"When you can measure what you are talking about and express in numbers, you know something about it".
- Lord Kelvin
Necessity of measurement
'Science' is a Latin word which means 'to know'.
During the time of Aristotle, scientific knowledge was confined to observations made by our senses only, which varied from person to person. This is called 'Subjective Observation'.
The observation which is same for all persons is called 'Objective Observation' which
involves 'measurement'.
* A science which involves measurement is called 'Quantitative Science'. For observations to be same for all persons, Quantitative study should be made.
PHYSICS is the study of 'Matter' and 'Energy'. It involves precise measurement and hence it is a quantitative science.
* Something that can be measured is called a 'Quantity'. Quantities that occur in Physics are called 'Physical Quantities' which can be measured with certain standard instruments.
e.g.: Length of a rod, mass of sugar, volume of water, speed of a car, force applied on a body, charge of an electron, time taken to read this lesson etc.
Physical Quantities are classified as i) Fundamental (base) quantities and
ii) Derived Quantities.
Fundamental Quantities: Physical Quantities length, mass, time which can exist independently without depending on one another are called Fundamental Quantities.
Derived Quantities: Physical Quantities like volume, density, velocity, force, specific heat, surface tension, moment of inertia, magnetic moment, electric field strength etc., are the physical quantities which are expressed in terms of fundamental quantities are called Derived (obtained by using fundamental quantities) Quantities.
How to express a Physical Quantity?
Physical Quantity length is expressed as 50 meters; mass as 15 kilograms; time as 2 hours; velocity as 25 kilometers per hour; force as 12 newtons; density as 5 kilograms per meter3.
If you observe carefully the above illustrations, it is clear that every Physical Quantity should be expressed (or represented) by i) a pure number (50, 15, 2 etc.)
Note: 50, 54.52, are pure numbers, but not numbers like which is a complex number.
ii) A name associated with the number called 'UNIT' (meter, kilogram, hour, kilogram per meter3 etc.)
Both are necessary: The description of any Physical Quantity is complete only in terms of both the above.
The length of an object cannot be mentioned as 50, because it may be 50 meters or 50 feet or 50 centimeters.
Similarly, if length is mentioned merely in meters, it may be 10 meters or 100 meters.
'Butterfly' can be a Physical Quantity!
Every Physical Quantity should be expressed in terms of fundamental quantities. Even a term like 'butterfly' can be a Physical Quantity and find place in Physics, if it is expressed in terms of fundamental quantities. If not a term like force (The central theme of Physics) has no place in this branch of Science.
Earlier there are only 'Three'!
In earlier centuries, when the subject Physics was in a formative stages, there were only three fundamental quantities namely 'length', 'mass' and 'time'.
UNITS: Historically, units have been devised by men to communicate the meaning of measurements to one another. The process of measurement is basically a 'comparison' process. Measuring any thing means comparing it with some standard to see how many times as big, as small it is.
Unit is a standard for measuring a Physical Quantity.
* The units of fundamental quantities are called fundamental units which are independent of other units.
e.g.: Unit of length 'meter'.
Unit of mass 'kilogram'.
Unit of time 'second'.
* The units of derived quantities are called 'derived units' which are expressed in terms of fundamental units.
For example, the unit of speed is expressed by dividing the unit of distance (length), meter by the unit of time, the second. Hence the unit of speed is meter/ second or ms-1 which is a derived unit.
Systems of Units:
There are four systems of units.
i) British or F.P.S. system
ii) French or C.G.S. or Gaussian system (metric system)
iii) M.K.S. System
iv) SI (System Internationale)
LENGTH: To certain extent it is a visible Physical Quantity. We can see the length of a wire, height of a person, distance travelled by a car.
The unit of length in British System is 'FOOT'.
Anyway, whose 'Foot' it was?
* The earliest form of culture for humanity was 'Agriculture'. Agriculture needs a fertile land and a perennial supply of water. Thus the earlier civilizations developed on the banks of rivers like Nile in Egypt and Ganges in India. Agricultural land was allotted for each family in the village by the 'Village Head'. The Village Head demarcates the land for each family according to their needs by measuring the length and breadth by his 'foot (being 'Head' he never bends and measures by hand) and posting stones at the four corners of the plots.
Though the rivers are supplying water for the fields and for the drinking and other domestic purposes, now and then their fury was shown in the form of floods which used to wash away not only the crops but also stone markings in the fields. Again the village Head should come and measure the land after the receding of the flood for each family with his 'foot' and make a fresh allotment.
Once it happened when rivers flooded, along with the crop. Few people of the village including the Village Head were washed away. Then the rest of the community required a standard tool for the measurement of 'length' (and even breadth) of the land. 'Necessity is the mother of invention'. Thus they have settled with a long stick which was chosen arbitrarily for measuring 'Length' and called it 'Foot' to honour the Village Head. Thus, the first measurement made was that of 'Length' of the earth. Geometry - 'Geo' means earth and 'metry' is the measurement.
Centuries later, in British system of measurement of length, the standard tool was a brass rod chosen arbitrarily and is called the 'British Imperial rod', and one third of which is a 'Foot'. So, 'foot' in British system is not the foot of any British emperor!
* Fundamental units of 'length' in British and French systems are defined in terms of length of long rods. If one third of the brass rod (called British Imperial yard) kept between two Gold plugs (royal colour) called 'foot' is the basic unit for length in British system, the length of a Platinum-Iridium rod kept in a vault at 0°C, at bureau of weights and measures at Severes, Paris is called 'meter' and 1/100th of which is 'centimeter', the basic unit for length in French system.
MASS: 'Mass' before Isaac Newton is defined as quantity of substance contained in a body Newton defined 'Mass' in terms of inertia of a body (which you will learn in chapter of DYNAMICS). Fundamental units of 'Mass' both in British and French systems are defined in terms of lump of metals.
Mass of lump of Platinum, taken arbitrarily is named as 'Pound' (lb) is the unit of mass
in British system (Pounding means making flour out of grain by using a heavy stone. Probably, the 'pound' may be used for this purpose). Another lump of Platinum-Iridium kept at Severes is called kilogram, and 1/1000 th mass of the same, called gram is the basic unit of mass in French system.
* Mass of a body cannot be seen, but it can be felt.
TIME: The length of a body can be realised visually. Mass of a body can be felt by experiencing its 'weight' by lifting the body. But what about 'Time'? In fact what is 'Time'?
One can answer the question 'What is time now'? But, it is not so easy to answer the question 'What is Time'? 'Time' is an abstract term compared to 'length' and 'mass'. The universe is full of 'events'. A fruit about to fall from a tree is an event. The same fruit striking the ground is another event. Between these two events, some physical quantity is lapsing. One can visit Delhi and Kolkata. But, one cannot be at both the places at the same instance. The physical quantity that lapses between events may be defined as'TIME'.
In defining the unit for time, one cannot consider any two events like one going to the college from home and same person returning home after the college. Such events will not occur regularly and systematically.
Motion of Earth: Hence for defining the unit for time, the motion of Earth was considered. But we are not experiencing the Earth's motion. Hence the motion of the Sun (In fact Sun does not move with respect to Earth) is observed. Motion of Sun we observe is due to the motion of the Earth.
When Sun reaches meridian (which we can know by the shortest shadow cast by the Sun. Which is at midday or noon) as one event, and the Sun consecutively reaching the meridian again in the same direction as the second event, the time lapsed between these consecutive events is called 'Solar day' and making many such observations, the mean solar day is calculated. Now × 60 × 60 = part of the mean solar day is called 'Second' which is defined as the basic unit of time in British system.
¤ As there are no British Sun and French Sun separately, French system also adopted
'Second' as the fundamental unit of time!!
Characteristics of a standard Unit
i) It must be indestructible.
ii) It must be reproducible to a high degree of accuracy anywhere in the world.
iii) It must not change with time.
iv) It should not change with the changes in temperature, pressure etc.
* The units of F.P.S. as well as C.G.S. systems could not satisfy these conditions to a very large extent.
But, the 'atomic standard' adopted by Committee of Weights and Measures, CGPM in 1960 satisfied the above aspects and the system of units is called International system or SI (Systems de Internationale) which will replace all other systems in use for scientific purposes.
SI Units
SI has distinct advantage over other systems of units.
Its, advantages are that it is, comprehensive, coherent and rational.
It is comprehensive because that its seven fundamental units cover all disciplines of science and engineering i.e, mechanical, electrical, thermodynamic and optical units.
* It is coherent because all the derived units are obtained by multiplication or division from certain set of basic units, known as fundamental units. For example in a coherent system, the unit of area meter2 (m2) is obtained when a fundamental unit of length meter (m) is multiplied by another unit of length meter (m).
SI is rational because it has absorbed in itself the rationalised Meter, Kilogram, Second (M.K.S.), Ampere System (RMKSA System) devised by Prof. Giorgi for use in electro technical sciences (Ampere is the unit for electric current).
SI - a common language: Earlier there was a communication gap between scientists, engineers and technologists. While scientists used absolute Units (independent of gravitational force), the engineers used gravitational units (depending on gravitational force which are more Practical Units) and technologists used both. Now an absolute system like SI would bridge this gulf and provide common language in which they can communicate with each other.
Easy for Problems: The use of SI units eliminates odd conversion factors which we come across in other systems of units. In framing problems using SI units, the emphasis can be on the basic principles than on choosing numbers which get cancelled out with conversion factors.
The Unit of velocity meter per second (ms-1) results when the unit of length meter (m) is divided by unit of time second (s).
The coherent unit of velocity is meter per second (ms-1) and not kilometer per hour (km hr-1).
SI contains three categories of units
i) Seven Fundamental Units ii) Two Supplementary Units
iii) Derived Units
Fundamental SI Units
i) Unit of length: metre (m)
It is 1650 763 73 wavelengths of the orange-red line of Krypton-86 at 15 °C and 76 cm of mercury corresponding to the transitions between the levels 2 p10 and 5d5.
By this unit nano lengths (10-9 m) can be measured precisely.
But in 1983, CGPM adopted a new definition for 'meter' in terms of velocity of light in vacuum.
'meter' is the distance travelled by light in vacuum in @@@@@@555 second.
ii) Unit of mass: kilogram (kg)
It is the mass of a cylinder of Platinum-Iridium alloy kept in the International Bureau of Weights and Measures at Severes, near Paris and is called 'International Prototype Kilogram'
An atomic standard of mass has not yet been adopted, because it is not yet possible to measure the masses on atomic scale as great precision as on macroscopic scale.
iii) Unit of time: second (s)
It is the time taken for 9192631770 vibrations of Caesium-133 atom.
It is called Caesium (atomic) clock.
In M.K.S. system second is measured in terms of rotation of the Earth about its axis. But this rotation rate is not constant with time. Moreover, in atomic Physics and Electronics, we have to measure time intervals of the order of a nano second (10-9 s) or less. To achieve this, we must use recurring atomic phenomena.
iv) Unit of electric current - ampere (A)
One 'ampere' is that current which when flowing in each of two parallel conductors of infinite length and negligible cross section and placed one meter apart in air or vacuum, causes each conductor to experience a force of 2 × 10-7 newtons per meter of length.
v) Unit of temperature: kelvin (K)
The kelvin unit of thermodynamic temperature is the fraction of 1/273.16 of the thermo- dynamic temperature of the 'triple point' of water.
Triple point of water is 273.16 K at 610 Pa (or) 4.6 mm of mercury. It means that at 273.16 K and at a pressure of 610 pascals (Pa), ice, water and steam coexist in equilibrium with one another. The Unit is kelvin and not degree kelvin.
vi) Unit of Luminous intensity - candela (cd)
It is the luminous intensity in a direction normal to the surface of 1/6 × 10-5 m2 of a
black body at the temperature of freezing Platinum (2046 K) at a pressure of 101325 newtons per square meter.
Recent Definition: The candela is the luminous intensity in a given direction, of a source
that emits monochromatic (Single colour) radiation of frequency 540 × 1012 hertz and that has a radiant intensity of 1/683 watt per Steradian in that direction.
vii) Unit of amount of substance: mole (mol)
'mole' is merely a number, just as 'pair' is a number which denotes 'two' and 'dozen' is a number which denotes 'twelve' entities. Similarly mole is a number which denotes 6.02252 × 1023 entities (Avogadro no.).
'mole' is the amount of the substance of a system that contains as many elementary entities as there are atoms in 0.012 kg of Carbon-12.
The elementary entities must be specified which may be an atom, a molecule, an ion,
an electron, a photon etc. (Potatoes, tomatoes or number of spectators witnessing a cricket
match can also be denoted in terms of moles, but it is not practicable, as mole is a very
big number).
Why Carbon-12 is considered? Because in nature Carbon-12 is abundant both in living
and non living things. 12 (0.012 kg) of Carbon is considered as molar mass (molecular weight) of Carbon is 12 gms.
Supplementary Units
i) Radian: It is the unit of 'Plane angle'.
'Radian' is the angle subtended at the centre of a circle by an arc whose length is equal to its radius.
ii) Steradian: It is the unit of solid angle.
It can be seen at the tip of an ice cream cone and at the corner of a room where the adjacent walls and roof (or floor) meet. Solid angle is considered at the source from which a light beam or magnetic flux is emerging (a conical shape, the tip of which subtends a solid angle). 
'The ratio of the area of the spherical surface intercepted by the solid angle to the square of the radius is the measure of the solid angle in steradian'.
If Δs be the area of spherical cap intercepted by the solid angle Δω, then where R is the radius of the spherical surface.
SI-an orthodox system!!
Rules and conventions followed in SI
i) Full names of Units, even when they are named after a scientist are not written with a capital letter.
e.g.: newton and not Newton; ampere and not Ampere; meter and not Meter; mole and
not Mole.
ii) Symbol for a unit, named after a scientist has a capital letter.
e.g.: N for newton; A for ampere
iii) Symbols for other units also are not written with capital letters.
e.g.: m for meter and not M.
iv) Units may be written in full or agreed symbols, but no other abbreviations may be used.
e.g.: meter or m and not mt. second or s and not sec.
v) Symbols for units do not take plural form.
e.g.: 50 kg and not as 50 kgs; 25 N and not as 25 Ns; 10 m and not as 10 ms
vi) No full stop or other punctuation marks should be used within or at the end of symbols for units.
e.g.: 10 m and not as 10 m., 20 ms-1 and not as 20 m.s-1
SI units and not as S.I Units
vii) Space is to be left between the numerical and symbol and also between the symbols for compound units such as work, power etc.
e.g.: 10 s and not as 10s
N m and not as Nm
viii) solidus (/) should not be used. e.g.: ms-1 and not m/s
ix) The word 'specific' is confined to 'per unit mass'.
e.g.: Specific Latent heat means Latent heat per unit mass.
The term specific gravity should no more be used. Instead relative density should be used.
x) The decimal point is a level point.
e.g.: 12.1 and not 12.1
Do you know?
* Light year is not the unit of time but it is a unit of distance.
* Light year is the distance travelled by the light in one year at a speed of 3 × 108 ms-1
1 Light year = 365 × 86400 × 3 ×108
= 9.46 × 1015 m
* Par second (pc) or parallactic second is the unit of length used to express astronomical distances.
* One astronomical unit (AU) is the mean sunearth distance = 1.496 × 1011 m
1 Par second = 2.062 × 105 AU
1 Light year = 6.32 × 104 AU
1 Light year = 0.31 Par second
1 Par second = 3.26 Light years
= 3.08 × 1016 m
1 angstrom (A°) = 10-10 m
1 micron = 10-6 m
1 nanometer = 10-9 m
1 fermi = 10-15 m
Dimensional formula and Dimensions
A code language!
Mathematics is a language of brevity. Small is beautiful. Mathematics is a tool for Physics. Nature's secrets are revealed through Physics briefly and effectively by Mathematical equations.
Mathematics uses a code language such as , which means therefore; means since; = means equal to etc. Imitating the style of Mathematics, Physics developed a code language to express the Derived Physical quantities in terms of fundamental units (and not fundamental quantities as some text books mention - "please note") and these code expressions are called 'Dimensional formula' and the indices of the powers to which the fundamental units are raised are called 'Dimensions'.
Dimensional formula testifies the authenticity of a Physical quantity and its uses are manifold.
Naming the fundamental units
unit of length is represented as [ L ]
unit of mass as [M ]
unit of time as [ T ]
unit of electric current as [ I ]
unit of temperature as [ K ]
unit of luminous intensity as [ C ]
Warning: [L] It is not length, it is Unit of Length; so the other terms.
¤ The Dimensional formula (or equation) of a Physical quantity is an expression showing
its relation to the fundamental units.
* The Dimensions of a Physical quantity are the powers to which the fundamental units are raised to obtain the unit of derived quantity.
If [Q] is the unit of derived quantity it can be expressed in terms of the fundamental units in the form [Q] = [ Ma Lb Tc ]
Where L, M, T stand for the unit of length, mass and time respectively.
The unit Q is said to have 'a' dimensions of mass, 'b' dimensions of length and 'c' dimensions of time where a, b, c are pure numbers.
e.g.: Derive the dimensional formula and dimensions of pressure.
Thus, Dimensional formula for pressure is [ M1 L-1 T-2 ]
The square bracket indicates 'dimensions of'
Dimensions of pressure are 1 in M, -1 in L and -2 in T.
Dimensionless quantities
Variables like velocity, force etc., have dimensions. But variables like strain, angle, relative density coefficient of friction are dimensionless, as they are ratios of two quantities having same dimensional formula.
Constants like velocity of light and gravitational constant have dimensions. But constants such as π, numbers etc., are dimensionless 'mole' is dimensionless as it is merely a number.
Quantities having 'same' dimensional formula
i) frequency and angular velocity
ii) work and torque
iii) plancks constant and angular momentum
iv) pressure and Young's modules
v) impulse and momentum
Principle of homogeneity of dimensions
This principle states that the dimensions of each term on the two sides of an equation must be the same.
Uses of Dimensional formula
Dimensional formula is used
i) to convert one system of units into another
ii) to check the correctness of an equation
iii) to derive possible relationship between different Physical Quantities
Note: Please refer 'Intermediate Exercises' for the detailed study of the above.
