Units and Measurements - Detailed NCERT based notes Class 11 Physics Chapter 2

2. UNITS AND MEASUREMENTS

2.1 INTRODUCTION

Units: The certain basic, arbitrarily chosen, internationally accepted reference standards are called units. Example : kilogram, metre, metre per second, etc.



Measurement: The comparision of a physical quantity with a certain basic, arbitrarily chosen, internationally accepted reference standard is called its measurement.



Classification of units:

(i) Fundamental or base units: The units for the fundamental or base quantities are called Fundamental or base units. They are not derived from/independent on other units. Example : metre, kilogram, second, etc.

(ii) Derived units: The units which can be expressed as the combination of the base units are called derived units. Example : metre per second, kilogram metre per square of second, etc.

NOTE : The complete set of these units, both the base units and derived units, is known as the system of units.

2.2 THE INTERNATIONAL SYSTEM OF UNITS

International System of Units (SI) : The system of units which is internationally aceepted for measurement at present is the International System of Units abbreviated as SI from the French name Le Systeme International d' Unites.



➤The SI system has seven base units as given in Table 2.1.

Table 2.1: SI Base quantities and Units

Basic quantity	SI Units
	Name	Symbol	Definition
Length	metre	m	The metre is the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second. (1983)
Mass	kilogram	kg	The kilogram is equal to the mass of the international prototype of the kilogram (a platinum-iridium alloy cylinder) kept at international Bureau of Weights and Measures, at Serves, near Paris, France. (1889)
Time	second	s	The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium -133 atom. (1967)
Electric current	ampere	A	The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2x10$^{-7}$ newton per metre of length. (1948)
Thermo dynamic temperature	kelvin	K	The kelvin, is fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (1967)
Amount of substance	mole	mol	The mole is the amount of substance of a system, which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12. (1971)
Luminous intensity	candela	cd	The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540x10$^{12}$ hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. (1979)

NOTE : Beside these basic units, there are two more units that are defined for .

(i) Plane angle: The angle subtend by a given arc of a circle at its center is called plane angle. Its SI unit is radian.


Source:NCERT Book

Mathematically,

Plane angle(d$\theta$)=$\frac{arc\hspace{0.1cm}length(ds)}{radius(r)}$

(ii) Solid angle: The angle subtend by a given surface of a sphere at its center is called solid. Its SI unit is steradian.


Source:NCERT Book

Mathematically,

Solid angle(d$\Omega$)=$\frac{intercepted\hspace{0.1cm}area(dA)}{square\hspace{0.1cm}of\hspace{0.1cm}radius(r^2)}$

The earlier systems of units for measurement that were in use extensively till recently:

(i) CGS system : The base units for length, mass and time in this system are centimeter, gram and second respectively.

(ii) FPS system : The base units for length, mass and time in this system are foot, pound and second respectively.

(iii) MKS system : The base units for length, mass and time in this system are meter, kilogram and second respectively.

Table 2.2 : Some units retained for general use (Though outside SI)

Name	Symbol	Value in SI Unit
Minute	min	60 s
hour	h	60 min = 3600 s
day	d	24 h = 86400 s
year	y	365.25 d = 3.156 x 10$^7$ s
degree	°	1° = ($\pi$/180) rad
litre	L	1 dm$^3$ = 10$^{-3}$ m$^3$
tonne	t	10$^3$ kg
carat	c	200mg
bar	bar	0.1 MPa = 10$^5$ Pa
curie	Ci	3.7 x 10$^{10}$ s$^{-1}$
roentgen	R	2.58 x 10$^{-4}$ C/kg
quintal	q	100kg
barn	b	100 fm$^2$ = 10$^{-28}$ m$^2$
are	a	1 dam$^2$ = 10$^2$ m$^2$
hectare	ha	1 hm$^2$ = 10$^4$ m$^2$
standard atmospheric pressure	atm	101325 Pa = 1.013 x 10$^5$ Pa

➤ Some Collected Important Facts:

♞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.

2.3 MEASUREMENT OF LENGTH

Methods of measurements:

(i) Direct Method: The method through which measurement of the target is done directly with measuring instruments like metre sacle, vernier calipers, screw gauge,etc is called direct method of measurements. For example :

• A metre scale is used for length from 10$^{-3}$ m to 10$^{2}$ m.


• A vernier callipers is used for lengths to an accuracy of 10$^{-4}$ m.


• A screw gauge and a spherometer can be used to measure lengths as less as to 10$^{-4}$ m.

(ii) Indirect method: The method through which measurement of the needed is done by using different measurements is called indirect method of measurements. For example :

• Measurement of large distance with parallax method.

2.3.1 Measurement of Large Distance

➤An important method that is used in measurement of large distances such as the distance of a planet or a star from the earth is Parallax Method.



Some important terms related to parallax method:

(i) Parallax : The displacement in the apparent position of an object as observed from two different points not on a straight line is called parallax.

(ii) Basis : The distance between the two points of observation is called basis.

(iii) Parallax angle/Parallatic angle : The angle between the two directions along which the object is observed from the two different points is called Parallax angle/Parallatic angle.

Estimation of large distance by parallax method:



Source:NCERT Book

Say,

S : The planet to be observed.
A,B : The points of observation.
D : The distance of the planet that to be measured.
b : The basis.
$\theta$ : The parallax angle.

As the planet is very far away, $\frac{b}{D}$ << 1, and therefore $\theta$ is ver small.
Then AB can be considered as arc of length b of a circle with centre at S and the radius D.
So,

b = D $\theta$ => D = $\frac{b}{\theta}$

Estimation of angular size of a distant object by parallax method:

Say,

d : The diameter (AB) of the planet.
D : The distance of the end points of the diameter from point of observation (O).
$\alpha$ : The angular size of the planet.

Similarly,

$\alpha$ = $\frac{d}{D}$ => d = D $\alpha$

2.3.2 Estimation of Very Small Distances : Size of a Molecule

➤ An electron microscope (resolution : 0.6A°) uses electron beam as waves focussed by properly designed electric and magnetic field and can almost resolve atoms and molecues in a material.


➤In recent time, a tunnelling microscope has ability to estimate the size of molecules.


Procedure for the formation of mono-molecular layer of oleic acid on water surface:

➤ Oleic acid is a soapy liquid with large molecular size of the order of 10$^{-9}$m.



(i) 1 cm$^3$ of oleic acid is dissolved in alcohol to make a solution of 20 cm$^3$.

(ii) Then 1 cm$^3$ of this solution is diluted to 20 cm$^3$ using alcohol making its concentration $ \frac{1}{20*20}$ cm$^3$ of oleic acid/cm$^3$.

(iii) Next, some lycopodium powder is sprinkled lightly on the surface of water in a large trough and one drop of this solution is put in the water.

(iv) Leave the oleic acid drop to spread into a thin, large and roughly circular film of molecular thickness on water surface.

Estimation of the molecular size/diameter of oleic acid:

Say,

A : Area of the thin film of oleic acid solution formed on the water surface(in cm$^2$).
n : Number of drops of oleic acid solution dropped on the water surface.
V : Approximate volume of each drop of the solution(in cm$^3$).

Volume of n drops of solution = nV cm$^3$

Amount of oleic acid in this solution = nV ($\frac{1}{20*20}$)cm$^3$

So, the thickness of the film,t = $ \frac{Volume\hspace{0.1cm} of\hspace{0.1cm} the\hspace{0.1cm} film}{Area\hspace{0.1cm} of\hspace{0.1cm} the\hspace{0.1cm} film}$

or, t = $ \frac{nV}{20*20*A}$ cm.

Since, the film has mono-molecular thickness.


So, the size/diameter of a molecule of oleic acid, d = t = $ \frac{nV}{20*20*A}$ cm

➤ Some Collected Important Facts:

♞ An optical microscope uses visible light and hence the resolution to which it can be used is the wavelength of light (about 4000A° to 7000A°). This is not used to resolve the size of atoms and molecules.

2.3.3 Range of Lengths

➤Has a wide range from the order of 10$^{-15}$m (size of a proton) to the order of 10$^{26}$m (size of observable universe).

Table 2.3 : Range and order of lenghts

Size of object or distance	Length (m)
Size of a proton	10$^{-15}$
Size of atomic nucleus	10$^{-14}$
Size of hydrogen atom	10$^{-10}$
Length of typical virus	10$^{-8}$
Wavelength of light	10$^{-7}$
Size of red blood corpuscle	10$^{-5}$
Thickness of a paper	10$^{-4}$
Height of the Mount Everest above sea level	10$^{4}$
Radius of the Earth	10$^{7}$
Distance of moon from the Earth	10$^{8}$
Distance of the Sun from the Earth	10$^{11}$
Distance of Pluto from the Earth	10$^{13}$
Size of our galaxy	10$^{21}$
Distance to Andromeda galaxy	10$^{22}$
Distance to the boundary of observable universe	10$^{26}$

Table 2.4 : Certain special length units for short and large lengths

Name	Symbol	Value in SI unit
Fermi	f	1f = 10$^{-15}$m
Angstrom	A°	1 A° = 10$^{-10}$m
astronomical unit	AU	1AU (average distance of the Sun from the Earth) = 1.496 x 10$^{11}$m
light year	ly	1 ly (distance that light travels with velocity of 3 x 10$^8$ ms$^{-1}$ in 1 year) = 9.46 x 10$^{15}$m
parsec		1 parsec (the distance at which average radius of earth's orbit subtends an angle of 1 arc second) = 3.08 x 10$^{16}$m

2.4 MEASUREMENT OF MASS

➤ Mass of commonly available objects can be determined by a common balance.

➤ Large masses in the universe like planets, stars, etc., based on Newton's law of gravitation can be measured using gravitational method.

➤ For measurement of small masses of atomic/subatomic particles etc., we make use of mass spectrograph in which radius of the trajectory is proportional to the mass of a charged particle moving in uniform electric field and magnetic field.

NOTE : An important standard unit of mass called unified atomic mass unit is used in case of atoms and molecules.

1 unified atomic mass unit=1u
=(1/12) of the mass of an atom of carbon-12 isotope ($^{12}_6C$) including the mass of the electrons = 1.66 x 10$^{-27}$ kg

2.4.1 Range of masses

➤The masses of the objects vary over a very wide range. These may vary from tiny mass of the order of $10^{-30}$ kg of an electron to the huge mass of about $10^{55}$ kg of the known universe.

Table 2.5 : Range and order of masses

Object	Mass (kg)
Electron	10$^{-30}$
Proton	10$^{-27}$
Uranium atom	10$^{-25}$
Red blood cell	10$^{-13}$
Dust particle	10$^{-9}$
Rain drop	10$^{-6}$
Mosquito	10$^{-5}$
Grape	10$^{-3}$
Human	10$^{2}$
Automobile	10$^{3}$
Boeing 747 aircraft	10$^{8}$
Moon	10$^{23}$
Earth	10$^{25}$
Sun	10$^{30}$
Milky way galaxy	10$^{41}$
Observable universe	10$^{55}$

➤ Some Collected Important Facts:

♞ The prototypes of the International standard kilogram supplied by the International Bureau of Weights and Measures (BIPM) are available at the National Physical Laboratory (NPL), New Delhi in India.

2.5 MEASUREMENT OF TIME

Clock : The device that is used to measure time is called clock.



Cesium clock/Atomic clock : The clock in which an atomic standard of time is used and the second is taken as the time nedded for 9,192,631,770 vibrations of the radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium-133 atom.



➤The efficient cesium atomic clock are so accurate that they impart the uncertainity in the time realisation as ±1x$10^{-13}$ (the maximum uncertainity gained/losed 3μs in one year) .

➤ Some Collected Important Facts:

♞ A cesium atomic clock is used at the National Physical Laboratory (NPL), New Delhi to maintain the indian standard of time.

2.5.1 Range of time intervals

➤ Vary over a very wide range. These may vary from the time intervals of the order of $10^{-24}$ s of life - span of most unstable particle to the huge time intervals of order of $10^{17}$ s of age of the universe.

Table 2.6 : Range and order of time intervals

Event	Time interval(s)
Life-span of most unstable particle	10$^{-24}$
Time required for light to cross a nuclear distance	10$^{-22}$
Period of x-rays	10$^{-19}$
Period of atomic vibrations	10$^{-15}$
Period of light wave	10$^{-15}$
Life time of an excited state of an atom	10$^{-8}$
Period of radio wave	10$^{-6}$
Period of a sound wave	10$^{-3}$
Wink of eye	10$^{-1}$
Time between successive human heart beats	10$^{0}$
Travel time for light from moon to the earth	10$^{0}$
Travel time for light from the Sun to the earth	10$^{2}$
Time period of satellite	10$^{4}$
Rotation period of the Earth	10$^{5}$
Rotation and revolution periods of the moon	10$^{6}$
Revolution period of the Earth	10$^{7}$
Travel time for light from nearest star	10$^{8}$
Average human life-span	10$^{9}$
Age of Egyptian pyramids	10$^{11}$
Time since dinosaurs became extinct	10$^{15}$
Age of the universe	10$^{17}$

2.6 ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN MEASUREMENT

Accuracy : The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity.



Precision : Precision tells us to what resolution or limit the quantity is measured.



Error : The uncertainity contained in the result of a measurement is called error.



Classification of errors:

(1) Classification based on its cause:

(i) Systematic errors : Those errors that tend to be in one direction, either positive or negative are called systematic errors. Their causes are known to us.



Sources of systematic errors:

(a) Instrumental errors : Those 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.

(b) Imperfection in experimental technique or procedure: Those errors that arise due to the effect of external conditions (such as changes in temperature, humidity, wind velocity, etc.) during the experiment. For example, the temperature of a human body read by a thermometer placed under the armpit will always give a temperature lower than actual value of the body temperature.

(c) Personal errors : Those 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.

Minimising of systematic errors:

➤These errors can be minimised by improving experimental techniques, selecting better instruments and removing personal bias as far as possible.


➤For a given set-up, these errors may be estimated to a certain extent and the necessary corrections may be applied to the readings.

(ii) Random errors: Those errors which occur irregularly and hence are random with respect to sign and size. Their causes are unknown to us and hence is not possible to remove them completely.

Minimising of random errors:

➤These errors can be minimised by repeating the observation a large number of times and taking the arithmetic mean of all the observations.The mean would be very close to the most accurate reading..


Say $a_1$, $a_2$, ... , $a_n$ be n number of observations and $a_{mean}$ be their arithmetic mean. Then

$a_{mean}$ = $ \frac{a_1+a_2+ ... +a_n}{n}$, or

$a_{mean}$ =$ \sum \limits_{i=1}^{n}a_{i}/n $

Least Count Error :

Least count : The smallest value that can be measured by the measuring instrument is called its least count.



Least count error : The error associated with the resolution of the instrument is called least count error.

NOTE : Least count error belongs to the category of random errors but within a limited size; It occurs with both systematic and random errors.

(2) Classification based on its magnitude:

(i) Absolut error : The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error.



➤In absence of any other method of knowing true value, arithmatic mean is considered as true value.


Say $a_i$ be the $i^{th}$ observation and $a_{mean}$ be the arithmetic mean of all the observations and $|\Delta a_i|$ be the absolute error in $i^{th}$ observation. Then

$|\Delta a_i|$ = $|a_i - a_{mean}|$

➤The arithmetic mean of all the absolute errors is taken as the final or mean absolute error. So,

mean absolute error,

$\Delta a_{mean}$ = $\sum \limits_{i=1}^n |\Delta a_i|/n$

➤If we do a single measurement, the value we get may be in the range $a_{mean}$±$\Delta a_{mean}$

i.e. a = $a_{mean}$±$\Delta a_{mean}$

or,

$a_{mean}$-$\Delta a_{mean} \leq a \leq$$a_{mean}$+$\Delta a_{mean}$

(ii) Relative error : The ratio of the mean absolute error to the mean value of the quantity measured is called relative error.

Relative error = $\Delta a_{mean}/a_{mean}$

(iii) Percentage error : The relative error is expressed in percent, it is called percentage error.

Percentage error,

$\delta a$ = ($\Delta a_{mean}/a_{mean}$) x 100%

2.6.1 Combination of Errors

(a) Error of a sum or difference

➤When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

Say,

$\Delta A$, $\Delta B$, $\Delta Z$ : Absolute error in the mesaurement of the physical quantities A, B and Z respectively. And,

let Z = A ± B, Then,

Z ± $\Delta Z$ = (A ± $\Delta A$) ± (B ± $\Delta B$)
Z ± $\Delta Z$ = (A ± B) ± ( ± $\Delta A$ ± $\Delta B$)

As Z = A ± B

Z ± $\Delta Z$ = Z ± ( ± $\Delta A$ ± $\Delta B$)

± $\Delta Z$ = ± $\Delta A$ ± $\Delta B$

Thus, the maximum value of the error $\Delta Z$ = $\Delta A$ + $\Delta B$

(b) Error of a product

➤When two quantities are multiplied, the relative error in the final result is the sum of the relative errors in the individual quantities.

Say,

$\Delta A$, $\Delta B$, $\Delta Z$ : Absolute error in the mesaurement of the physical quantities A, B and Z respectively. And,

let Z = AB, Then,

Z ± $\Delta Z$ = (A ± $\Delta A$) (B ± $\Delta B$)
Z ± $\Delta Z$ = AB (1 ± $ \frac { \Delta A}{A}$) (1 ± $\frac { \Delta B}{B}$)

As Z = AB

Z ± $\Delta Z$ = Z (1 ± $ \frac { \Delta A}{A}$) (1 ± $\frac { \Delta B}{B}$)

1 ± $\frac{\Delta Z}{Z}$ = 1 ± $\frac{\Delta A}{A}$ ± $\frac{\Delta B}{B}$ ± $\frac{\Delta A}{A}$ $\frac{\Delta B}{B}$

$\frac{\Delta A}{A}$ $\frac{\Delta B}{B}$ is a small quantity, so can be neglected.

± $\frac{\Delta Z}{Z}$ = ± $\frac{\Delta A}{A}$ ± $\frac{\Delta B}{B}$

$\frac{\Delta Z}{Z}$ = ± ($\frac{\Delta A}{A}$ + $\frac{\Delta B}{B}$)

Possible values of $\frac{\Delta Z}{Z}$ are $\frac{\Delta A}{A}$ + $\frac{\Delta B}{B}$, $\frac{\Delta A}{A}$ - $\frac{\Delta B}{B}$, -$\frac{\Delta A}{A}$ + $\frac{\Delta B}{B}$ and -$\frac{\Delta A}{A}$ - $\frac{\Delta B}{B}$

Thus, the maximum relative error $\frac{\Delta Z}{Z}$ = ($\frac{\Delta A}{A}$ + $\frac{\Delta B}{B}$)

(c) Error of a quotient

➤When two quantities are divided, the relative error in the final result is the sum of the relative error in the individual quantities.

Say,

$\Delta A$, $\Delta B$, $\Delta Z$ : Absolute error in the mesaurement of the physical quantities A, B and Z respectively. And,

let Z = $\frac{A}{B}$, Then,

Z ± $\Delta Z$ = $\frac{(A ± \Delta A)}{(B ± \Delta B)}$
Z ± $\Delta Z$ = $\frac{A(1 ± \frac { \Delta A}{A})}{B (1 ± \frac { \Delta B}{B})}$

As Z = $\frac{A}{B}$

Z ± $\Delta Z$ = Z (1 ± $ \frac { \Delta A}{A}$) (1 ±\ $\frac { \Delta B}{B}$)$^{-1}$

As $\frac{\Delta B}{B} << 1$, so expanding binomially

Z ± $\Delta Z$ = Z (1 ± $ \frac { \Delta A}{A}$) (1 ∓ $\frac { \Delta B}{B}$)

1 ± $\frac{\Delta Z}{Z}$ = 1 ± $\frac{\Delta A}{A}$ ∓ $\frac{\Delta B}{B}$ ± $\frac{\Delta A}{A}$ $\frac{\Delta B}{B}$

$\frac{\Delta A}{A}$ $\frac{\Delta B}{B}$ is a small quantity, so can be neglected.

± $\frac{\Delta Z}{Z}$ = ± $\frac{\Delta A}{A}$ ∓ $\frac{\Delta B}{B}$

$\frac{\Delta Z}{Z}$ = ± ($\frac{\Delta A}{A}$ - $\frac{\Delta B}{B}$)

Possible values of $\frac{\Delta Z}{Z}$ are $\frac{\Delta A}{A}$ + $\frac{\Delta B}{B}$, $\frac{\Delta A}{A}$ - $\frac{\Delta B}{B}$, -$\frac{\Delta A}{A}$ + $\frac{\Delta B}{B}$ and -$\frac{\Delta A}{A}$ - $\frac{\Delta B}{B}$

Thus, the maximum relative error $\frac{\Delta Z}{Z}$ = ($\frac{\Delta A}{A}$ + $\frac{\Delta B}{B}$)

(d) Error in case of a measured quantity raised to a power

➤The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.

Say,

$\Delta A$, $\Delta Z$ : Absolute error in the mesaurement of the physical quantities A and Z respectively. And,

Let Z = $A^k$, Then,

ln Z = k lnA

differentiating both sides,

$ \frac{dZ}{Z}$ = k $\frac{dA}{A}$

In terms of fractional error,

± $\frac{\Delta Z}{Z}$ = ± k($\frac{\Delta A}{A}$ )

Thus, the maximum relative error $\frac{\Delta Z}{Z}$ = k$\frac{\Delta A}{A}$

Similarly,

for Z = $\frac{A^k}{B^m}$

the maximum relative error $\frac{\Delta Z}{Z}$ = k$\frac{\Delta A}{A}$+ m$\frac{\Delta B}{B}$

2.7 SIGNIFICANT FIGURES

Significant figures/digits : All reliable digits in a measurement plus the first uncertain digit together form significant figures.



Rules for counting significant figures:

(i) All the non-zero digits are significant. For example, in 2457 has four significant digits.

(ii) All the zeroes between two non-zero digits are significant, no matter where the decimal point is, if at all. For xample, both 6034 and 3.024 has 4 significant figures.

(iii) If the number is less than 1, the zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant. For example, in 0.00 2308, the underlined zeroes are not significant.

(iv) The terminal or trailing zero(s) in a number without a decimal point are not significant. For example, 123000 has 3 significant figures. When some value is recorded on the basis of actual measurement, then the terminal or trailing zero(s) in a number without a decimal point are significant. For example, the length is reported to be 4700mm in a measurement, here 4700mm has 4 significat figures.

(v) The terminal or trailing zero(s) in a number with a decimal point are significant. For example, 1.500 has 4 significant figures.

(vi) Change in the units of measurement of a quantity does not change the number of significant figures. For example, 123m = 12300cm both 123m and 12300cm has 3 significant figures

(vii) The power of ten in scientific notation are not significant. For example, 1.40 x $10^5$ (scientific notation) has 3 significant figures.

(viii) The multiplying or dividing factors which are neither rounded numbers nor numbers representing measured values are exact numbers and have infinite number of significant digits. For example in r=d/2 or s=2$\pi$r, the factor is an exact number and it can be written as 2.0, 2.00,... and hence has infinite number of significant digits.

NOTE :There can be confusion regarding trailing zero(s) as discussed in (iv). To remove such confusion, the best way is to report the measurement in scientific notation [the number expressed as a x $10^b$, where 1$\leq$a < 10 and b is a poitive or negative exponent called order of magnitude].

2.7.1 Rules for Arithmetic Operations with Significant Figures

(i) In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures. For example, say mass=4.237 g (four signifacnt figures) and volume be 2.51 $cm^3$ (3 significant figures), then its density should be reported to 3 significant figures as 1.69 g $cm^{-3}$

(ii) In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places. For example, the sum of the numbers 436.32, 227.2 and 0.301 should be evaluated as 663.8 g because the least precise measurement (227.2 g) is correct to only one decimal place.
Similarly, 12.9 - 7.06 should be evaluated as 5.8g.
While 0.347 m - 0.344m should be evaluated as 0.003m=3 x $10^{-3}$m.

NOTE : In the last example, it should not be evaluted as 3.00 x $10^{-3}$m because it does convey the precision of measurement properly. For addition and subtraction, the rule is in terms of decimal places

2.7.2 Rounding off the Uncertain Digits

(i) The preceding digit is raised by 1 if the insignificant digit to be dropped is more than 5, and is left unchanged if the latter is less than 5. For example 2.746 rounded off to 3 significant figures is 2.75, while 2.743 would be 2.74.

(ii) If the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by 1. For example, 2.745 rounded off to 3 significant figures is 2.74 while 2.715 would be 2.72.

2.7.3 Rules for Determinig the Uncertainity in the results of Arithmatic Calculations

(i) If a set of experimental data is specified to n significant figures, a result obtained by combining the data will also valid to n significant figures. For example, if l=16.2 ± 0.1 cm and b=10.1 ± 0.1 cm, then A=lb=163.62 ± 2.6 $cm^2$ but it will be valid upto 164 ± 3 $cm^2$

(ii) Intermediate results in a multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement to avoid rounding errors. For example, the reciprocal of 9.58, calculated (after rounding off) to the same number of significant figures (3) is 0.104, but the reciprocal of 0.104 calculated to 3 significant figures is 9.62. However, if we had wriiten 1/9.58=0.1044 and then taken the reciprocal to 3 significant figures, we would have retrived the original value of 9.58.

2.8 DIMENSIONS OF PHYSICAL QUANTITIES

Dimensions of a physical quantity : The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity (denoted with square brackets, [ ]). For example, force=mass*acceleration=mass*(length)/$(time)^2$=>the dimensions of force are $[MLT^{-2}]$.

2.9 DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS

Dimensional formulae : The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity. For example, the dimensional formula of the volume is $[M^0L^3T^0]$.



Dimensional equation : An equation obtained by equating a physical quantity with its dimensional formulae is called the dimensional equation of the physical quantity. For example, [V]=$[M^0L^3T^0]$

2.10 DIMENSIONAL ANALYSIS AND ITS APPLICATIONS

Dimensional analysis : Dimensional analysis describes the the relationships between different physical quantities based on their basic units and dimensions.



Its application:

(i) Checking the dimensional consistency of equation : If the dimensions of each term on both sides of a equation are same, then it is called dimensionally correct/consistent equation and this type of equation need not be actually a correct equation but a dimensionally wrong or inconsistent equation must be wrong.

Principle of homogeneity of dimensions: This principle states that the dimessions of all the terms in a physical equation should be same. For example, in the physical expression $x=x_0+v_0t+(1/2)at^2$, the dimession of $x, x_0, v_0t and (1/2)at^2$ all are same.

(ii) Deducing relation among the physical quantities. .