Contributed by:
The highlights are:
1. Physical Quantity – Fundamental & Derived Quantities
2. Unit – Fundamental & Derived Units
3. Characteristics of Standard Unit
4. FPS, CGS, MKS & SI System of Units
5. Definition of Fundamental SI units
6. Measurement of Length – Large Distances and Small Distances
7. Measurement of Mass and Measurement of Time
8. Accuracy, Precision of Instruments, and Errors in Measurements
9. Systematic Errors and Random Errors
10. Absolute Error, Relative Error, and Percentage Error
11. Combination of Errors, in Addition, Subtraction, Multiplication, Division, and Exponent.
12. Significant Figures, Scientific Notation, and Rounding off Uncertain Digits
13. Dimensions, Dimensional Formulae and Dimensional Equations
14. Dimensional Analysis – Applications and Demerits
1.
UNITS AND MEASUREMENT
1. Physical Quantity – Fundamental & Derived Quantities
2. Unit – Fundamental & Derived Units
3. Characteristics of Standard Unit
4. fps, cgs, mks & SI System of Units
5. Definition of Fundamental SI units
6. Measurement of Length – Large Distances and Small Distances
7. Measurement of Mass and Measurement of Time
8. Accuracy, Precision of Instruments and Errors in Measurements
9. Systematic Errors and Random Errors
10. Absolute Error, Relative Error and Percentage Error
11. Combination of Errors in Addition, Subtraction, Multiplication, Division and
Exponent.
12. Significant Figures, Scientific Notation and Rounding off Uncertain Digits
13. Dimensions, Dimensional Formulae and Dimensional Equations
14. Dimensional Analysis – Applications- I, II & III and Demerits
Next
2.
Physical Quantity
A quantity which is measurable is called ‘physical quantity’.
Fundamental Quantity
A physical quantity which is the base and can not be derived from any
other quantity is called ‘fundamental quantity’.
Examples: Length, Mass, Time, etc.
Derived Quantity
A physical quantity which can be derived or expressed from base or
fundamental quantity / quantities is called ‘derived quantity’.
Examples: Speed, velocity, acceleration, force, momentum,
torque, energy, pressure, density, thermal conductivity,
resistance, magnetic moment, etc.
Home Next Previous
3.
Measurement of any physical quantity involves comparison with a
certain basic, arbitrarily chosen, internationally accepted reference
standard called unit.
Fundamental Units
The units of the fundamental or base quantities are called fundamental
or base units.
Examples: metre, kilogramme, second, etc.
Derived Units
The units of the derived quantities which can be expressed from the
base or fundamental quantities are called derived units.
Examples: metre/sec, kg/m3, kg m/s2, kg m2/s2, etc.
Home Next Previous
4.
System of Units
A complete set of both fundamental and derived units is known as the
system of units.
Characteristics of Standard Units
A unit selected for measuring a physical quantity must fulfill the following
i) It should be well defined.
ii) It should be of suitable size i.e. it should neither be too large nor too
small in comparison to the quantity to be measured.
iii) It should be reproducible at all places.
iv) It should not change from place to place or time to time.
v) It should not change with the physical conditions such as temperature,
pressure, etc.
vi) It should be easily accessible.
Home Next Previous
5.
Various System of Units
In earlier time, various systems like ‘fps’, ‘cgs’ and ‘mks’ system of units were
used for measurement. They were named so from the fundamental units in
their respective systems as given below:
Quantity Dimension System of units
fps cgs mks
Length L foot centi metre metre
Mass M pound gramme kilogramme
Time T second second second
Systeme Internationale d’ unites (SI Units)
The SI system 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.
This is the system of units which is at present accepted internationally.
SI system uses decimal system and therefore conversions within the system
are quite simple and convenient. Home Next Previous
6.
Fundamental Units in SI system
Quantity Symbol SI unit Symbol
Length L metre m
Mass M kilogramme kg
Time T second s
Electric Current A ampere A
Main units Thermodynamic K kelvin K
Temperature
Light Intensity Cd candela cd
Amount of mole mole mol
substance
Plane angle dθ radian rad
ary units Solid angle dΩ steradian sr
Home Next Previous
7.
Definition of SI Units
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)
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 Sevres, near Paris, France. (1889)
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)
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 2×10–7 newton per metre of length. (1948)
Home Next Previous
8.
The kelvin, is the fraction 1/273.16 of the thermodynamic
temperature of the triple point of water. (1967)
The candela is the luminous intensity, in a given direction, of a
source that emits monochromatic radiation of frequency 540×1012
hertz and that has a radiant intensity in that direction of 1/683 watt
per steradian. (1979)
The mole is the amount of substance of a system, which contains as
many elementary entities as there are atoms in 0.012 kilogramme of
carbon-12. (1971)
Home Next Previous
9.
Plane angle
Plane angle ‘dθ’ is the ratio of arc ‘ds’ to the radius ‘r’. Its SI unit is ‘radian’.
ds
r
ds
dθ dθ =
r r
O
Solid angle
Solid angle ‘dΩ’ is the ratio of the intercepted area ‘dA’ of the spherical
surface described at the apex ‘O’ as the centre, to the square of its radius ‘r’.
Its SI unit is ‘steradian’.
dA
r
r
dΩ dA
dΩ =
r2
O
Home Next Previous
10.
The following conventions are adopted while writing a unit:
(1) Even if a unit is named after a person the unit is not written in
letters. i.e. we write joules not Joules.
(2) For a unit named after a person the symbol is a capital letter e.g. for
joules we write ‘J’ and the rest of them are in lowercase letters e.g.
second is written as ‘s’.
(3) The symbols of units do not have plural form i.e. 70 m not 70 ms or
10 N not 10 Ns.
(4) Punctuation marks are not written after the unit
e.g. 1 litre = 1000 cc not 1000 c.c.
Home Next Previous
11.
Some Units are 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 107 s
degree 0
10 = (π / 180) rad
litre l 1 dm3 = 10-3 m3
tonne t 103 kg
carat c 200 mg
bar bar 0.1 MPa = 105 Pa
curie ci 3.7 x 1010 s-1
roentgen r 2.58 x 10 -4 C/kg
quintal q 100 kg
barn b 100 fm2 = 10-28 m2
are a 1 dam2 = 102 m2
hectare ha 1 hm2 = 104 m2
standard atmosphere atm 101325 Pa = 1.013 x 105 Pa
pressure Home Next Previous
12.
MEASUREMENT OF LENGTH
The order of distances varies from 10-14 m (radius of nucleus) to 1025 m
(radius of the Universe)
The distances ranging from 10-5 m to 102 m can be measured by direct
methods which involves comparison of the distance or length to be
measured with the chosen standard length.
i) A metre rod can be used to measure distance as small as 10-3 m.
ii) A vernier callipers can be used to measure as small as 10-4 m.
iii) A screw gauge is used to measure as small as 10-5 m.
For very small distances or very large distances indirect methods are used.
Home Next Previous
13.
Measurement of Large Distances P
The following indirect methods may be used
to measure very large distances: θ
1.Parallax method
D D
• Let us consider a far away planet ‘P’ at a
distance ‘D’ from our two eyes.
• Suppose that the lines joining the planet to
the left eye (L) and the right eye (R) subtend
an angle θ (in radians).
L b R
• The angle θ is called ‘parallax angle’ or
‘parallactic angle’ and the distance LR = b is
called ‘basis’.
LR b
θ= =
• As the planet is far away, b/D << 1, and D D
therefore θ is very small.
b
D=
•Then, taking the distance LR = b as a θ
circular arc of radius D, we have
Home Next Previous
14.
Measurement of the size or angular diameter of an astronomical
If ‘d’ is the diameter of the planet and ‘α’ is
d
the angular size of the planet (the angle
subtended by d at the Earth E), then
α = d/D
The angle α can be measured from the same
D D
location on the earth.
α
It is the angle between the two directions
when two diametrically opposite points of the
planet are viewed through the telescope.
E
Since D is known, the diameter d of the planet
can be determined from
d=αD
Home Next Previous
15.
Echo method or Reflection method
υ×t =2S
S= • This method is used to measure the
υ ×t
2
distance of a hill.
• A gun is fired towards the hill and the
time interval between the instant of firing
the gun and the instant of hearing the echo
of the gun shot is noted.
• This is the time taken by the sound to S
travel from the observer to the hill and S
back to the observer. Sound
wave
• If v = velocity of sound;
Gun fire Echo received
S = the distance of hill from the observer
and
T = total time taken, then
vxT
S=
2
Home Next Previous
16.
• In place of sound waves, LASER can be used to
measure the distance of the Moon from the Earth.
LASER is a monochromatic, intense and unidirectional
• If ‘t’ is the time taken for the LASER beam in going
to and returning from the Moon, then the distance can
be calculated from the formula
cxt
S= where c = 3 x 108 m s-1
2
Home Next Previous
17.
Estimation of Very Small Distances
1. Using Electron Microscope:
For visible light the range of wavelengths is from about 4000 Å to 7000 Å
(1 angstrom = 1 Å = 10-10 m). Hence an optical microscope cannot resolve
particles with sizes smaller than this.
Electron beams can be focused by properly designed electric and magnetic
The resolution of such an electron microscope is limited finally by the fact
that electrons can also behave as waves.
The wavelength of an electron can be as small as a fraction of an angstrom.
Such electron microscopes with a resolution of 0.6 Å have been built. They
can almost resolve atoms and molecules in a material.
In recent times, tunneling microscopy has been developed in which again
the limit of resolution is better than an angstrom. It is possible to estimate
the sizes of molecules.
Home Next Previous
18.
2. Avogadro’s Method:
A simple method for estimating the molecular size of oleic acid
is given below.
Oleic acid is a soapy liquid with large molecular size of the
order of 10–9 m.
The idea is to first form mono-molecular layer of oleic acid on
water surface.
We dissolve 1 cm3 of oleic acid in alcohol to make a solution of
20 cm3 (ml). Then we take 1 cm3 of this solution and dilute it to
20 cm3, using alcohol.
1
So, the concentration of the solution is cm3 of oleic
20 x 20
acid per cm3 of solution.
Next we lightly sprinkle some lycopodium powder on the
surface of water in a large trough and we put one drop of this
solution in the water.
The oleic acid drop spreads into a thin, large and roughly
circular film of molecular thickness on water surface.
Home Next Previous
19.
Then, we quickly measure the diameter of the thin film to get its
area A.
Suppose we have dropped ‘n’ drops in the water.
Initially, we determine the approximate volume of each drop
(V cm3).
Volume of n drops of solution = nV cm3
1
Amount of oleic acid in this solution = nV c
20 x 20 m3
This solution of oleic acid spreads very fast on the surface of
water and forms a very thin layer of thickness ‘t’.
If this spreads to form a film of area ‘A’ cm2, then the thickness
of the film
Volume of the film nV
t= or t= cm
Area of the film 20 x 20 x A
If we assume that the film has mono-molecular thickness, then
this becomes the size or diameter of a molecule of oleic acid. The
value of this thickness comes out to be of the order of 10 –9m.
Home Next Previous
20.
Range of Lengths
The size of the objects we come across in the Universe varies over a very
wide range.
These may vary from the size of the order of 10–14 m of the tiny nucleus of an
atom to the size of the order of 1026 m of the extent of the observable
Universe.
We also use certain special length units for short and large lengths which are
given below:
Unit Symbol Value Definition
1 fermi 1f 10–15 m
1 angstrom 1Å 10–10 m
1 Astronomical 1 AU 1.496 × 1011 m Average distance of the Sun from the Earth
Unit
1 light year 1 ly 9.46 × 1015 m The distance that light travels with speed of
3 × 108 m s–1 in 1 year
1 parsec 3.08 × 1016 m The distance at which average radius of
(3.26 ly) Earth’s orbit subtends an angle of 1 arc
second
Home Next Previous
21.
Range and Order of Lengths (Llongest : Lshortest = 1041 : 1)
S.No Size of the object or distance Length (m)
1 Size of proton 10-15
2 Size of atomic nucleus 10-14
3 Size of the Hydrogen atom 10-10
4 Length of a typical virus 10-8
5 Wavelength of a light 10-7
6 Size of the red blood corpuscle 10-5
7 Thickness of a paper 10-4
8 Height of the Mount Everest from sea level 104
9 Radius of the Earth 107
10 Distance of the moon from the earth 108
11 Distance of the Sun from the earth 1011
12 Distance of the Pluto from the Sun 1013
13 Size of our Galaxy 1021
14 Distance of the Andromeda galaxy 1022
15 Distance of the boundary of observable universe 1026
Home Next Previous
22.
MEASUREMENT OF MASS
The SI unit of mass is kilogram (kg).
The prototypes of the International standard kilogramme supplied by the
International Bureau of Weights and Measures (BIPM) are available in many
other laboratories of different countries.
In India, this is available at the National Physical Laboratory (NPL), New
While dealing with atoms and molecules, the kilogramme is an inconvenient
unit. In this case, there is an important standard unit of mass, called the
unified atomic mass unit (u), which has been established for expressing the
mass of atoms as
1 unified atomic mass unit = 1 u
One unified mass unit is equal to (1/12) of the mass of an atom of Carbon-12
isotope (12C6 ) including the mass of electrons.
1 u = 1.66 × 10–27 kg
Home Next Previous
23.
Methods of measuring mass
(i) By using a common balance.
(ii) Large masses in the Universe like planets, stars, etc.,
based on Newton’s law of gravitation can be measured by
using gravitational method.
(iii) 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 and magnetic
field.
Range of Masses
The masses of the objects, we come across in the
Universe, 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 1055 kg of the
known Universe.
Home Next Previous
24.
Range and Order of Masses (Mlargest : Msmallest = 1085 : 1 ≈ (1041)2)
S.No. Object Mass (kg)
1 Electron 10-30
2 Proton 10-27
3 Uranium atom 10-25
4 Red blood cell 10-13
5 Dust particle 10-9
6 Rain drop 10-6
7 Mosquito 10-5
8 Grape 10-3
9 Human 102
10 Automobile 103
11 Boeing 747 108
12 Moon 1023
13 Earth 1025
14 Sun 1030
15 Milky way galaxy 1041
16 Observable Universe 1055 Home Next Previous
25.
MEASUREMENT OF TIME
We use an atomic standard of time, which is based on the periodic
vibrations produced in a cesium atom. This is the basis of the cesium
clock, sometimes called atomic clock, used in the national standards.
In the cesium atomic clock, the second is taken as the time needed 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 vibrations of the cesium atom regulate the rate of this cesium atomic
clock just as the vibrations of a balance wheel regulate an ordinary
wristwatch or the vibrations of a small quartz crystal regulate a quartz
A cesium atomic clock is used at the National Physical Laboratory
(NPL), New Delhi to maintain the Indian standard of time.
Home Next Previous
26.
Range and Order of Time Intervals (Tlongest : Tshortest = 1041 : 1)
S.No. Event Time Intervals (s)
1 Life span of most unstable particle 10-24
2 Time required for light to cross a nuclear distance 10-22
3 Period of X-rays 10-19
4 Period of atomic vibrations 10-15
5 Period of light wave 10-15
6 Life time of an excited atom 10 -8
7 Period of radio wave 10-6
8 Period of sound wave 10-3
9 Wink of eye 10-1
10 Time between successive human heart beats 100
11 Travel time for light from the Moon to the Earth 100
12 Travel time for light from the Sun to the Earth 102
13 Time period of a satellite 104
14 Rotation period of the Earth 105
15 Rotation and revolution periods of the Moon 106
16 Revolution period of the Earth 107
17 Travel time for light from the nearest star 108
18 Average human life span 109
19 Age of Egyptian pyramids 1011
20 Time since dinosaurs became extinct 1015
21 Age of the Universe 1017
Home Next Previous
27.
ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN
The result of every measurement by any measuring instrument contains
some uncertainty. This uncertainty is called error.
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.
Suppose the true value of a certain length is near 2.874 cm.
In one experiment, using a measuring instrument of resolution 0.1 cm, the
measured value is found to be 2.7 cm, while in another experiment using a
measuring device of greater resolution, say 0.01 cm, the length is
determined to be 2.69 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. Home Next Previous
28.
In general, the errors in measurement can be broadly classified as
(I) Systematic errors and (II) Random errors
I. 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:
(a) Instrumental errors:
The instrumental errors that arise from the errors due to imperfect design
or calibration of the measuring instrument, zero error in the instrument,
etc.
(i)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);
(ii)In a vernier callipers the zero mark of vernier scale may not coincide with
the zero mark of the main scale;
(iii) An ordinary metre scale may be worn off at one end. Home Next Previous
29.
(b) 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.
(c) Personal errors:
The personal errors 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.
If you 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.
Systematic errors can be minimized by
(i) improving experimental techniques,
(ii) selecting better instruments and
(iii) removing personal bias as far as possible.
Home Next Previous
30.
II. 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, personal errors by the observer taking readings,
When the same person repeats the same observation, it is very
likely that he may get different readings every time.
Least count error
Least count:
The smallest value that can be measured by the measuring instrument is
called its least count.
The least count error is the error associated with the resolution of the
Home Next Previous
31.
(i) A Vernier callipers has the least count as 0.01 cm;
(ii) A spherometer may have a least count of 0.001 cm.
Using instruments of higher precision, improving experimental techniques,
etc., we can reduce the least count error.
Repeating the observations several times and taking the arithmetic mean of
all the observations, the mean value would be very close to the true value of
the measured quantity.
Note:
Least count error belongs to Random errors category but within a limited
size; it occurs with both systematic and random errors.
Home Next Previous
32.
Absolute Error, Relative Error and Percentage Error
Absolute error
The magnitude of the difference between the individual measurement
value and the true value of the quantity is called the absolute error of the
This is denoted by |Δa|.
Note: In absence of any other method of knowing true value, we consider
arithmetic mean as the true value.
The errors in the individual measurement values from the true value are:
Δa1 = a1 - amean
Δa2 = a2 - amean
----------------
----------------
Δan = an - amean
The Δa calculated above may be positive or negative.
But absolute error |Δa| will always be positive.
Home Next Previous
33.
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 Δamean.
Δamean = (|Δa1|+|Δa2 |+|Δa3|+...+ |Δan|)/n
n
= ∑ |Δai|/n
i=1
If we do a single measurement, the value we get may be in the range
amean ± Δamean
This implies that any measurement of the physical quantity a is
likely to lie between
(amean + Δamean) and (amean - Δamean)
Home Next Previous
34.
Relative
The relative error is the ratio of the mean absolute error
Δamean to the mean value amean of the quantity measured.
Mean absolute error
Relative error =
True value or Arithmetic Mean
Δamean
Relative error =
amean
Percentage error
When the relative error is expressed in per cent, it is
called the percentage error (δa).
Mean absolute error
Percentage error = x 100%
True value or Arithmetic Mean
Δamean
Percentage error δa = x 100%
amean
Home Next Previous
35.
Combination of Errors
In an experiment involving several measurements, the errors in all the
measurements get combined.
Density is the ratio of the mass to the volume of the substance.
If there are errors in the measurement of mass and of the sizes or
dimensions, then there will be error in the density of the substance.
(a) Error of a Sum:
Suppose two physical quantities A and B have measured values
A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.
Let Z=A+B
When two quantities
Z ± ΔZ = (A ± ΔA) + (B ± ΔB) are added, the
absolute error in
= (A + B) ± (ΔA + ΔB) the final result is the
sum of the
= Z ± (ΔA + ΔB) individual errors.
± ΔZ = ± (ΔA + ΔB) or ΔZ = (ΔA + ΔB)
Home Next Previous
36.
(b) Error of a Difference:
Suppose two physical quantities A and B have measured values
A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.
Let Z=A-B
Z ± ΔZ = (A ± ΔA) - (B ± ΔB)
= (A - B) ± ΔA ΔB
±
= Z ± (ΔA + ΔB) (since ± and are the
±
same)
± ΔZ = ± (ΔA + ΔB) When two quantities
are subtracted, the
or ΔZ = (ΔA + ΔB) absolute error in
the final result is the
sum of the
individual errors.
Rule:
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.
Home Next Previous
37.
(c) Error of a Product:
Suppose two physical quantities A and B have measured values
A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.
Let Z=AxB
Z ± ΔZ = (A ± ΔA) x (B ± ΔB)
Z ± ΔZ = AB ± A ΔB ± B ΔA ± ΔA ΔB
Dividing LHS by Z and RHS by AB we have,
ΔZ ΔB ΔA ΔA ΔB
1± = 1± ± ±
Z B A AB
ΔZ ΔB ΔA ΔA ΔB
± = ± ± is very small and hence negligible
Z B A AB
When two quantities are
ΔZ ΔA ΔB multiplied, the relative error in
or = + the final result is the sum of the
Z A B relative errors of the individual
quantities.
Home Next Previous
38.
Error of a Product: ALITER
Suppose two physical quantities A and B have measured values
A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.
Let Z=AxB
Applying log on both the sides, we have
log Z = log A + log B
Differentiating, we have
ΔZ ΔA ΔB
= +
Z A B
Home Next Previous
39.
(d) Error of a Quotient:
Suppose two physical quantities A and B have measured values
A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.
ΔB ΔA ΔA ΔB
Let Z = A Z ± ΔZ = A ± A x ± ± x
B B B B B B B
(A ± ΔA) Dividing LHS by Z and RHS by A / B and
Z ± ΔZ = simplifying we have,
(B ± ΔB)
(A ± ΔA) ΔZ ΔB ΔA
± = ± ±
Z ± ΔZ = Z B A
ΔB
B 1± ΔA ΔB
B is negligible
B2
(A ± ΔA) ΔB
-1 ΔZ ΔA ΔB
or = +
Z ± ΔZ = 1± Z A B
B B
When two quantities are
ΔA ΔB divided, the relative error in the
1
±
Z ± ΔZ = A ± final result is the sum of the
B B B
relative errors of the individual
(by Binomial Approximation) quantities. Home Next Previous
40.
Error of a Quotient: ALITER
Suppose two physical quantities A and B have measured values
A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors.
Let Z = A
B
Applying log on both the sides, we have
log Z = log A - log B
Differentiating, we have
ΔZ ΔA ΔB
= -
Z A B
Logically an error can not be nullified by making another error. Therefore
errors are not subtracted but only added up.
ΔZ ΔB Math has to be bent to satisfy
ΔA
= + Physics in many situations!
Z A B Think of more such situations!!
When two quantities are multiplied or divided, the relative error in the
final result is the sum of the relative errors in the individual quantities.
Home Next Previous
41.
(e) Error of an Exponent (Power):
Suppose a physical quantity A has measured values A ± ΔA where ΔA is its
absolute error.
Let Z = Ap where p is a constant.
Z = A x A x A x ………x A (p times)
Z ± ΔZ = (A ± ΔA) x (A ± ΔA) x (A ± ΔA) x ……. x (A ± ΔA) (p times)
ΔZ ΔA ΔA ΔA ΔA
= + + + ……… + (p times as per the
Z A A A A
product rule for errors)
ΔZ ΔA Note:
or =p If p is negative, |p| is taken because errors
Z A due to multiple quantities get added up.
The relative error in a physical quantity raised to the power p is the
p times the relative error in the individual quantity.
Home Next Previous
42.
(f) Error of an Exponent (Power): ALITER
Suppose a physical quantity A has measured values A ± ΔA where ΔA is its
absolute error.
Let Z = Ap where p is a constant.
Applying log on both the sides, we have
log Z = |p| log A (Whether p is positive or negative
errors due to multiple quantities get
Differentiating, we have added up only)
ΔZ ΔA
= |p|
Z A
Ap x Bq
In general, if Z = , then
C r
Note:
ΔZ ΔA ΔB ΔC
= p + q +r Cr is in Denominator, but the
Z A B C relative error is added up.
Home Next Previous
43.
SIGNIFICANT FIGURES
The reported result of measurement is a number that includes all digits in
the number that are known reliably plus the first digit that is uncertain.
The reliable digits plus the first uncertain digit are known as significant
digits or significant figures.
(i) The period of oscillation of a simple pendulum is 2.36 s; the digits 2 and
3 are reliable and certain, while the digit 6 is uncertain. Thus, the measured
value has three significant figures.
(ii) The length of an object reported after measurement to be 287.5 cm has
four significant figures, the digits 2, 8, 7 are certain while the digit 5 is
A choice of change of different units does not change the number of
significant digits or figures in a measurement.
Eg. The length 1.205 cm, 0.01205, 12.05 mm and 12050 μm all have four SF.
Home Next Previous
44.
Rules for determining the number of significant figures
(i) All the non-zero digits are significant.
(ii) All the zeros between two non-zero digits are significant, no matter
where the decimal point is, if at all.
(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.
(iv) The terminal or trailing zero(s) in a number without a decimal point are
not significant.
(v) The trailing zero(s) in a number with a decimal point are significant.
Home Next Previous
45.
Scientific Notation
Any given number can be written in the form of a×10b in many ways;
for example 350 can be written as 3.5×102 or 35×101 or 350×100.
a×10b means "a times ten raised to the power of b", where the exponent b is an
integer, and the coefficient a is any real number called the significand or
mantissa (the term "mantissa" is different from "mantissa" in common
If the number is negative then a minus sign precedes a (as in ordinary
decimal notation).
In normalized scientific notation, the exponent b is chosen such that the
absolute value of a remains at least one but less than ten (1 ≤ |a| < 10).
For example, 350 is written as 3.5×102.
This form allows easy comparison of two numbers of the same sign in a, as
the exponent b gives the number's order of magnitude.
Home Next Previous
46.
Rules for Arithmetic Operations with Significant Figures
In arithmetic operations the final result should not have more
significant figures than the original data from which it was
(1)Multiplication or division:
The final result should retain as many significant figures as are
there in the original number with the least significant figures.
(2) Addition or subtraction:
The final result should retain as many decimal places as are there
in the number with the least decimal places.
Home Next Previous
47.
Rounding off the Uncertain Digits
Rounding off a number means dropping of digits which are not
significant. The following rules are followed for rounding off the
1.If the digits to be dropped are greater than five, then add one to the
preceding significant figure.
2. If the digit to be dropped is less than five then it is dropped without
bringing any change in the preceding significant figure.
3.If the digit to be dropped is five, then the preceding digit will be left
unchanged if the preceding digit is even and it will be increased by
one if it is odd.
4.In any involved or complex multi-step calculation, one should retain,
in intermediate steps, one digit more than the significant digits and
round off to proper significant figures at the end of the calculation.
Home Next Previous
48.
DIMENSIONS OF PHYSICAL QUANTITIES
The nature of a physical quantity is described by its dimensions.
All the physical quantities can be expressed in terms of the seven base or
fundamental quantities viz. mass, length, time, electric current,
thermodynamic temperature, intensity of light and amount of substance,
raised to some power.
The dimensions of a physical quantity are the powers (or exponents) to which
the fundamental or base quantities are raised to represent that quantity.
Using the square brackets [ ] around a quantity means that we are dealing
with ‘the dimensions of’ the quantity.
i)The dimensions of volume of an object are [L3]
ii)The dimensions of force are [MLT -2]
iii)The dimensions of energy are [ML2T-2]
Home Next Previous
49.
Dimensional Quantity
Dimensional quantity is a physical quantity which has dimensions.
For example: Speed, acceleration, momentum, torque, etc.
Dimensionless Quantity
Dimensionless quantity is a physical quantity which has no dimensions.
For example: Relative density, refractive index, strain, etc.
Dimensional Constant
Dimensional constant is a constant which has dimensions.
For example: Universal Gravitational constant, Planck’s constant, Hubble
constant, Stefan constant, Wien constant, Boltzmann constant, Universal
Gas constant, Faraday constant, etc.
Dimensionless Constant
Dimensionless constant is a constant which has no dimensions.
For example: 5, -.0.38, e, π, etc.
Home Next Previous
50.
DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS
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.
(i)The dimensional formula of the volume is [M° L3 T°],
(ii)The dimensional formula of speed or velocity is [M° L T -1]
(iii) The dimensional formula of acceleration is [M° L T –2]
An equation obtained by equating a physical quantity with its
dimensional formula is called the dimensional equation of the physical
(i) [V] = [M° L3 T°]
(ii) [v] = [M° L T-1]
(iii) [a] = [M° L T–2]
Home Next Previous
51.
Quantities having the same dimensional formulae
1. Impulse and momentum
2. Work, energy, torque, moment of force
3. Angular momentum, Planck’s constant, rotational impulse
4. Stress, pressure, modulus of elasticity, energy density
5. Force constant, surface tension, surface energy
6. Angular velocity, frequency, velocity gradient
7. Gravitational potential, latent heat
8. Thermal capacity, entropy, universal gas constant and Boltzmann’s
const.
9. Force, thrust
10. Power, luminous flux
Dimensional formulae for physical quantities often used in Physics are
given at the end. (From Slide 63)
Home Next Previous
52.
DIMENSIONAL ANALYSIS AND ITS
Dimensional analysis is a tool to find or check relations among physical
quantities by using their dimensions.
When magnitudes of two or more physical quantities are multiplied, their
units should be treated in the same manner as ordinary algebraic symbols.
We can cancel identical units in the numerator and denominator.
Similarly, physical quantities represented by symbols on both sides of a
mathematical equation must have the same dimensions.
Dimensional Analysis can be used-
1.To check the dimensional consistency of equations
(Principle of homogeneity of dimensions).
2. To convert units in one system into another system.
3. To derive the relation between physical quantities based on certain
reasonable assumptions.
Home Next Previous
53.
I. Checking the Dimensional Consistency of Equations
The principle of homogeneity of dimensions:
The magnitudes of physical quantities may be added together or
subtracted from one another only if they have the same dimensions.
For example, initial velocity can be added to or subtracted from final
velocity because they have same dimensional formula [M0LT -1] .
But, force and momentum can not be added because their dimensional
formulae are different and are [MLT -2] and [MLT -1] respectively.
Home Next Previous
54.
1. To check the dimensional consistency of v2 = u2 + 2as
The dimensions of the quantities involved in the equation are:
[u] = [M0LT -1]
[v] = [M0LT -1]
[a] = [M0LT -2]
[s] = [M0LT 0]
Substituting the dimensions in the given equation,
[M0LT -1]2 = [M0LT -1]2 + [M0LT -2] [M0LT 0] (Note that the constant 2 in the term
‘2as’ does not have dimensions)
[M0L2T-2] = [M0L2T-2] + [M0L2T-2]
Each term of the above equation is having same dimensions.
Therefore, the given equation is dimensionally correct or dimensionally
consistent.
Home Next Previous
55.
If an equation fails the consistency test, it is proved wrong;
But if it passes, it is not proved right.
Thus, a dimensionally correct equation need not be actually an
exact (correct) equation, but a dimensionally wrong (incorrect) or
inconsistent equation must be wrong.
Example: Equations v2 = u2 - 2as or v2 = u2 + ½as are dimensionally
consistent but are incorrect equations in mechanics.
Albert Einstein tried his famous mass-energy equation as
E = m / c2, E = m2 / c, E = m2 c, etc.
Finally he settled with E = m c2 using dimensions and then proved it
with the help of Calculus.
Home Next Previous
56.
2. To check the dimensional consistency of ½ mv2 = mgh
The dimensions of the quantities involved in the equation are:
[m] = [ML0T0]
[v] = [M0LT -1]
[g] = [M0LT -2]
[h] = [M0LT 0]
Substituting the dimensions in the given equation,
[ML0T0] [M0LT -1]2 = [ML0T0] [M0LT -2] [M0LT 0]
(Note that the constant ½ in the term
‘½ mv2 ’ does not have dimensions)
[ML2T-2] = [ML2T-2]
Each term of the above equation is having same dimensions.
Therefore, the given equation is dimensionally correct or dimensionally
Home Next Previous
57.
II. Conversion of units in one system into another system
Units are derived from the dimensions and the dimensions are derived from
the actual formulae of physical quantities.
If the dimensions are known for a physical quantity, then it is easy to
express it in fps, cgs, mks, SI systems or any other arbitrary chosen system.
n1[M1aL1bT1c] = n2[M2aL2bT2c]
a b c
M1 L1 T1
n 2 = n1
M2 L2 T2
n1 and n2 are the magnitudes in the respective systems of units.
Smaller the unit bigger the magnitude of a physical quantity and vice versa.
For example, 1 m = 100 cm (m is the bigger unit and cm is the smaller one)
1 N = 105 dynes (Newton is bigger and dyne is smaller)
Home Next Previous
58.
1. To convert 1 joule in erg.
‘joule’ is unit of energy or work in SI system and ‘erg’ is the unit in cgs system.
The dimensional formula of energy or work is [ML2T-2].
The units from dimensions in SI and cgs systems are kg m2 s-2 and g cm2 s-2
Let n1 joule = n2 erg
SI System cgs System
Magnitude n1 = 1 n2 = ?
Mass (M) 1 kg (=1000 g) 1g
Length (L) 1 m (= 100 cm) 1 cm
Time (T) 1s 1s
[MaLbTc] = [ML2T-2] Therefore, a=1, b=2, c=-2
a b c n2 = 1 (1000)1 (100)2 (1)-2
M1 L1 T1
n2 = n1
M2 L2 T2 n2 = 107
1 2 -2
1000 g 100 cm 1s 1 joule = 107 erg
n2 = 1
1g 1 cm 1s Home Next Previous
59.
2. To convert 1 newton into a system where mass is measured in
mg, length in km and time in minute
‘newton or kg m s-2’ is unit of force in SI system and ‘mg km min-2’ is the unit
in the new system.
The dimensional formula of force is [MLT -2].
Let n1 newton = n2 mg km min-2
SI System New System
Magnitude n1 = 1 n2 = ?
Mass (M) 1 kg (=106 mg) 1 mg
Length (L) 1 m (= 1/1000 km) 1 km
Time (T) 1 s (= 1/60) 1s
[MaLbTc] = [MLT -2] Therefore, a=1, b=1, c=-2
a b c
M1 L1 T1 n2 = 1 (106 )1 (10-3)1 (60)2
n2 = n1
M2 L2 T2 n2 = 3.6 x 106
1 1 -2
106 mg 1/1000 km 1/60 s
n2 = 1 1 newton =3.6x106mg km min-2
1 mg 1 km 1s
Home Next Previous
60.
III. Deducing Relation among the Physical Quantities
l
T =k
g The method of dimensions can sometimes be used to deduce relation
among the physical quantities.
For this we should know the dependence of the physical quantity on
other quantities (upto three physical quantities or linearly independent
variables) and consider it as a product type of the dependence.
Example:
1. Consider a simple pendulum, having a bob attached to a string that
oscillates under the action of the force of gravity. Suppose that the period
of oscillation of the simple pendulum depends on its length (l), mass of the
bob (m) and acceleration due to gravity (g). Derive the expression for its
time period using method of dimensions.
The dependence of time period T on the quantities l, g and m as a
product may be written as:
T = k l x my g z
where k is dimensionless constant and x, y and z are the exponents.
Home Next Previous
61.
The dimensions of the quantities involved in the equation are:
[m] = [ML0T0]
[l] = [M0LT 0]
[g] = [M0LT -2]
[T] = [M0L0T]
By substituting dimensions on both sides of T = k l x my gz, we have
[M0L0T] = [M0LT 0]x [ML0T0]y [M0LT -2]z
[M0L0T] = [M]y [L]x+z [T]-2z
On equating the dimensions on both sides, we have
y=0
x+z=0
–2z = 1
So that x = ½ , y = 0, z = -½
Then, T = k l½ g–½
l The value of k is 2π l
Or T = k and T = 2π
g g
determined from
other Home Next Previous
62.
Demerits of Dimensional Analysis
The dimensional analysis can not be used in the following cases:
1.The value of constants in an equation can not be determined as the
constants do not have dimensions.
2.Only dimensional consistency and not the physical consistency can be
3.Dimensions can be found from the physical quantity, but physical
quantity can not be always guessed from dimensions because two or more
quantities may have same dimensions.
4.The equation containing the dependency on more than 3 quantities can
not be determined using only M, L and T.
(Note that if 4 independent quantities are involved, then 4 variables
and hence 4 simultaneous equations are required; hence there must
be 4 fundamental dimensions)
5. The equation containing exponential, trigonometric, logarithmic
functions, etc. can not be derived as they do not have dimensions.
6.The equations having the relations other than products / quotients can
not be derived. Home Next Previous
63.
Dimensional formulae for some physical quantities
Home Next Previous
67.
V
V
dv
dx
Home Next Previous
68.
change in temperature
distance
S Q / T
Home Next Previous
69.
heat energy
area x time x temperature 4
Home Next Previous
70.
potential difference
current
Q 1Q 2
o
4Fd 2
Home Next Previous
71.
4Fd 2
o
m1m 2
Home Next Previous
73.
]
Acknowledg
ement
Physics Part I for Class XI by NCERT
Home End Previous
+1-315-636-4485
+91 98151-74050 
