units of measurements in physics

units of measurement class 11

units of measurements are

units of measurements all

a unit of measurement accepted universally

Fundamental and Derived Units

The exact specification of the measurement of a physical quantity requires (i) the standard or unit in which the quantity is

measured.

(ii) the numerical value representing the number of times the quantity contains that unit

The physical quantities which do not depend upon other quantities are called fundanental quantities.

In M.K.S. system the fundamental quantities are mass, length and time, while in more general standard

Internationl (S.I.) system the Fundamental quanti

length, time, temperature,

ties

are

mass,

illuminating power (or luminous intensity), cur rent and amount of substance. The units of funda- mental quantities are called fundamental units.

The units of physical quantities which may be derived from fundamental units are called derived units.

2. Systems of Units

There are following principal systems of units: 1. C.G.S. System: In this system the unit of length is centimetre (cm), that of mass is gram (g) and that of time is second (s).

2. FP.S. Systemn : In this system the unit of length, mass and time are foot (t), pound (lb) and second (s) respectively.

3. M.K.S. System: In this system the units of length, mass and time are metre (m), kilogram (kg) and second (s) respectively.

4. S.I. System: In this system there are seven fundamental quantities whose units and symbols are as follows

as follows

S. N. Fundamental Quantity Unit Symbol

Length

metre

Mass

kilogram kg

Time

second

Temperature

kelvin

Luminous Intensity

candela

Electrie Current

ampere

Amount of Substance

mole

mol

In S.I. system there are two supplementary units.

1. Radian (rad): Unit of plane angle.

2. Steradian (st): Unit of solid angle.

ndUcte

7. Uses of Dimensional Equation

The following are the inmportant dimensional equations 1. Conversion of units of one system to another: This is based on the fact that the specification of a physical quantity requires a proper unit (u) and a numerical value representing the number (n) of units contained in that physical quantity and the product of numerical value contained in and the unit chosen of, physical quantity always remains constant, whatever the system of units may be Le.

uses

the

n lul = constant

[ML T and if (derived) units of that physical

If a physical quantity X has dimensional formula,

quantity in two systems are [M fLPTS] and [ML2T21 respectively end nj and ng be the numerical values in the two systems respectively, then

n1 lul = na lugl

.e.

n M Li T] = ng M L2 T

M no1 M2

Thus knowing the values of fundamental units in two systems and numerical value in one system, the numerical value in other system may be evaluated.

2. To check the correctness of a physical relation: This is based on the principle of homoge- neity of dimensions. According to which the dimensions of all the terms of the tuwo sides of an equation must be

the same.

3. To derive the new relations: If a physical quantity X depends on other physical quantities P, Q and R (say), then we may write

X o pa QbR°

.. (1)

Prefixes for Power of ten

Prefix

Abbreviation Power of Ten

10-18

atto

femto

10-15

pico

109

nano

micro

milli

10-3

centi

102

kilo-

mega-

giga

109

tera

1012

peta

1015

exa

1018

Examples:

1 micro second 1 us = 10s

l nano second = 1 ns = 10 s

1 kilo-metre = 1 km 10" m

4. Some Frequently used Units

1. Micrometer (um) = 1um = 10 m 2. Angstrom 1A = 10 m 3.

Astronomical unit: This is the mean distance of earth from sun

1 AU = 1496 x 10 m = 15x10 m

Light Year: It is the distance traversed by light in vacuum in 1 year i.e., 1 light year

365 x 24 x 60 x 60 x 3x 10°

9.45 x 10 m

Par second: 1 par second

= 3-07 x 106 m = 3.26 light years

6. X-ray Unit (XU) =1 XU = 107 m 8 1 bar = 10° N/m=10° Pa 7. 1 atmosphere (atm) 1-013x 10° Pa 9

1 torr = 1 mm of mg = 133-3 Pa 10. 1 barn = 1028 m 11.

l horse power = 746 watt

-3)

1 ft= 0 3048 13. 1 Pound= 483g=0 46I kg 14 1 poiseuille (lP)= 10 poine 1 metrie ton 10 quintal1000 k

15 16 1 Chandra Shelkhar Limit (CSL)- 14 M, wh

M=mass of sun.

17 1 Shake = 8 S.

5. Dimensions

The dimensions of a physical quantity are powers raised on fundamental1 derived unit of that physical quantity. If a physi- quantity ] has derived unit (mass)", (length)", a (time), then the dimensions of X are symbolica expressed as

units

obtain

X= [M"LbT

The dimensions are calculated as follows

Area = length x breadth = [LJ[L] = [L1

Volume = length x breadth x height

[LI[LI|L = [L']

Velocity displacement [L]

= [LT

time

Acceleration =velocity_[LT 1- ILT-21

time

Force = mass x acceleration = [M][LT 1

= [MLT

Impulse = force x time = [MLT 1[T = [MLT

TT17 esTOS uf ŠUTe Pinysic

www

ysicairGuantitres

Units

decided a

In 1971 CGPM held its meeting and decided

system of units which is known as the International

System of Units. It is abbreviated as SI from the

French name Le Systéme International d'Unités. This system is widely used throughout the world.

Table (1.1) gives the fundamental quantities and their units in SI.

from the

Table 1.1 : Fundamental or Base Quantities

Quantity

Name of the Unit

Symbol

Length

metre

Mass

kilogranm

Time

second

Electric Current

ampere

Thermodynamic Temperature kelvin

Amount of Substance

mole

mol

Luminous Intensity

candela

Besides

the

seven

fundamental

units

two

supplementary units are defined. They are for plane

angle and solid angle. The unit for plane angle is radian with the symbol rad and the unit for the solid

angle is steradian with the symbol sr

SI Prefixes

The magnitudes of physical quantities vary over a wide range. We talk of separation between two protons inside a nucleus which is about 10 15 m m and the distance of a quasar from the earth which is about

10 m. The mass of an electron is 9:1x 10 kg and

- 31

that of our galaxy is about 2:2x 10" kg.

CGPM recommended standard prefixes for certain powers of 10. Table (1.2) shows these prefixes.

Table 1.2: SI prefixes

Power of 10 Prefix Symbol

E exa

exa

peta

tera

giga

mega

kilo

hecto

deka

deci

- 1

- 2

centi

- 3

milli

micro

- 9

nano

- 12

pico

- 15

femto

- 18

atto

1.4 DEFINITIONS OF BASE UNITS

Any standard unit should have the following twc properties

a) Invariability: The standard unit must be invariable. Thus, defining distance between the tip of the middle finger and the elbow as a unit of length is not invariable.

(b) Availability: The standard unit should be easily made available for Comparing with otheer quantities.

The procedures to define a standard value as a unit are quite often not very simple and use modern equipments. Thus, a complete understanding of these procedures cannot be given in the first chapter. We briefly mention the definitions of the base units which may serve as a reference if needed.

Metre

It is the unit of length. The distance travelled by

light in vacuum 299,792,458 second is called 1 m.

Kilogram

The mass of a cylinder made of platinum-iridium alloy kept at International Bureau of Weights and

Measures is defined as 1 kg.

Second

Cesium-133 atom emits electromagnetic radiation of several wavelengths. A particular radiation is selected which corresponds to the transition between the two hyperfine levels of the ground state of Cs-133.

Each radiation has a time period of repetition of certain characteristics.

9,192,631,770 time periods of the selected transition is defined as 1 s.

Ampere

Suppose two long straight wires with negligible cross-section are placed parallel to each other in vacuum at a separation of 1m and electric currents are established in the two in same direction. The wires attract each other. If equal currents are maintained in the two wires so that the force between them is

2x 10 newton per metre of the wires, the current in any of the wires is called 1 A. Here, newton is the SI unit of force.

Kelvin

The fraction 1/273 16

of the

thermodynamic temperature of triple point of water is called 1 K

Mole

The amount of a substance that contains as many elementary entities (molecules or atoms if the

substance is monatomic) as there are number of atoms

in 0012 kg of carbon-12 is called a mole. This number (number of atoms in 0'012 kg of carbon-12) is called

Avogadro constant and its best value available is 6:022045 x 10 with an uncertainty of about 0-000031x 10

Candela

The SI unit of luminous intensity is 1 cd which is the luminous intensity of a blackbody of surface area

m placed at the temperature of freezing

600,000 m platinum and at a pressure of 101,325 N/m', in the direction perpendicular to its surface.

1.5 DIMENSION

All the physical quantities of interest can be derived from the base quantities. When a quantity is expressed in terms of the base quantities, it is written as a product of different powers of the base quantities.

The exponent of a base quantity that enters into the expression, is called the dimension of the quantity in that base. To make it clear, consider the physical quantity force. As we shall learn later, force is equal to mas times acceleration. Acceleration is change in velocity divided by time interval. Velocity is length divided by time interval. Thus,

All

force = mass x acceleration

velocity

mass x

time

length/time

= masS X

time

= mass x length x (time)-2

Thus, the dimensions of force are 1 in mass, 1 in length and-2 in time. The dimensions in all other base quantities are zero. Note that in this type of calculation the magnitudes are not considered. It is equality of the type of quantity that enters. Thus, change in velocity, initial velocity, average velocity, final velocity all are equivalent in this discussion, each one is length/time.

For convenience the

It is

base

quantities

are

represented by one letter symbols. Generally, mass is denoted by M, length by L, time by T and electric current by I. The thermodynamic temperature, the amount of substance and the luminous intensity are denoted by the symbols of their units K, mol and cd respectively. The physical quantity that is expressed in terms of the base quantities is enclosed in square brackets to remind that the equation is among the dimensions and not among the magnitudes. Thus

equation (1.2) may be written as [force]= MLT

Such an expression for a physical quantity in terms

of the base quantities is called the dimensional

Thus, the dimensional formula of force is

formula.

MLT The two versions given below are equivalent and are used interchangeably.

(a) The dimensional formula of force is MIT 2

(b) The dimensions of force are 1 in mass, 1 in length and-2 in time.

Example 1.1

Calculate the dimensional formula of energy from the

equation E=mu

Solution : Dimensionally, E = mass x (velocity), since is

a number and has no dimension.

or, E]-Mx =MLT

1.6 USES OF DIMENSION

A. Homogeneity of Dimensions in an Equation

An equation contains several terms which are separated from each other by the symbols of equality plus or minus. The dimensions of all the terms in an equation must be identical. This is another way of saying that one can add or subtract similar physical quantities. Thus, a velocity cannot be added to a force

Or an electric current cannot be subtracted from the thermodynamic temperature. This simple principle 1s called the principle of homogeneity of dimensions in a equation and is an extremely useful method to check whether an equation may be correct or not. If the dimensions of all the terms are not same, the equatio must be wrong. Let us check the equation

X=ut+1/2at<2

for the dima

the dimensional homogeneity. Here x is the distan.

travelled by a particle in time t which starts at a speed u and has an accelerationa along the direction of

e distance

motion.

x]=L

ut]=velocity x time =- time

length

lengn time =L

at lat = acceleration x (time)

velocity (time) 2 - length/time x (time)" = L

ime

time

Thus the equation is correct as far as the dimensions

are concerned.

Limitation of the Method

oe nat the dimension of - at is same as that

at Fure numbers are dimensionless. Dimension aoes not depend on the maonitude. Due to this reason

Lhe equation x =ut + at is also dimensionally correct.

Thus, a dimensionally correct equation need not be actually correct but a dimensionally wrong equauo must be wrong.

Example 1.2

Test dimensionaly if the formula t =2 TDi may be

correct, where t is time period, m is mass, F is force and x is distance.

Solution: The dimension of force is MLT. Thus, the dimension of the right-hand side

MLT/L

The left-hand side is time period and hence the dimension is T. The dimensions of both sides are equal

and hence the formula may be correct.

B. Conversion of Units

When we choose to work with a different set of units for the base quantities, the units of all the derived quantities must be changed. Dimensions can be useful in finding the conversion factor for the unit of a derived physical quantity from one system to other. Consider an example. When SI units are used, the unit of pressure is pascal. Suppose we choose 1 cm as the unit of length, I g as the unit of mass and 1 s as the unit of ime (this system is still in wide use and is called CGS system). The unit of pressure will he different in this system. Let us call it for the time- being 1 CGS pressure. NOW, how many CGS pressure

is equal to 1 pascal ?

Let us first write the dimensional formula of

pressure.

We have

P=F/A

Thus. [P]=[F]/[A]=MLT^-2/L^2=ML^-1T^-2

Thus,

1 pascal = (1 kg) (1 m) (1 s) -2

So,

Thus, 1 CGS pressure 1 kg1m 1s)

1 CGS pressure = (1 g) (1 cm) (1 s) (1s 1s)

and

1 pascal

1g1 cm = (10^3) (10^2)^-1= 10

or,

1 pascal = 10 CGS pressure.

Thus, knowing the conversion factors for the hase quantities, one can work out the conversion factor for any derived quantity if the dimensional formula of the derived quantity is known.

C. Deducing Relation among the Physical Quantities

Sometimes dimensions can be used to deduce aa relation between the physical quantities. If one knows the quantities on which a particular physical quantity depends and if one guesses that this dependence is of product type, method of dimension may be helpful in the derivation of the relation. Taking an example, suppose we have to derive the expression for the time period of a simple pendulum. The simple pendulum has a bob, attached to a string, which oscillates under the action of the force of gravity. Thus, the time period may depend on the length of the string, the mass of the bob and the acceleration due to gravity. We assume that the dependence of time period on these quantities is of product type, that is,

t = kl^am^bg^c

(1.3)

where k is a dimensionless constant and a, b and c

are exponents which we want to evaluate. Taking the dimensions of both sides,

T L"M 'LT 2°=L°+eM 'T -2

Since the dimensions on both sides must be identical,

we have

a+c= 0

b = 0

and

-2c = 1

giving a = b=0 and c=-

Putting these values in equation (1.3)

t=k(l/g)^1/2

Thus, by dimensional analysis we can deduce tha t the tinme period of a simple pendulum is independenu of its mass, is proportional to the square root of the length of the pendulum and is inversely proportional to the square root of the acceleration due to gravity au the place of observation.

Limitations of the Dimensional Method

Although dimensional analysis is very useful in far.

deducing certain relations, it cannot lead us too far.

First of all we have to know the quantities on which a particular physical quantity depends. Even then the

method works only if the dependence is of the product type. For example, the distance travelled by a uniformly accelerated particle depends on the initial velocity u, the acceleration a and the time t. But the

method of dimensions cannot lead us to the correct expressi0n for x because the expression is not of of product type. 1t is equal to the sum of two terms as

ut +at

Secondly, the

dimensions cannot be deduced by the method of

numerical constants having no

dimensions. In the example of time period of a simple pendulum, an unknown constant k remains in equation (1.4). One has to know from somewhere else that this

constant is 2t.

Thirdly, the method works only if there are as

many equations available as there are unknowns. In mechanical quantities, only three base quantities length, mass and time enter. So, dimensions of these three may be equated in the guessed relation giving at most three equations in the exponents. If a particular quantity (in mechanics) depends on more than three quantities we shall have more unknowns and less equations. The exponents cannot be determined uniquely in such a case. Similar constraints are present for electrical or other nonmechanical quantities.

ORDER OF MAGNITUDE

In physics, we come across quantities which vary over a wide range. We talk of the size of a mountain and the size of the tip of a pin. We talk of the mass of our galaxy and the mass of a hydrogen atom. We talk of the age of the universe and the time taken by an electron to complete a circle around the proton in a hydrogen atom. It becomes quite difficult to get a feel of largeness or smallness of such quantities. To express such widely varying numbers, one uses thee powers of ten method.

In this method, each number is expressed as

ax 10 where 1 sa < 10 and b is a positive or negative integer. Thus the diameter of the sun is expressed as

1-39 x 10 m and the diameter of a hydrogen atom as

1-06 x 10 10 m. To get an approximate idea of the number, one may round the number a to 1 if it is less than or equal to 5 and to 10 if it is greater than 5.

The number can then be expressed approximately as

10. We then get the order of magnitude of that number. Thus, the diameter of the sun is of the order of 10 m and that of a hydrogen atom is of the order of 10 a representation is called the order of magnitude of that quantity. Thus, the diameter of the sun is 19 orders of magnitude larger than the diameter of a hydrogen atom. This is because the order of magnitude

m. More precisely, the exponent of 10 in such

of 10 is 9 and of 10 is 10. The difference is

9--10) = 19.

To quickly get an approximate value of a quantity in a given physical situation, one can make an order

magnitude calculation. In this all numbers are approximated to 10 form and the calculation is made

Let us estimate the number of persons that mav sit in a circular field of radius 800 m. The area of the field is

A =Tr=3:14 x (800 m) 10 m2

The average area one person occupies in sitting

sitting

50 cm x 50 cm =0:25 m = 2:5 x 10 m = 10 m 2

The number of persons who can sit in the field is

N 10m2 = 10

10 1 m

Thus of the order of 10 persons may sit in the field

THE STRUCTURE OF WORLD

Man has always been interested to find how the world is structured. Long long ago scientists suggested that the world is made up of certain indivisible small particles. The number of particles in the world is large but the varieties of particles are not many. Old ndian philosopher Kanadi derives his name from this proposition (In Sanskrit or Hindi Kana means a smal particle). After extensive experimental work people arrived at the conclusion that the world is made up of

just three types of ultimate particles, the proton, the

neutron and the electron. All objects which we have

around us, are aggregation of atoms and molecules,

The molecules are composed of atoms and the atoms have at their heart a nucleus containing protons and neutrons. Electrons move around this nucleus in

special arrangements. It is the number of protons,

neutrons and electrons in an atom that decides all the

properties and behaviour of a material. Large number of atoms combine to form an object of moderate or large size. However, the laws that we generally deduce for these macroscopic objects are not always applicable to atoms, molecules, nuclei or the elementary particles.

These laws known as classical physics deal with large size objects only. When we say a particle in classical physics we mean an object which is small as compared to other moderate or large size objects and for which the classical physics is valid. It may still contain millions and millions of atoms in it. Thus, a particle of dust dealt in classical physics may contain about

18 10 atoms.

Twentieth century experiments have revealed another aspect of the construction of world. There are perhaps no ultimate indivisible particles. Hundreds of elementary particles have been discovered and there are free transformations from one such particle to the other. Nature is seen to be a well-connected entity.

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units of measurements in physics

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