4 × 107 = 107 𝑚. Thus, originally, metre was defined as
4
one ten millionth of the distance of North Pole to the equator of Earth. This is in fact
a very inconvenient definition of metre. So, in 1889, metre was defined as the distance
between two scratches on a platinum-iridium rod preserved, near Paris, in France. In
our country, the standard metre is preserved at National Physical Laboratory in Delhi.
The platinum-iridium rod is primary standard of length. Because of inaccessibility,
copies of the primary standard are made. These copies are called secondary standards.
These secondary standards are distributed to the standardizing agencies of different
nations. The wooden metre sticks are made by manufacturing specifications derived
from the secondary standard. In 1960, the standard metre was defined in terms of the
wavelength of orange light emitted by atoms of a single pure isotope of Krypton (Kr-
86) and is independent of time. Thus, the wavelength of light is truly accessible and
accurate standard. In addition to metre, there are many other units of length.
Following units are used to measure very large distances.

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 Physics Notes I Sem. (NEP) Unit – 01, Chapter – 01

1) light year (ly): “It is defined as the distance travelled by light, at a speed of 3 ×
108 𝑚𝑠 −1, in one year”.
“It may also be defined as the distance travelled by light in one year”
1 𝑙𝑖𝑔ℎ𝑡 𝑦𝑒𝑎𝑟 = 3 × 108 𝑚𝑠 −1 × 1 𝑦𝑒𝑎𝑟
1
= 3 × 108 𝑚𝑠 −1 × 365 𝑑𝑎𝑦
4
= 3 × 108 𝑚𝑠 −1 × 365.25 × 24 × 60 × 60 𝑠𝑒𝑐𝑜𝑛𝑑
= 9.467 × 1015 𝑚 ≈ 9.5 × 1015 𝑚
𝟏 𝒍𝒚 = 𝟗. 𝟓 × 𝟏𝟎𝟏𝟓 𝒎
Alpha centauri, the nearest star outside the solar system, is 4.3 𝑙𝑦 away from
the Earth. The Milky Way (Akash Ganga) is nearly 105 𝑙𝑦 in diameter.
2) Astronomical Unit (AU): “It is defined as the mean distance between the Sun
from Earth”.
𝟏 𝑨𝑼 = 𝟏. 𝟒𝟗𝟔 × 𝟏𝟎𝟏𝟏 𝒎 = 𝟏. 𝟓 × 𝟏𝟎𝟏𝟏 𝒎
3) parsec: It is an abbreviation of parallactic second.
“It is the distance at which one astronomical unit subtends an angle of one
second of arc”. In other words, “it is the distance corresponding to an annual
parallax of one second of arc”.
Annual parallax 𝜃 is the angle at which the semi-
major axis of the Earth’s orbit at right angles to the star’s 𝜃
direction is seen from the star.
𝟏 𝒑𝒂𝒓𝒔𝒆𝒄 = 𝟑. 𝟐𝟔 𝒍𝒚 E

Let us express persec in terms of metre by making use of

𝑙 𝑙
𝜃 = or 𝑟 =
𝑟 𝜃
𝜋
𝑙 = 1.496 × 1011 𝑚, 𝜃 = 1 𝑠𝑒𝑐𝑜𝑛𝑑 = 𝑟𝑎𝑑𝑖𝑎𝑛
60×60×180
1.496×1011 ×60×60×180
or 1 𝑝𝑎𝑟𝑠𝑒𝑐 = 𝑚 ∴ 𝟏 𝒑𝒂𝒓𝒔𝒆𝒄 = 𝟑 × 𝟏𝟎𝟏𝟖 𝒎
𝜋
[Which of these is largest – astronomical unit, light year and parsec?
1 𝑝𝑎𝑟𝑠𝑒𝑐 > 1 𝑙𝑖𝑔ℎ𝑡 𝑦𝑒𝑎𝑟 > 1 𝐴𝑈
Following units are used to measure very small distances:
i. 1 micron (1 𝜇𝑚) = 10−6 𝑚
ii. 1 nanometre (1 𝑛𝑚) = 10−9 𝑚
iii. 1 angstrom (1 Å) = 10−10 𝑚
iv. 1 fermi or 1 femtometre (1 𝑓𝑚) = 10−15 𝑚

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 Physics Notes I Sem. (NEP) Unit – 01, Chapter – 01

Guidelines for Writing of Units and their Symbols:

1. The units named after scientists are not written with a capital initial letter.
As an illustration, the unit of force named after Newton is written as newton
and not Newton.
2. The symbol of the units named after the scientists have initial capital letters.
As an illustration, the symbol for newton is N.
3. If the symbol of a unit is not derived from a proper name, then small letters
are used as symbols of units.
4. No full stop should be placed after the symbol.
5. Index of notation should be used for expressing derived units in terms of
basic units. As an example, it is preferred write 𝑚𝑠 −1 instead of 𝑚⁄𝑠.
6. Units and their symbols do not take a plural form. As an example,
‘centimetre’ is written as ‘centimetre’. Its symbol is written as cm and not
cms.
7. Some space is always to be left between number and the symbol of the unit.
As an example, it be incorrect to write 2.3m. The correct representation is
2.3 m.

Need for Measurement:

Physics deals with the study of natural phenomena. In order to approach the
subject qualitatively, it is very essential that we make measurements. As an example,
we know that the Earth rotates not only about its own axis but also about the sun.
What is the period of revolution of Earth about its own axis? And what is the period
of revolution of Earth about Sun? To answer these questions, we have to make
measurements. What is the temperature at which a given solid melts? What is the
velocity with which a car is moving in a given time? To answer these questions,
measurements are necessary.

Length Measurement:
In order to measure very small distances such as inter-molecular distances, size
of molecules, radius of atom etc., we cannot use measuring instruments such as
vernier calipers, screw gauge, spherometer etc. Even an optical microscope shall not
be of much use. In optical microscope, we use visible light whose wavelength if of the
order of 6 × 10−7 𝑚. So, an object whose dimension is less than 6 × 10−7 𝑚 cannot
be viewed with an optical microscope because light would pass through such an object
instead of being scattered. There is another type of microscope called electron
microscope in which, instead of visible light, we use an accelerated beam of electrons.
Instead of lenses, we use suitable electric and magnetic field for the purpose of
focusing. Electron microscopes having resolution as low as 0.6 Å have been
developed. Such electron microscopes can resolve atoms and molecules in a material.
Now a days, even tunneling microscopes have been developed which are able to
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