Physics Notes First Year Class

PHYSICS REVIEW NOTES: XI
COMPREHENSIVE PHYSICS REVIEW NOTES FOR CLASS FIRST YEAR
BY
DR. RAM CHAND RAGUEL
PHD(PHYSICS)
Principal & Head of the Physics Department
Government Girls Degree College, Jhudo
District Mirpurkhas
0233878056, ggdcjhudo@gmail.com
http://www.facebook.com/ggdcjhudo
2017
RAM’S OUTLINE SERIES

Copyright c 2017, Department of Physics, Government Girls Degree College, Jhudo
COMPOSED BY DR. RAM CHAND RAGUEL
This manuscript is written in LATEX. The diagrams and images are created in open-source
applications IPE, LatexDraw, VUE and Blender 3D.
The author is a visiting scientist to Aspen Center for Physicist, USA, the University of
Malaya, Kuala Lumpur, Malaysia, the International Center for Theoretical Physics (ICTP),
Italy and the Chinese Academy of Sciences, Beijing, China. The author is also a member
of American Association of Physics Teachers (AAPS), USA. The author’s research proﬁle
can be found at his Google Scholar page.
ram_r25@hotmail.com, raguelmoon@gmail.com
http://www.facebook.com/ramcraguel
@RamCRaguel
Research page: https://sites.google.com/site/thecomphys/research-1/Soft–Condensed-Matter-Theory
First printing, February 2017

Dedication
This manuscript is dedicated to my dear students who emphasized me to write this work.
Typos, errors and omissions will be removed in next revised edition. More comprehen-
sive and conceptual ideas, sketches and diagrams will be added. I would highly appreciate
students’ comments for further revision of the manuscript.

Contents
1 SCOPE OF PHYSICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 SCALARS AND VECTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 MOTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 MOTION IN TWO DIMENSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 STATICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 GRAVITATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 WORK, POWER AND ENERGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8 WAVE MOTION AND SOUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
9 NATURE OF LIGHT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10 GEOMETRICAL OPTICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Ram’s Outline Series

1. SCOPE OF PHYSICS
1.1 Science
The desire to know about things, events and natural phenomenon around us is called
Science. The word Science actually comes from the Latin word Scientia meaning "to
know". Science is the sum of human knowledge which man has gained through ages.
In past, this knowledge was limited but as the knowledge expands, its complete study
becomes impossible. Therefore science is divided into two main branches:
Physical Science: It deals with the non-living objects and those functions of living objects
which are similar to non-living objects.
Biological Science: It deals with living objects only.
1.2 What is Physics?
The word physics is derived from Greek word fusis which means "nature". Therefore,
physics means "Laws of nature". Physics is defined as that branch of science, which
studies natural phenomena in terms of basic laws and physical quantities. The whole
natural universe consists of two basic quantities : (i) matter and (ii) energy. Therefore we
define physics as:
Physics is the branch of physical science which deals with the study of
matter, energy, and the interaction between them.
Definition
• Physics is a systematic study of the universe.
• It is experimental science.

8 Chapter 1. SCOPE OF PHYSICS
• It is quantitative science.
Two principles thrust in the study of Physics are:-
— Unification. which means explaining different physical phenomena by using few
laws and concepts.
— Reductionism. which means explaining complex phenomena by breaking them
into smaller constituents and studying simpler parts.
1.3 Classification Of Physics
Physics can be classified into three main categories:
Classical Physics Also known as Macroscopic Physics (or Newtonian Physics) which
deals with the study of astronomical and other big elements. The
macroscopic world contains the things we can see with our eyes.
For example, mechanics, thermodynamics, optics etc. Father of
classical physics is Newton.
Mesoscopic Physics Which deals with the study of hundreds of atoms or molecules. The
mesoscopic world is in between the microscopic and the macro-
scopic world. This branch is very new and originated some ten
years ago. Application of this field is largely found in the electron-
ics industry. Industry makes money out of the miniaturization of
transistors, which switch the electrical current on a computer chip.
Modern Physics Modern physics is a branch of physics that deals the topic where
matter and energy are not separate, but it is an alternate form of
each other. It often involves extreme conditions: either very small
things for example atoms and sub-atomic particles OR relativistic
effects which involve velocities compared to the speed of light.
Father of modern physics is Albert Einstein.
1.4 Branches of Physics
There are many branches of physics. The oldest branch of physics is Mechanics. Detail of
all branches is given below:
Mechanics: Mechanics is the branch of Physics which deals with the study of particles or
bodies when they are at rest or in motion. Mechanics is further divided into: -
1. Statics: Statics is the study of objects at rest; this requires the idea of forces in
equilibrium.
2. Dynamics: Dynamics is the study of moving objects. It comes from the
Greek word dynamis which means power. Dynamics is further subdivided into
kinematics and kinetics.
(a) Kinematics is the study of the relationship between displacement, velocity,
acceleration and time of a given motion, without considering the forces
that cause the motion.
(b) Kinetics deals with the relationship between the motion of bodies and
forces acting on them.

1.4 Branches of Physics 9
Thermodynamics: The study of relationship between heat and other forms of energy.
Thermodynamics is only concerned with large scale observations.
Electromagnetism: The study of properties of electric current and magnetism and their
relationship: -
1. Electrostatics: The study of electric charges at rest,
2. Electrodynamics: The study of moving electric charges.
3. Magneto-statics: The study of magnetic poles at rest.
Fluid Dynamics: The mechanics of motion in fluids in both liquid and gaseous states are
investigated in studies of fluid dynamics.
Aerodynamics: The branch of dynamics that deals with the motion of air and other
gaseous fluids and with the forces acting on bodies in motion relative to such fluids.
Atomic Physics: Atomic physics is the branch of physics which deals with the composi-
tion of atom apart from nucleus. It is mainly concerned with the arrangement and
behaviour of electrons in the shells around the nucleus.
Nuclear Physics: The physics of atomic nuclei and their interactions, especially in the
generation of nuclear energy.
Quantum Mechanics: The study of discrete (quantized) nature of phenomena at the
atomic and subatomic level.
Plasma Physics: The study of fourth state of matter - Plasma. Sun is made of plasma.
Condensed Matter Physics: The study of properties of condensed materials (solid, liquid
and those intermediate between them and dense gas) with the ultimate goal and
developing new materials with better properties; it is extension of Solid State Physics.
Statistical Mechanics: The branch of physics that attempts to relate the properties of
macroscopic systems to their atomic and molecular constituents.
Optics: Optics is the branch of physics, which deals with the propagation, behaviour and
properties of light.
Acoustics: The study of production and propagation of sound waves.
Geophysics: Geophysics is the branch of physics which deals with the study of earth. It is
mainly concerned with the shape, structure and composition of earth. It also studies
the gravitational force, magnetic fields, earthquakes, magmas, eruption of volcanoes
etc.
Biophysics: The interdisciplinary study of biological phenomena and problems, using the
principles and techniques of physics.
Astronomy: The branch of science which deals with celestial objects, space, and the
physical universe as a whole.
Astrophysics: Astrophysics is concerned with the study of universe i.e., stars, galaxies
and planets using the laws of physics.
Cosmology: The study of the origin of the universe as a whole, of the contents, structure,
and evolution of the universe from the beginning of time to the future.
Particle Physics: The branch of physics that deals with the properties and behavior of
elementary particles. Also known as High Energy Physics.
Cryogenics: Also known as Cryophysics, is the study of matter at extremely low temper-
atures.

1.5 Physical Quantities
Physics is based on measurement. We discover physics by learning how to measure the
quantities that are involved in physics and we call it as physical quantities. Physical
quantities are quantities that can be measured. All physical quantities have magnitudes
with suitable (standardized) units. These can be classified as:
Figure 1.1: Physical Quantity
Basic Quantities: Those physical quantities which can not be derived and are basic in
nature. Seven basic quantities are chosen for their convenience.
Derived Quantities: All other quantities are derived from one or more of the basic quanti-
ties. These can be expressed in terms of fundamental physical quantities, e.g., speed
= distance/time.
1.5.1 Category of Physical Quantities
Dimensional Costants These are the quantities which possess dimensions and have a
fixed value. For example, Gravitational constants G, mass of earth ME, speed of
light c etc.
Dimensional Variables These are the quantities which possess dimensions and do not
have a fixed value; for example, velocity v, acceleration a, force F etc.
Dimensionless Constants These are the quantities which do not possess dimensions and
have a fixed value; for example, π.
Dimensionless Variables These are the quantities which are dimensionless and do not
have a fixed value; for example, Strain, Steradian, Specific Gravity etc.
1.6 Unit
All physical quantities are measured with respective to standard magnitude of the same
physical quantity and these standards are called UNITS; e.g., second, meter, kilogram, etc.
1.6.1 International System of Units (SI)
The International System of Units (French: Système international d’unités, SI) is the
modern form of the metric system, and is the most widely used system of measurement
throughout the world.
1.6.2 SI Standards
Length
Length is defined as the distance between two points. The SI unit of length is metre.
"One standard metre is equal to 1650763.73 wavelengths of the orange-red light emitted
by the individual atoms of Krypton−86 in a krypton discharge lamp."

1.6 Unit 11
Table 1.1: SI units
Quantity Unit Name Symbol
Length meter m
Mass kilogram kg
Time second s
Thermodynamic Temperature Kelvin K
Electric Current Ampere A
Amount of substance mole mol
Luminous Intensity candela cd
Mass
Mass is the quantity of matter contained in a body. It is independent of temperature and
pressure. It does not vary from place to place. The SI unit of mass is kilogram.
"The kilogram is equal to the mass of the international prototype of the kilogram (a
plantinum-iridium alloy cylinder) kept at the International Bureau of Weights and Measures
at Sevres, near Paris, France."
Time
In 1967, an atomic standard was adopted for second, the SI unit of time.
"One standard second is defined as the time taken for 9192631770 periods of the radiation
corresponding to unperturbed transition between hyperfine levels of the ground state of
Cesium−133 atom. Atomic clocks are based on this." In atomic clocks, an error of one
second occurs only in 5000 years.
Ampere
"The ampere is the constant current which, flowing through two straight parallel infinitely
long conductors of negligible cross-section, and placed in vacuum 1m apart, would produce
between the conductors a force of 2×10−7 newton per unit length of the conductors."
Kelvin
"The Kelvin is the fraction of 1
273.16 of the thermodynamic temperature of the triple point
of water"
Candela
"The candela is the luminous intensity in a given direction due to a source, which emits
monochromatic radiation of frequency 540×1012Hz and of which the radiant intensity in
that direction is 1
683 watt per steradian."
Mole
"The mole is the amount of substance which contains as many elementary entities as there
are atoms in 0.012kg of carbon-12."

1.7 Dimension
The word dimension means the physical nature of a quantity. It is used to find the nature
of equation or expression in terms of fundamental quantities.
Dimension is an expression of the character of a derived quantity in relation to funda-
mental quantities, without regard for its numerical value.
For example, the distance between two points in space can be measured in feet, meters,
or miles, which are different ways of expressing the dimension of length. In any system of
measurement, such as the metric system, certain quantities are considered fundamental,
and all others are considered to be derived from them. The symbols that we use to specify
the dimensions of length, mass, and time are L, M, and T, respectively. The expression
of any particular quantity in terms of fundamental quantities is known as dimensional
analysis and often provides physical insight into the results of a mathematical calculation.
During dimensional analysis on given equation, the following rules may apply:
1. Dimensions on left hand side (LHS) of the equation must be equal to the dimensions
on the right hand side (RHS):
LHS = RHS
2. Dimensions can be treated as algebraic quantities.
3. During dimension analysis, coefficients of any kind in equations should be ignored.
For example:
vf = vi +at
vf = m/s = L/T; vi = m/s = L/T; a = m/t2
= L/T2
=⇒
L
T
=
L
T
+
L
&&T2 T =
L
T
+
L
T
= 2
L
T
But 2 in above equation is coefficient which must be dropped:
L
T
= ¡¡2
L
T
Therefore,
L
T
=
L
T
=⇒ M0
L1
T−1
= M0
L1
T−1
In above dimension analysis, there is no mass involved so we write M0, power to
mass shows zero, while L has one power and T has negative one power.
4. Such quantities can be added or subtracted only if they have the same dimensions.
5. Convert all subtractions into additions. Dimensions can never be subtracted but al-
ways be added and finally coefficient should be dropped:
vf = vi −gt
In above equation, we can not subtract dimensions, but we have to add them.
6. Dimensionally correct equation does not mean that the equation is correct.
7. The correct equation always be dimensionally correct.

1.8 Scientific Notation 13
Physical Quantities and their Dimensions
Following physical quantities with their dimensions are given. Note that power to dimen-
sion shows how many times the physical quantity is used in the equation or formula. If no
physical quantity (no unit) is used then power to that dimension is given as zero.
PHYSICAL QUANTITIES HAVING SAME DIMENSIONAL FORMULA
• Distance, Displacement, radius, light year, wavelength, radius of gyration (L) .
• Speed, Velocity, Velocity of light .
• acceleration, acceleration due to gravity (g), intensity of gravitational field, cen-
tripetal acceleration .
• Impulse, Change in momentum
• Force, Weight, Tension, Thrust
• Work, Energy, Moment of force or Torque, Moment of couple
• Force constant, Surface Tension, Spring constant, Energy per unit area
• Angular momentum, Angular impulse, Plank’s constant, Angular velocity, Fre-
quency, Velocity gradient, Decay constant, rate of disintegration
• Stress, Pressure, Modulus of Elasticity, Energy density
• Latent heat, Gravitational potential
• Specific heat, Specific gas constant
• Thermal capacity, Entropy, Boltzmann constant, Molar thermal capacity,
• wave number, Power of a lens, Rydberg constant
• Time, RC, L R ,
• Power, Rate of dissipation of energy,
• Intensity of sound, Intensity of radiation
• Expansion coefficient, Temperature, coefficient of resistance
• Electric potential, potential difference, electromotive force
• Intensity of magnetic field, Intensity of magnetization
1.8 Scientific Notation
Scientific notation is part of the language physics which allows us to deal with a vast array
of numbers, large and small.
Scientific notation is defined as a standardized way to represent any number as the
product of a real number and a power of 10.
a×10b
In this form, a is called the coefficient and b is the exponent.
The coefficient is the value of any numerical expression in real number.
1.8.1 Multipliers and Prefix
In Physics, multipliers are defined in powers of 10 from 10−24 to 1024, proceeding in
increments of three orders of magnitude (103 or 1,000). These multipliers are denoted in
Table No. 1.2 and in Table No. 1.3.

Quantity Unit Dimension
Area (A) m2 M0L2T0
Volume (V) m3 M0L3T0
Density (ρ) kg/m3 M1L−3T0
Velocity (v) m/s M0L1T−1
Acceleration (a) m/s2 M0LT−2
Momentum (p) kg.m/s M1L1T−1
Force (F) N (kg.m/s2) M1L1T−2
Pressure (p) Pa (kg/m.s2) M1L−1T−2
Energy (E) J (kg.m2/s2) M1L2T−2
Power (P) W (kg.m2/s3) M1L2T−3
Frequency (ν) Hz (1/s) M0L0T−1
Strain (ε) No unit M0L0T0
Stress (σ) Pas (N/m2) M1L−1T−2
Surface Tension (γ) N/m M1L0T−2
Entropy (S) J/K M1L2T−2K−1
Power of Lens (P) Diaptors M0L−1T0
Electric Resistance (R) Ω(V/A) M1L2T−3I−2
Electric Capacity (C) Farad M−1L−2T4I2
Electric Charge (Q) Coulomb M0L0T1I1
Electric Potential (E) V M1L2T−3I−1
Refractive Index (n) unit-less M0L0T0
Magnetic Flux (φ) weber M1L2T−2I−1
Coefﬁcient of linear expansion
(α)
1/K M0L0T0K−1
Magnetic ﬁeld (B) Tesla M1L0T−2I−1
Resistivity (ρ) Ω−m M1L3T−3I−1
Plank’s constant (h) J −s M1L2T−1

1.9 Errors and Significant Figures 15
Table 1.2: Bigger multipliers
Prefix Symbol Multiplier
Yotta Y 1024
Zetta Z 1021
Exa E 1018
Peta P 1015
Tera T 1012
Giga G 109
Mega M 106
Hektokilo hk 105
Myria ma 104
Kilo k 103
Hekto h 102
Deka D 101
UNIT 1 100
1.9 Errors and Significant Figures
• The uncertainty in the measurement of a physical quantity is called an 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 limit the quantity is measured.
The errors in measurement can be classified as: -
(i) Systematic errors and (ii) Random errors
• SYSTEMATIC ERRORS: These are the errors that tend to be either positive or
negative. Sources of systematic errors are
– Instrumental errors
– Imperfection in experimental technique or procedure
– Personal errors
• RANDOM ERRORS :Those errors which occur irregularly. These errors arise due
to unpredictable fluctuations in experimental conditions
• Least count error is the error associated with the resolution of the instrument.
• The magnitude of the difference between the individual measurement and the true
value of the quantity is called the absolute error of the measurement.
∆a = |a−amean|
• The relative error or the percentage error is the ratio of the mean absolute error to
the mean value of the quantity measured. When the relative error is expressed in per
cent it is called the percentage error: -
Relative Error = ∆amean
amean
; and Percentage Error = ∆amean
amean
×100

Table 1.3: Smaller multipliers
Prefix Symbol Multiplier
Yocto y 10−24
Zepto z 10−21
Atto a 10−18
Femto f 10−15
Pico p 10−12
Nano n 10−9
Micro µ 10−6
milli m 10−3
Centi c 10−2
Deci d 10−1
1.9.1 Calculation of errors
Error of sum or difference
• Errors are always added.
• When two quantities are added or subtracted, the absolute error in the final result is
the sums of the absolute errors in the individual quantities: -
– If C = A+B, then maximum possible error in C is ∆C = ∆A+∆B.
– If C = A−B, then maximum possible error in C is ∆C = ∆A+∆B.
Error of product or division
• Even though quantities are multiplied or divided, the errors are always added.
• When two quantities are multiplied or divided the relative error is the sum of the
relative errors in the multipliers: -
– If C = A×B, then maximum possible error in C = ∆C/C = (∆A/A+∆B/B).
– If C = A/B, then maximum possible error in C = ∆C/C = (∆A/A+∆B/B).
Error of power
• The relative error in a physical quantity raised to the power k is the k times the
relative error in the individual quantity :
Suppose C = Ak, then error in C = ∆C/C = k(∆A/A).
1.9.2 Significant Figures
The reliable digits plus the first uncertain digit in a measurement are called Significant
Figures.
Rules for finding significant figures in a measurement : -
• There are three rules on determining how many significant figures are in a number:
– Non-zero digits are always significant.

1.10 Questions and answers 17
– Any zeros between two significant digits are significant.
– A final zero or trailing zeros in the decimal portion ONLY are significant.
• If the number is less than 1, the zero(s) on the right side of decimal point but to the
left of the first non-zero digit are not significant.
For example: In 0.00035 the underlined zeros are not significant.
• The final or trailing zeros in a number without a decimal point are not significant :
For example: 1885m = 188500cm = 1885000mm has four significant figures.
• The trailing zeros in a number with a decimal point are significant :
For example: The numbers 75.00 or 0.06700 have four significant figures each.
• Zeroz between any significant figures are significant. For example: in 406, the
number 4 and 6 are significant so is zero. This is sometimes called "captured zero".
• Trailing zeros in a whole number: 200 is considered to have only ONE significant
figure if this is based on the way each number is written. When whole number
are written as above, the zeros, BY DEFINITION, did not require a measurement
decision, thus they are not significant.
• If 200 really has two or three significant figures then it must be written in scientific
notation. If 200 has two significant figures, then 2.0×102 is used. If it has three,
then 2.00×102 is used. If it has four, then 200.0 is sufficient.
1.10 Questions and answers
Q:1 Define following? (i) Supplementary Units (ii) Radian (iii) Steradian
Ans 1. Supplementary Units: The General Conference on Weights and Measures has
not yet classified certain unit of SI under either base or derived units. These
SI units are called derived supplementary units. Radian and Steradian are
supplementary units. See Table.
Table 1.4: Supplementary units
Quantity Unit Name Symbol
Plane Angle radian rad
Solid Angle steradian sr
2. Radian: The 2D angle between two radii of a circle corresponding to the arc
length of one radius on its circumference is called radian.
3. Steradian: It is the 3D angle subtended at the center of the sphere correspond-
ing to its surface area equal to the square of radius of sphere.
Q:2 What are practical units?
Ans larger number of units are used in general life for measurement of different quantities
in comfortable manner. But they are neither fundamental units nor derived units.
Generally, the length of a road is measured in mile. This is the practical unit of
length. Some practical units are given below :
1. 1fermi = 1 fm = 10−15m
2. 1 X-ray unit = lxu = 10−13m
3. 1angstrom = 1 ˙A = 10−10m

4. 1micron = 1µm = 10−6m
5. 1 astronomical unit = 1Au = 1.49×1011m [Average distance between sun and
earth, i.e., radius of earth’s orbit]
6. 1 light year = 1LY = 9.46 × l015m [Distance that light travels in 1 year in
vacuum]
7. 1parsec = 1pc = 3.08×1016m = 3.26 light year [The distance at which a star
subtends an angle of parallex of 1 s at an arc of 1 Au].
8. One shake = 10−8 second.
9. One slug = 14.59kg
10. One pound = 453.6 gram weight
11. One metric ton = 1000kg
12. 1barn = 10−28m2
13. 1 atmospheric pressure = 1.013×105N/m2 = 760mm of Hg
14. 1 bar = 105N/m2 or pascal
15. 1 torr = lmm of Hg = 133.3N/m2
16. 1 mile = 1760yard = 1.6 kilometre
17. 1 yard = 3 ft
18. 1ft = 12 inches
19. 1 inch = 2.54cm

1.10 Questions and answers 19
RAM’S EXCLUSIVE
Converting physical quantity from one system to another system
Dimensional formula is useful to convert the value of a physical quantity from one
system to the other. Physical quantity is expressed as a product of numerical value
and unit.
In any system of measurement, this product remains constant. By using this fact,
we can convert the value of physical quantity from one system to another. Let n1 is
the numerical value of the system u1 and let n2 is the numerical value of another
system u2, then:
n1[u1] = n2[u2]
Example: Convert one Joule into Erg.
Solution Joule and erg are units of work. The dimensions formula for work are:
[ML2T−2].
u1 = [M1L2
1T−2
1 ],u2 = [M2L2
2T−2
2 ]
Where u1 for SI and u2 for CGS. According conversion equation:
n1[u1] = n2[u2] =⇒ n1[M1L2
1T−2
1 ] = n2[M2L2
2T−2
2 ]
Here M1 = kg, L1 = meter and T1 = second. For u2: M2 = grams, L2 = cm
and T2 = sec.
But M1 = 1000M2, L1 = 100L2, T1 = T2 and n1 = 1, so
(1)[1000M2][100L2]2
[T−2
2 ] = n2[M2][L2
2][T−2
2 ]
n2 = [
1000M2
M2
][
100L2
L2
]2
[
T2
T2
]−2
n2 = [1000][100]2
[1]−2
= 1000×10000 = 10000000 = 107
∴ 1Joule = 107
erg.

2. SCALARS AND VECTORS
2.1 INTRODUCTION
There are many physical quantities in nature. For proper measurement and calculation
each of these quantities requires one or more dimensions to describe it mathematically.
Here we can divide them up into two types according to how many dimensions it uses to
describe. These are called vectors and scalars.
2.1.1 Scalars
Scalars are used to describe one dimensional quantities, that is, quantities which require
only one number to completely describe them. A scalar tells you how much of something
there is.
A scalar is a physical quantity that has only a magnitude (size) along with a unit.
Deﬁnition
Scalar quantities change when their magnitudes change.
2.1.2 Vectors
Vectors are used to describe multi-dimensional quantities. Multi-dimensional quantities are
those which require more than one number to completely describe them. Vectors, unlike
scalars, have two characteristics, magnitude and direction. (If there are more than two
dimensions then we use tensor). A vector tells you how much of something there is and
which direction it is in.

22 Chapter 2. SCALARS AND VECTORS
A vector is a physical quantity that has both a magnitude and a direction.
Definition
• Distance is a scalar quantity that refers to "how much ground an object has covered"
during its motion.
• Displacement is a vector quantity that refers to "how far out of place an object is"; it
is the object’s overall change in position.
• The magnitude of v is written |v|.
• Properties of Vectors:
1. Vectors are equal if they have the same magnitude and direction.
2. Vectors must have the same units in order for them to be added or subtracted.
3. The negative of a vector has the same magnitude but opposite direction.
4. Subtraction of a vector is defined by adding a negative vector:
A−B = A+(−B)
• Vector quantities change when:
1. their magnitude change
2. their direction change
3. their magnitude and direction both change
• Electric current, velocity of light have both magnitude and direction but they do not
obey the laws of vector addition. Hence they are scalars.
Table 2.1: Comparison
Aspect Scalar Vector
Mathematics arithmetic: addition,
subtraction sum, dif-
ference multiplica-
tion
trigonometry: vector addition, vector subtraction resul-
tant or net (∑), change (δ) dot product, cross product
Represent a number with a unit
• a number and a direction angle, both with units
OR
• a number with a unit for each unit vector (î, ˆj, ˆk)
OR
• an arrow drawn to scale in a specific direction
2.1.3 Vector notation
Vectors are different to scalars and must have their own notation. There are many ways of
writing the symbol for a vector. Vectors can be shown by symbols with an arrow pointing
to the right above it. For example, force can be written as: F.

2.2 TYPES OF VECTORS 23
Graphical representation of vectors
Vectors are drawn as arrows. An arrow has both a magnitude (how long it is) and a direction
(the direction in which it points). The starting point of a vector is known as the tail and the
end point is known as the head.
Figure 2.1: Vector
2.2 TYPES OF VECTORS
2.2.1 Real Vector OR Polar Vector
If the direction of a vector is independent of the coordinate system, then it is called a polar
vector. Example : linear velocity, linear momentum, force, etc.
2.2.2 Pseudo Vector OR Axial Vector
Vectors associated with rotation about an axis and whose direction is changed when the
co-ordinate system is changed from left to right, are called axial vectors (or) pseudo
vectors.
Example : Torque, Angular momentum, Angular velocity, etc.
2.2.3 Position Vector
It is a vector that represents the position of a particle with respect to the origin of a
co-ordinate system. The Position Vector of a point (x,y,z) is r.
2.2.4 Unit Vector
It is a vector whose magnitude is unity (one). A unit vector is used to show the direction of
a given vector. Mathematically, it can be deﬁned as: ˆa = A
A
.
2.2.5 Equal vectors
Two vectors are said to be equal if they have the same magnitude and same direction,
wherever be their initial positions.
2.2.6 Like vectors
Two vectors are said to be like vectors, if they have same direction but different magnitudes.
2.2.7 Unlike vectors
The vectors of different magnitude acting in opposite directions are called unlike vectors.

2.2.8 Opposite vectors OR negative vector
The vectors of same magnitude but opposite in direction, are called opposite OR negative
vectors.
2.2.9 Null vector or zero vector
A vector whose magnitude is zero, is called a null vector or zero vector. It is represented by
O and its starting and end points are the same. The direction of null vector is not known.
2.2.10 Proper vector
All the non-zero vectors are called proper vectors.
2.2.11 Co-initial vectors
Vectors having the same starting point are called co-initial vectors. A and B start from the
same origin O. Hence, they are called as co-initial vectors.
2.2.12 Coplanar vectors
Vectors lying in the same plane are called coplanar vectors and the plane in which the
vectors lie are called plane of vectors.
RAM’S MIND MAP
2.3 VECTOR ADDITION
When adding vector quantities remember that the directions have to be taken into account.

2.3 VECTOR ADDITION 25
• The result of adding vectors together is called the resultant.
• When adding two vectors together:
1. the greatest (maximum) resultant is equal to their sum
2. the smallest (minimum) resultant is equal to their difference
3. the resultant can have any value between these limits depending on the angle
between the two vectors
• Pythagorean theorem is used to determine magnitude of the vector.
• The tangent function is used to determine direction of the vector.
• In problems, vectors may be added together by scale diagram or mathematically.
Addition of Vectors by Graphical Method
A process in which two or more vectors are added is called addition of vectors. Parallel or
anti-parallel vectors are added by simple arithmetic rules. For non-parallel vectors, vectors
are not added and subtracted by simple arithmetic rules. For this process vectors are added
and subtracted by head to tail method.
Head to tail Rule
Consider two vectors, A and B. In order to add we can place the tail of B so that it meets
the head of A. The sum, A+B, is the resultant vector from the tail of A to the head of B.
Figure 2.2: Head to tail Rule
Adding Parallel Vectors
If the vectors you want to add are in the same direction, they can be added using simple
arithmetic. Consider two vectors P and Q which are acting along the same line. To add
these two vectors, join the tail of Q with the head of P.
The resultant of P and Q is R = P+Q. The length of the line AD gives the magnitude of R.
R acts in the same direction as that of P and Q.
Parallelogram Law
In order to ﬁnd the sum of two vectors, which are inclined to each other, parallelogram law
of vectors, can be used.
According to the parallelogram law of vector addition:
“If two vector quantities are represented by two adjacent sides or a parallelogram then
the diagonal of parallelogram will be equal to the resultant of these two vectors.”
Consider two vectors A and B. To add A and B using the parallelogram method, place
the tail of B so that it meets the tail of A. Take these two vectors to be the ﬁrst two
adjacent sides of a parallelogram, and draw in the remaining two sides. The vector sum,
A+B, extends from the tails of A and B across the diagonal to the opposite corner of the
parallelogram.
If the vectors are perpendicular and unequal in magnitude, the parallelogram will be a

Figure 2.3: (left) Vector Addition. (Right) Parallelogram Method.
rectangle. If the vectors are perpendicular and equal in magnitude, the parallelogram will
be a square.
Adding Perpendicular Vectors
Consider two vectors A and B which are perpendicular to each other. Addition of these
vectors can be performed by head to tail rule and the magnitude of resultant vector A and
B can be calculated by using Pythagorean Theorem.
Triangle law of vectors
To ﬁnd the resultant of two vectors P and Q which are acting at an angle θ, following laws
are used:
Figure 2.4: Law of Sines and Law of Cosine
1. Law of Cosine: Magnitude of resultant of two vectors P and Q can be obtained by
Law of Cosine:
R2
= P2
+Q2
−2PQcos(180o
−θ) |R| = P2 +Q2 +2PQcosθ

2.4 VECTOR SUBTRACTION 27
2. Law of Sines: This law is used to find the direction of the resultant of these vectors:
P
sinβ
=
Q
sinα
=
R
sin(180o −θ)
Properties of addition of vectors
1. Commutative law (The order of addition is unimportant.): A+B = B+A
2. Associative law : A+(B+C) = (A+B)+C
3. Distributive law : m(A+B) = mA+mB. Where m is a scalar
4. Binary operation: Vector addition is a binary operation. (Only two vectors can be
added at a time.)
2.4 VECTOR SUBTRACTION
Let’s take the two vectors A and B as shown in figure.
To subtract B from A, take a vector of the same magnitude as B (negative of vector), but
Figure 2.5: Vectors A and B
pointing in the opposite direction, and add that vector to A, using either the head-to-tail
method or the parallelogram method.
Figure 2.6: Vectors Subtraction
2.5 VECTOR COMPONENTS
• Angled Vector which is not along x-axis, y-axis or z-axis can be thought of as having
an influence in three different directions. Each part of a 3-dimensional vector is
known as a component.

• The combined influence (Resultant) of the three components is equivalent to the
influence of the single 3-dimensional vector.
2.5.1 Resolution of vectors and rectangular components
A vector directed at an angle with the co-ordinate axis, can be resolved into its components
along the axes. This process of splitting a vector into its components is known as resolution
of a vector.
Explanation
Consider a vector R = OA making an angle θ with x−axis. The vector R can be resolved
into two components along X −axis and y−axis respectively. Draw two perpendiculars
from A to X and Y axes respectively. The intercepts on these axes are called the scalar
components |Rx| and |Ry|.
Then, OP is |Rx|, which is the magnitude of x component of R and OQ is |Ry|, which is the
magnitude of y component of R.
From OPA,
cosθ = OP
OA = |Rx|
|R|
OR
|Rx| = |R|cosθ
sinθ = OQ
OA =
|Ry|
|R|
OR
|Ry| = |R|sinθ
And |R2| = |Rx
2
|+|Ry
2
|
Also, R can be expressed as:
R = |Rx|î+|Ry| ˆj
where î and ˆj are unit vectors. In terms of Rx and Ry , θ can be expressed as:
θ = tan−1 |Ry|
|Rx|
.
2.6 MULTIPLICATION OF VECTORS
There are two forms of vector multiplication: one results in a scalar, and one results in a
vector.
2.6.1 Scalar product OR Dot product of two vectors
If the product of two vectors is a scalar, then it is called scalar product. If A and B are
two vectors, then their scalar product is written as A.B and read as A dot B. Hence scalar
product is also called dot product. This is also known as INNER or DIRECT PRODUCT.
The scalar product of two vectors is a scalar, which is equal to the product of magnitudes
of the two vectors and the cosine of the angle between them. The scalar product of two
vectors A and may be B expressed as:
A.B = |A||B|cosθ
where |A| and |B| are the magnitudes of A and B respectively and θ is the angle between A
and B. The magnitude of A or B can be calculated by using Pythagoras Theorem.

2.6 MULTIPLICATION OF VECTORS 29
In Scalar Product, units vectors can be calculated as:
î.î = ˆj. ˆj = ˆk.ˆk = 1 î. ˆj = î.ˆk = ˆj.ˆk = 0
NOTE: Dot Product of unit vectors always yield zero(0) OR one (1).
2.6.2 Vector product or Cross product of two vectors
If the product of two vectors is a vector, then it is called vector product. If A and B are
two vectors, then their vector product is written as A×B and read as A cross B. This is
also called as outer product because the resultant vector is out of the plane containing two
vectors.
The vector product or cross product of two vectors is a vector whose magnitude is equal to
the product of their magnitudes and the sine of the smaller angle between them and the
direction is perpendicular to a plane containing the two vectors.
If θ is the smaller angle through which A should be rotated to reach B, then the cross

product of A and B is expressed as,
C = A×B = |A||B|sinθ ˆn
where |A| and |B| are the magnitudes of A and B respectively and ˆn is a unit vector
perpendicular to both A and B. The resultant product can be expressed as C. The direction
of C is perpendicular to the plane containing the vectors A and B.
The magnitude of the cross product vector is equal to the area made by a parallelogram
of A and B. In other words, the greater the area of the parallelogram, the longer the cross
product vector.
The resultant product C can be expressed in î, ˆj, ˆk form if A and B are given in unit vector
form:
C = A×B =
î ˆj ˆk
Ax Ay Az
Bx By Bz
The magnitude of A, B or C can be calculated by using Pythagoras Theorem.
In Vector Product, units vectors can be calculated as:
î× î = ˆj × ˆj = ˆk × ˆk = 0
While combination of different unit vectors can be expressed as:
î× ˆj = ˆk ˆj × ˆk = î ˆk × î = ˆj ˆj × î = −ˆk î× ˆk = − ˆj ˆk × ˆj = −î
NOTE: Cross Product of unit vectors always yield zero(0) OR another unit vector.

2.6 MULTIPLICATION OF VECTORS 31
Figure 2.7: Technique of cross product
Points to Note:
• Vector does not obey the laws of simple algebra.
• Vector obeys the laws of vector algebra.
• Vector does not obey division law. e.g. A
B
is meaningless.
• Division of a vector by a positive scalar quantity gives a new vector
whose direction is same as initial vector but magnitude changes.
• A scalar quantity never be divided by a vector quantity.
• The angle between two vectors is always lesser or equal to 180o. (i.e.,
0 < θ < 180o)
• A vector never be equal to scalar quantity.
• The magnitude or modulus of a vector quantity is always a scalar
quantity.
• Two vectors are compared with respect to magnitude.
• The minimum value of a vector quantity is always greater than or equal
to zero.
• The angle between like parallel vectors is zero and that of unlike parallel
vectors is 180o.
• The magnitude of parallel vectors may or may not be same. If the
magnitude of like parallel vectors are same, then the vectors are known
as equal vectors.

3. MOTION
3.1 KINEMATICS
In Kinematics we study the description of motion of bodies. We can describe the motion
of any body with its, distance, time, velocity, acceleration and time it takes.
3.1.1 Parameters used in Kinematics
Particle
A particle is ideally just a piece or a quantity of matter, having practically no linear
dimensions but only a position.
Rest
When a body does not change its position with respect to time and surroundings, then it is
said to be at rest.
Motion
Motion is the change of position of an object with respect to time and surroundings.
Distance and Displacement
The total length of the path is the distance traveled by the particle and the shortest distance
between the initial and ﬁnal position of the particle is the displacement.
The distance traveled is a scalar quantity and the displacement is a vector quantity.
SI unit of distance OR displacement is meter (m). The dimensions are : MoL1To

34 Chapter 3. MOTION
Comparison between distance and displacement
• For a moving particle in a given time interval distance can be many valued
function, but displacement would always be single valued function.
• Displacement could be positive, negative or zero, but distance would always
be positive.
• Displacement can decrease with time, but distance can never decrease with
time.
• Distance is always greater than or equal to the magnitude of displacement.
• Distance would be equal to displacement if and only is particle is moving
along straight line without any change in direction.
Speed
Distance covered by a body in unit time is known as speed.
Let a body covers a distance S in time t, then, mathematically:
v =
distance
time
v =
S
t
It is a scalar quantity. Its SI unit is meter ms−1. The dimensions are: MoLT−1
Average Speed
The average speed is defined as total distance traveled by a body in a particular time
interval divided by the time interval. Thus, the average speed OR total distance covered
divided by total time taken :
vavg =
total distance covered
t2 −t1
=
∆t
OR vavg =
total time taken
Velocity
The velocity of a particle is defined as the rate of change of displacement of the particle.
It is also defined as the speed of the particle in a given direction. The velocity is a vector
quantity. It has both magnitude and direction.
Its SI unit is ms−1 and its dimensional formula is M0LT−1.
Uniform Velocity
A particle is said to move with uniform velocity if it moves along a fixed direction and
covers equal displacements in equal intervals of time, however small these intervals of
time may be.
Instantaneous velocity
It is the velocity at any given instant of time or at any given point of its path. The
instantaneous velocity v is given by
v = lim
∆t→0
∆S
∆t

3.1 KINEMATICS 35
Average Velocity
Let S1 be the displacement of a body in time t1 and S2 be its displacement in time t2. The
average velocity during the time interval (t2 −t1) is defined as:
vavg =
S2 −S1
t2 −t1
=
∆S
∆t
— NOTE. velocity = speed + direction of motion.
— NOTE. Note that ∆ (delta) always means "final minus initial".
— NOTE. If the velocity of an object varies over time, then we must distinguish
between the average velocity during a time interval and the instantaneous velocity at a
particular time.
Acceleration
Time rate of change of velocity is called acceleration. Mathematically:
a =
v
t
Acceleration is a vector quantity. Whenever magnitude or direction of velocity or both
change then there is acceleration. SI unit of acceleration is ms−2. Dimensions of accelera-
tion are: MoLT−2.
Uniform acceleration
If the velocity changes by an equal amount in equal intervals of time, the acceleration is
said to be uniform.
Retardation or deceleration
If the velocity decreases with time, the acceleration is negative. The negative acceleration
is called retardation or deceleration.
Average Acceleration and Instantaneous Acceleration
In general, when a body is moving, its velocity is not always the same. A body whose
velocity is increasing is said to be accelerated.
Average acceleration is defined as change in velocity divided by the time interval.
Let us consider the motion of a particle. Suppose that the particle has velocity v1 at t = t1
and at a later time t = t2 it has velocity v2. Thus, the average acceleration during time
interval ∆t = t2 −t1 is :
vavg =
v2 −v1
t2 −t1
=
∆v
∆t
If the time interval approaches to zero, average acceleration is known as instantaneous
acceleration. Mathematically,
a = lim
∆t→0
∆v
∆t

3.1.2 Representing Speed, Velocity and Acceleration
• Speed v and Distance S are both always positive quantities, by deﬁnition. While
Velocity −→v has both magnitude and direction. Therefore for 1D motion (motion
along a straight line), we can represent the direction of motion with a +/– sign:
• Objects A and B have the same speed v = |−→v | = +10m/s, but they have different
velocities.
Figure 3.1:
• v = constant =⇒ ∆v = 0 =⇒ a = 0
• v increasing (becoming more positive) =⇒ a > 0
• v decreasing (becoming more negative) =⇒ a < 0
• In 1D, acceleration a is the slope of the graph of v vs. t
The direction of the acceleration
For 1D motion, the acceleration, like the velocity, has a sign ( + or – ). Just as with velocity,
we say that positive acceleration is acceleration to the right, and negative acceleration is
acceleration to the left.
• direction of a =direction of v.
• direction of a = the direction toward which the velocity is tending = direction of v.
3.1.3 GRAPHS AND NATURE
Graphs are pictorial representations of data. In other words, graphs can show us a picture
of data. It is straight line or curve which gives the relationship between two quantities.

3.1 KINEMATICS 37
Graphs tell us TWO things: SLOPE and AREA UNDER CURVE
Slope of a line
If we divide vertical value with horizontal value, we get slope. In other words, slope is the
rate of vertical line over horizontal line. For example, speed is slope of distance versus
time.
The negative slope means the magnitude of quantity is decreasing and positive means
it is increasing. Zero slope means the quantity is constant. The slope also shows how fast
or how slow is the rate.
Area under curve
If we multiply vertical value with horizontal value then we get area under the curve. For
example, in velocity-time graphs, if we multiply velocity (vertical) with time (horizontal),
we get area under curve which is total distance covered by the body.
Velocity-time Graph
The graph which shows variation of velocity of the body with respect to time is called
velocity-time graph.
CASE I: Graph of constant velocity: Consider a body which moves with constant ve-
locity, the acceleration of the body is zero. The velocity-time graph is horizontal

straight line parallel to the time-axis.
The area under curve gives the total distance covered by the body. This area can be
calculated by multiplying velocity with time:
area under curve = velocity×time
S = v×t
CASE II: Graph of uniform acceleration: When a velocity of a body increases with a
constant rate then the body is said to be moving with uniform or constant acceleration.
The velocity-time graph is straight line inclined to the time-axis (x-axis).
CASE III: Graph of variable acceleration: If the velocity of the body doesn’t increase
by equal amounts in equal intervals of time, it is said to have variable acceleration.
The shape of velocity-time graph is curve.
CASE IV: Graph of average acceleration: Whenever the acceleration is uniform or vari-
able, the average acceleration can be calculated by the relation:
aavg =
∆v
∆t
The slope of graph between two points A and B gives the average acceleration:
aavg =
v2 −v1
t2 −t1
=
∆v
∆t
Area under the curve gives the total distance covered by the body.
distance = S = area of ABC
But, the area of ABC = 1
2|Base|×|Height|
the area of ABC = 1
2AB×BC
distance = 1
2t ×v ( AB = t,BC = v)
S =
1
2
v×t

3.1 KINEMATICS 39
3.1.4 Equations of Motion
If a body moves in straight line then the motion is said to be linear motion.
Suppose a body is moving with a constant acceleration a along a straight line. Let the
initial velocity of the body be vi and ﬁnal velocity b vf after time interval t during which
distance covered is S. Then the equations of motion are given as follows:
(1) vf = vi +at
(2) S = vi +1/2 ×at2
(3) 2aS = v2
f −v2
i
(4) S =
vf +vi
2
×t
Distance traveled in nth second
Let Sn is the distance traveled in one second between t = n and t = n−1 seconds, then
equation for calculating the distance traveled in nth second would be:
Sn = vi +(2n−1)
a
2
Motion under gravity or free fall motion
The most familiar example of motion with constant acceleration on a straight line is
motion in a vertical direction near the surface of earth. If air resistance is neglected, the
acceleration of such type of particle is gravitational acceleration which is nearly constant
for a height negligible with respect to the radius of earth. The magnitude of gravitational
acceleration near surface of earth is g = 9.81m/s2 = 32ft/s2.
Case I: If particle is moving upwards : In this case applicable kinematics equations of
motion are:
(1) vf = vi −gt
(2) h = vi −1/2 ×gt2
(3) −2gh = v2
f −v2
i
Here h is the vertical height of the particle in upward direction. At maximum hight
the ﬁnal velocity vf = 0.
Case II: If particle is moving downward: In this case,
(1) vf = vi +gt
(2) h = vi +1/2 ×gt2
(3) 2gh = v2
f −v2
i
Here h is the vertical height of the particle in downward direction. In this case the
initial velocity (vi) of free fall body is taken as zero.

3.1.5 Force
Force is that agency which causes a body to change its state of motion or rest.
Force is vector quantity and it is denoted by F. The SI unit of force is newton (N).
The dimensions of F are MLT−2.
I Newton
Force which produces acceleration of 1m/s2 in a mass of 1kg is called 1 newton.
It is denoted by N. 1 newton = 1 kilogram × meter
second2
3.1.6 Types of forces
Forces can be categorizes in two types:
Contact Force
In which the two interacting objects are physically in contact with each other.
For example: friction force, normal force, spring force etc are contact forces.
Normal force: If two blocks come in contact, they exert force on each other. The compo-
nent of contact force perpendicular to the surface of contact is generally known as normal
reaction.

3.1 KINEMATICS 41
RAM’S MIND MAP
String and Tension: If a block is pulled by a string, the string is in the condition of
tension (T). Tension is also force which ﬂows through string. SI unit of Tension is same as

Force (i.e.; Newton). There are two types of strings:
(i) Massless String: In the case of massless string, the tension, every where remains the
same in it.
(ii) Massive String: The tension in massive rope varies point to point.
Action at a distance force
These forces (non- contact forces) are forces in which the two interacting objects are not in
physical contact which each other, but are able to exert a push or pull despite the physical
separation.
For example: Gravitational force, electric force, magnetic force etc are action at a distance
forces.
3.2 Newton’s Laws of Motion
Sir Isaac Newton’s three laws of motion describe the motion of massive bodies and how
they interact. Newton published his laws in 1687, in his book “Philosophiæ Naturalis
Principia Mathematica” (Mathematical Principles of Natural Philosophy).
3.2.1 Newton’s First Law of Motion
Newton’s ﬁrst law states that every object will remain at rest or in uniform motion in a
straight line unless compelled to change its state by the action of an external force.
If the sum of all the forces on a given particle is ∑F and its acceleration is a, the above
statement may also be written as
a = 0, if and only if ∑F = 0
In this case velocity of the body is zero or uniform. This law is also know as Law of Inertia.
There are many examples of ﬁrst law of motion in everyday life.
(i) A book lying on the table remains at rest unless it is lifted or pushed by exerting a force.
(ii) A satellite revolving around the Earth continues it motion forever with uniform velocity.
3.2.2 Newton’s Second Law of Motion
The acceleration of an object as produced by a net force is directly proportional to the
magnitude of the net force, in the same direction as the net force, and inversely proportional
to the mass of the object.
Consider a body of mass m on which a force F is applied. The body will be accelerated
in the direction of force and let the acceleration produced be a. then according to 2nd Law
of Motion:
a ∝ F —->(i)
a ∝ 1
m —->(ii)
Combining equations (i) and (ii), we get:
a = F
m
or
F = ma
This is mathematical form of 2nd law of motion.

3.2 Newton’s Laws of Motion 43
3.2.3 Newton’s Third Law of Motion
It states that for every action, there is an equal and opposite reaction.
whenever one body exerts a certain force on a second body, the second body exerts an
equal and opposite force on the first. Newton’s third law is sometimes called as the law of
action and reaction.
Let there be two bodies 1 and 2 exerting forces on each other. Let the force exerted on
the body 1 by the body 2 be F12 and the force exerted on the body 2 by the body 1 be F21 .
Then according to third law,
F12 = −F21
One of these forces, say F12 may be called as the action whereas the other force F21 may
be called as the reaction or vice versa. The action and reaction never cancel each other and
the forces always exist in pair.
The effect of third law of motion can be observed in many activities in our everyday
life. The examples are
(i) When a bullet is fired from a gun with a certain force (action), there is an equal and
opposite force exerted on the gun in the backward direction (reaction).
(ii) When a man jumps from a boat to the shore, the boat moves away from him. The force
he exerts on the boat (action) is responsible for its motion and his motion to the shore is
due to the force of reaction exerted by the boat on him.
(iii) We will not be able to walk if there were no reaction force. In order to walk, we push
our foot against the ground. The Earth in turn exerts an equal and opposite force. This
force is inclined to the surface of the Earth. The vertical component of this force balances
our weight and the horizontal component enables us to walk forward.
Weight
It is defined as the force by which earth attracts a body towards its centre. If body is situated
either on the surface of earth or near the surface of earth, then gravitational acceleration is
nearly constant and is equal to g = 9.8m/s2. The force of gravity (weight) on a block of
mass m is W = mg acting towards centre of earth.
Weight is denoted by W. The SI unit of force is same as that of force, i.e.; newton (N).
3.2.4 APPLICATION OF NEWTON’S LAWS
MOTION OF BODIES CONNECTED BY A STRING
(A) When the bodies move vertically: Consider two bodies of unequal masses m1 and m2
connected by the ends of a string, which passes over a frictionless pulley as shown in the
diagram.

Figure 3.2: Application of Newton’s Law.
If mass of body A is greater than the mass of body B, i.e., m1 > m2, the body ‘A’ will
move downward with acceleration a and the body ‘B’ will move up with same acceleration.
Here we have to ﬁnd the value of a and tension T.
There are two forces acting on A:
(i) Weight of body: W1 = m1g
(ii) Tension in the string = T
The net force acting on the body is
F = m1g−T
Net force acting on body ’A’ is given by Newton’s 2nd law as m1a. Thus we have the
equation for the motion of body "A" as:
m1a = m1g−T −−− > (i)
There are also two forces acting on B (i) Weight of body: W2 = m2g
(ii) Tension in the string = T
Since body "B" is moving up, the net force acting on body is
F = T −m2g
T −m2g = m2a −−− > (ii)
Adding (i) and (ii), we get:
m1g−m2g = m1a+m2a =⇒ (m1 −m2)g = (m1 +m2)a
a =
(m1 −m2)
m1 +m2
g
Putting the value of a in equation (ii) to ﬁnd the magnitude of T :
T −m2g = m2a = m2
(m1 −m2)
m1 +m2
g =
m2g(m1 −m2)
m1 +m2
+m2g
T =
m2g{(m1 −m2)+(m1 +m2)}
m1 +m2
=
m2g{m1 −¨¨m2 +m1 +¨¨m2 }
m1 +m2
T =
2m1m2g
(m1 +m2)

(B) One body placed on a horizontal surface and connected by another vertically falling
body:
Two bodies of different masses are attached at two ends of a light string passing over a
light pulley. The mass m2 is placed on a horizontal surface and m1 is hanging freely in air.
For vertical equilibrium m2: =⇒ N = m2g
For horizontal acceleration of m2: =⇒ T = m2a
For vertically downward acceleration of m1: =⇒ m1g−T = m1a
a =
m1
m1 +m2
g
T =
m1m2g
(m1 +m2)
(c) Motion on a smooth inclined plane:
m1g−T = m1a —> (1)
T −m2gsinθ = m2a —> (ii)
a =
m1 −m2 sinθ
m1 +m2
g
T =
m1m2(1+sinθ)g
(m1 +m2)
θ
+y
+x
θ
N
m2gsinθ
T
m2g
+
T
m1g
3.2.5 INCLINED PLANE
Any plane surface which makes an angle θ with the horizontal surface is called inclined
plane such that 0o < θ < 90o.
Inclined plane is an example of simple machine which is used to lift heavy bodies without
applying very huge force.
Motion of a body on inclined plane
Consider a block of mass m placed on an inclined plane, which makes an angle θ with the
horizontal plane. The weight W of the block is acting vertically downward. The weight of
the block can be resolved into two rectangular components:
W cosθ and W sinθ.
other forces acting on the block are:
(i) Normal reaction (R) which is perpendicular to the plane

(ii) Force of friction (f) acting opposite to the direction of motion of block.
Let us take x-axis perpendicular to the inclined plane. If the block is at rest, then W sinθ
acting down the plane balances the opposing frictional force. According to Newton’s First
Law of Motion:
Along x-axis:
∑Fx = 0
f −W sinθ = 0 −−− > (1)
and along y-axis:
∑Fy = 0
R−W cosθ = 0 −−− > (2)
Since there is no motion in the direction perpendicular to the inclined plane, therefore
W cosθ is balanced by R i.e. R = W cosθ. If block slides down with an acceleration equal
to a, then the resultant force is equal to ma and the force on block will be:
W sinθ − f
According to Newton’s 2nd Law:
W sinθ − f = ma
If the force of friction is negligible, then
W sinθ = ma
&&mgsinθ = &&ma( W = mg)
a = gsinθ
This expression shows that if friction is negligible the acceleration of a body on an inclined
plane is independent of mass but is directly proportional to sinθ.
Particular cases
When θ = 0o: In this case body is lying on the surface.
a = gsin0. Since sin0 = 0, so a = g×0
a = 0
When θ = 90o: In this case slope is perpendicular to the surface.
a = gsin90. Since sin90 = 1, so a = g×1
a = g
It means that body will move as free fall motion.

RAM’S MIND MAP
3.2.6 Momentum
The momentum of a body is deﬁned as the product of its mass and velocity. If m is the
mass of the body and v, its velocity, the linear momentum of the body is given by
P = mv
Momentum has both magnitude and direction and it is, therefore, a vector quantity. The
direction of momentum is same as that of velocity. The SI unit of momentum is kgms−1
and its dimensional formula is MLT−1.
When a force acts on a body, its velocity changes, consequently, its momentum also
changes. The slowly moving bodies have smaller momentum than fast moving bodies of
same mass.
Impulse of a force
The impulse I of a constant force F acting for a short time t is deﬁned as the product of the
force and time.
Impulse = Force × time
Impulse = Ft
Impulse of a force is a vector quantity and its SI unit is Ns. Examples of impulse: The
blow of a hammer, the collision of two billiard balls etc.

Impulse and Momentum
By Newton’s second law of motion, the force acting on a body is equal to ma where m is
the mass of the body and a is acceleration produced.
The impulse of the force = F ×t = (ma)t.
If u and v be the initial and ﬁnal velocities of the body then,
a =
v−u
t
Therefore, impulse of the force = m× (v−u)
t ×t = m(v−u) = mv−mu.
Impulse = ﬁnal momentum of the body−initial momentum of the body.
That is: Impulse of the force = Change in momentum
Impulse = P
Ft = P
F =
P
t
This equation is another form of Newton’s Second Law of Motion. It states that the force
is the rate of change of linear momentum.
3.2.7 Law of conservation of momentum
The law of conservation of momentum states that: When some bodies constituting an
isolated system act upon one another, the total momentum of the system remains constant.
Consider an isolated system of two bodies "A" & "B" having masses m1 & m2 moving
initially with velocities u1 & u2 respectively. They collide with each other and after the
impact their velocities become v1 & v2.
Total momentum of system before collision = m1u1 +m2u2
Total momentum of system after collision = m1v1 +m2v2
When the two bodies collide with each other, they come in contact for a short time t.
During this interval, let the average force exerted one of the bodies is F. We know that the
rate of change of linear momentum is equal to applied force, therefore:
FA = (m1v1 −m1u1)/t —-> (1)
FB = (m2v2 −m2u2)/t —-> (2)
According to the third law of motion :
FA = −FB
Therefore: (m1v1 −m1u1)/t = −(m2v2 −m2u2)/t
m1v1 −m1u1 = −(m2v2 −m2u2)
m1v1 −m1u1 = −m2v2 +m2u2
m1u1 +m2u2 = m1v1 +m2v2
This is known as the Law of Conservation of Momentum. This expression shows that the
total momentum of an isolated system before and after collision remains constant i.e. the
total momentum of the system is conserved.

3.2.8 COLLISION
When a body strikes against body or one body inﬂuences the other from a distance, collision
is said to be occur. Collisions are of two types :
Elastic collision
An elastic collision is that in which the momentum of the system as well as kinetic energy
of the system before and after collision is conserved.
Inelastic collision
An inelastic collision is that in which the momentum of the system before and af-
ter collision is conserved but the kinetic energy before and after collision is not con-
served.
— NOTE:. If the initial and ﬁnal velocities of colliding bodies lie along the same line
then it is known as head on collision.
Elastic collision in one dimension
Consider two non-rotating spheres of mass m1 and m2 moving initially along the line
joining their centers with velocities u1 and u2 in the same direction. Let u1 is greater than
u2. They collide with one another and after having an elastic collision start moving with
velocities v1 and v2 in the same directions on the same line.
Momentum of the system before collision = m1u1 +m2u2
Momentum of the system after collision = m1v1 +m2v2
According to the law of conservation of momentum:
m1u1 +m2u2 = m1v1 +m2v2
m1v1 −m1u1 = m2u2 −m2v2
m1(v1 −u1) = m2(u2 −v2) −−−− > (1)
Similarly
K.E of the system before collision = 1/2(m1u2
1)+1/2(m2u2
2)
K.E of the system after collision = 1/2(m1v2
1)+1/2(m2v2
2)
Since the collision is elastic, so the K.E of the system before and after collision is con-
served.
Thus
1/2(m1v2
1)+1/2(m2v2
2) = 1/2(m1u2
1)+1/2(m2u2
2
1/2(m1v2
1 +m2v2
2) = 1/2(m1u2
1 +m2u2
2)
m1(v1 +u1)(v1 −u1) = m2(u2 +v2)(u2 −v2) −−−− > (2)
Dividing equation (2) by equation (1)
¨¨m1 (v1 +u1)$$$$$
(v1 −u1)
¨¨m1 $$$$$
(v1 −u1)
= ¨¨m2 (v2 +u2)$$$$$
(v2 −u2)
¨¨m2 $$$$$
(v2 −u2)
v1 +u1 = u2 +v2
From the above equation
v1 = u2 +v2 −u1 −−−− > (a)

v2 = v1 +u1 −u2 −−−− > (b)
Putting the value of v2 in equation (1)
m1(v1 −u1) = m2(u2 −v2)
m1(v1 −u1) = m2u2 −(v1 +u1 −u2)
m1(v1 −u1) = m2u2 −v1 −u1 +u2
m1(v1 −u1) = m22u2 −v1 −u1
m1v1 −m1u1 = 2m2u2 −m2v1 −m2u1
m1v1 +m2v1 = m1u1 −m2u1 +2m2u2
v1(m1 +m2) = (m1 −m2)u1 −2m2u2
v1 =
(m1 −m2)u1
(m1 +m2)
+
2m2u2
(m1 +m2)
In order to obtain v2 putting the value of v1 from equation (a) in equation (1)
m1(v1 −u1) = m2(u2 −v2)
m1(u2 +v2 −u1 −u1) = m2(u2 −v2)
m1(u2 +v2 −2u1) = m2(u2 −v2)
m1u2 +m1v2 −2m1u1 = m2u2 −m2v2
m1v2 +m2v2 = 2m1u1 +m2u2 −m1u2
v2(m1 +m2) = 2m1u1 +(m2 −m1)u2
v2 =
2m1u1
(m1 +m2)
+
(m2 −m1)u2
(m1 +m2)
Table 3.1: Difference between Elastic and Inelastic Collision
S.No Perfectly elastic collisions Perfectly Inelastic collisions
1 Particles do not stick together after col-
lision
Particles stick together after collision.
2 Relative velocities of separation after
collision = relative velocities of ap-
proach before collision
Rel. vel. of separation after collision in
zero.
3 Coeff. of restitution, e = 1 Coeff. of restitution, e = 0
4 Linear momentum is conserved. Linear momentum is conserved.
5 K.E. is conserved. K.E. is NOT conserved.
3.2.9 FRICTION
The property by virtue of which the relative motion between two surfaces in contact is
opposed is known as friction.

Frictional Forces
Tangential forces developed between the two surfaces in contact, so as to oppose their
relative motion are known as frictional forces or commonly friction. It is denoted by f. SI
unit of frictional force is newton (N). Mathematically:
f = µN
Where µ is the coefficient of friction and N is normal reaction force which is equal to the
weight of the body. Coefficient of friction is dimensionless quantity.
3.2.10 Types of Frictional Forces
Frictional forces are of three types :-
1. Static frictional force
2. Kinetic frictional force
3. Rolling frictional force
Static Frictional Force
Frictional force acting between the two surfaces in contact which are relatively at rest,
so as to oppose their relative motion, when they tend to move relatively under the effect
of any external force is known as static frictional force. Static frictional force is a self
adjusting force and its value lies between its minimum value up to its maximum value.
It is denoted by fs, mathematically:
fs = µsN
Kinetic Frictional Force
Frictional force acting between the two surfaces in contact which are moving relatively,
so as to oppose their relative motion, is known as kinetic frictional force. It’s magnitude
is almost constant and is equal to µkN where µk is the coefficient of kinetic friction for
the given pair of surface and N is the normal reaction acting between the two surfaces in
contact. It is always less than maximum value of static frictional force.
Mathematically:
fk = µkN
Coefficient of kinetic friction is always less than the coefficient of static friction, i.e.,
µk < µs .
Rolling Frictional Force
Frictional force which opposes the rolling of bodies (like cylinder, sphere, ring etc.) over
any surface is called rolling frictional force. Rolling frictional force acting between any
rolling body and the surface is almost constant and is given by µrN. Where µr is coefficient
of rolling friction and N is the normal reaction between the rolling body and the surface.
Mathematically:
fr = µrN
Note:- Rolling frictional force is much smaller than maximum value of static and kinetic
frictional force.
fr << fk < fs(max) =⇒ µr < µk < µs

Points to Note:
• If a particle moves a distance at speed v1 and comes back with speed
v2, then.
vavg =
2v1v2
v1 +v2n
But average velocity would be zero : vavg = 0.
• If a particle moves in two equal intervals of time at different speeds v1
and v2 respectively, then vavg = v1+v2
2 .
• The average velocity between two points in a time interval can be
obtained from a position versus time graph by calculating the slope of
the straight line joining the co-ordinates of the two points.
• The area of speed-time graph gives distance.
• The area of velocity-time graph gives displacement.
• Speed can never be negative.
• Average velocity may or may not be equal to instantaneous velocity.
• If body moves with constant velocity, the instantaneous velocity is
equal to average velocity.
• The instantaneous speed is equal to modulus of instantaneous velocity.
• The area of velocity-time graph gives displacement.
• The area of speed-time graph gives distance.
• The slope of tangent at position-time graph at a particular instant gives
instantaneous velocity at that instant.
• The slope of velocity-time graph gives acceleration.
• The area of acceleration-time graph in a particular time interval gives
change in velocity in that time interval.
• Momentum depends on frame of reference.
• A body cannot have momentum without having energy but the body
may have energy (i.e., potential energy) without having momentum.
• The momentum of a body may be negative.
• The slope of p versus t curve gives the force.
• The area under F versus t curve gives the change in momentum.
• A meteorite burns in the atmosphere. Its momentum is transferred to
air molecules and the earth.
• The relation between momentum and kinetic energy KE :
KE =
p2
2m
Here p = momentum of the particle of the mass m.
• If light (m1) and heavy (m2) bodies have same momenta, then
KE1
KE2
=
m2
m1

• When two bodies of same mass are approaching each other with differ-
ent velocities and collide, then they simply exchange the velocities and
move in the opposite direction.
• When a heavy body moving with velocity u collides with a lighter body
at rest, then the heavier body remains moving in the same direction with
almost same velocity. The lighter body moves in the same direction
with a nearly velocity of 2u.
• When a body of mass M suspended by a string is hit by a bullet of
mass m moving with velocity v and embeds in the body, then common
velocity of the system:
v =
mv
m+M
The velocity of the bullet is:
v =
m+M
m
× 2gh
The height to which system rises is: h = v 2
2g
• Two bodies A and B having masses m1 and m2 have equal kinetic
energies. If they have velocities v1 and v2, then
v1
v2
=
m2
m1
,
p1
p2
=
m1
m2

4. MOTION IN TWO DIMENSION
4.1 PROJECTILE MOTION
A body thrown with some initial velocity and then allowed to move under the action of
gravity alone, is known as a projectile.
If we observe the path of the projectile, we ﬁnd that the projectile moves in a path,
which can be considered as a part of parabola. Such a motion is known as projectile
motion.
A few examples of projectiles are (i) a bomb thrown from an aeroplane (ii) a javelin or
a shot-put thrown by an athlete (iii) motion of a ball hit by a cricket bat etc.
The projectiles undergo a vertical motion as well as horizontal motion. The two com-
ponents of the projectile motion are (i) vertical component and (ii) horizontal component.
These two perpendicular components of motion are independent of each other.
The motion of the projectile can be discussed separately for the horizontal and vertical parts.
We take the origin at the point of projection. The instant when the particle is projected is
taken as t = 0. The plane of motion is taken as the X −Y plane. The horizontal line OX is
taken as the X −axis and the vertical line OY as the Y −axis. Vertically upward direction
is taken as the positive direction of the Y −axis.
We have vx = vcosθ; ax = 0
vy = vsinθ ; ay = −g.
4.1.1 Horizontal Motion
As ax = 0, we have
vx = vcosθ
and x = vcosθt.
The x-component of the velocity remains constant as the particle moves.

56 Chapter 4. MOTION IN TWO DIMENSION
4.1.2 Vertical Motion
The acceleration of the particle is g in the downward direction. Thus, ay = −g. The
y-component of the initial velocity is vy. In this case we can use three equations of motion.
The vertical motion is identical to the motion of a particle projected vertically upward
with speed vsinθ. The horizontal motion of the particle is identical to a particle moving
horizontally with uniform velocity vcosθ.
4.1.3 Time of Flight
Time of flight is the total time taken by the projectile from the instant of projection till it
strikes the ground.
As the projectile goes up and comes back to the same level, thus covering no vertical
distance i.e., S = h = 0. Thus the time of flight t can be find out by using 2nd equation of
motion:
S = viyt +
1
2
ayt2
Here S = h = 0, viy = vi sinθ and ay = −g,
0 = vi sinθt −
1
2
gt2
1
2
gt2
= vi sinθt
t =
2vi sinθ
g
This is the expression of time of flight of a projectile.
4.1.4 Maximum Height Reached
Consider a projectile is thrown upward with initial velocity vi making an angle θ with
horizontal. Initially, the vertical component of velocity is vi sinθ. At maximum height,
the value of vertical component of velocity becomes zero. If t is the time taken by the
projectile to attain the maximum height h, then by using 3rd equation of motion:
2ayh = v2
fy −v2
iy
Here vfy = 0, viy = vi sinθ and ay = −g,
−2gh = 0−v2
iy
h =
v2
i sin2
θ
2g
This is the expression of the height attained by the projectile during its motion.

4.2 UNIFORM CIRCULAR MOTION 57
4.1.5 Range (R)
Range of a projectile is the horizontal distance between the point of projection and the
point where the projectile hits the ground.
In projectile motion, the horizontal component of velocity remains same. Therefore the
range R of the projectile can be determine using formula:
R = vix ×t
where vix is the horizontal component of velocity and t is the time of ﬂight of projectile.
Putting the value of vix = vicosθ and t = 2vi sinθ
g into above equation we get:
R = vi cosθ ×
2vi sinθ
g
R =
v2
i
g
×2sinθ cosθ
According to trigonometric identities: 2sinθ cosθ = sin2θ
R =
v2
i
g
×sin2θ
Thus the range of projectile depends upon the velocity of projection and angle of projection.
Maximum Horizontal Range (Rmax)
It is seen from the equation that for the given velocity of projection, the horizontal range
depends on the angle of projection only. The range is maximum only if the value of sin2θ
is maximum.
Maximum value of sin2θ = 1, =⇒ 2θ = sin−1
(1)
The value of sin−1
(1) = 90o, hence,
2θ = 90o, =⇒ θ = 45o.
Therefore the range is maximum when the angle of projection is 45o.
Rmax =
v2
i
g
4.2 UNIFORM CIRCULAR MOTION
When an object moves in a circular path such that the magnitude of velocity is constant
then, the motion is called uniform circular motion.
4.2.1 ANGULAR DISPLACEMENT
The angle traveled by a body during its motion around a circular path is called its angular
displacement.
Consider a particle moves in a circular path from a point P1 to P2 in an interval of time
t. It travels an angle ∠P1OP2 = θ which is called angular displacement of the particle.
The direction of angular displacement is along the axis of rotation and is given by right
hand rule.

Radian
One radian is the angle traced by an arc of length equal to radius of circle.
Length of circular track of radius r is 2πr. Therefore numbers of radians in a circle of
radius r will be
= 2πr/r = 2π
Angle at the centre of circle in one complete rotation = 360o = 2πrad.
180o
= π or
1o
=
π
180o
rad
The length of arc S is directly proportional to angle θ subtended (measured in radians)
traced at the centre of circle by ends of the arc:
S ∝ θ
S = rθ
Where r is radius of circle.
4.2.2 ANGULAR VELOCITY
The rate of change of angular displacement is called the angular velocity of the particle.
Let θ be the angular displacement made by the particle in time t , then the angular velocity
of the particle is
ω =
θ
t
Its SI unit is rads−1 and dimensional formula is T−1. For one complete revolution, the
angle swept by the radius vector is 360o or 2π radians. If T is the time taken for one
complete revolution, known as period, then the angular velocity of the particle is:
ω =
θ
t
=
2π
T
If the particle makes f revolutions per second, then
ω = 2π
1
T
= 2π f
where f = 1
T is the frequency of revolution.
4.2.3 Average Angular Velocity
The ratio of total angular displacement of the total interval of time during circular motion
is called average angular velocity.
Let ∆θ is the angular displacement during the time interval ∆θ, the average angular velocity
during this interval is:
ωavg =
∆θ
∆t

4.2.4 Instantaneous Angular Velocity
The angular velocity of the object at any instant of time is called instantaneous angular
velocity.
If ∆θ is the angular displacement during the time interval ∆θ, then its instantaneous
angular velocity ωins is described by the relation:
ωins = lim
∆t→0
∆θ
∆t
In the limit when ∆t approaches zero, the angular displacement will be inﬁnitesimally
small. So it would be a vector quantity. Its direction will be along axis of rotation and is
given by right hand rule.
4.2.5 Angular Acceleration
The time rate of change of angular velocity is called angular acceleration. It is denoted by
α. Mathematically,
α =
ω
t
It is a vector quantity and its direction is along the axis of rotation. The SI unit of angular
acceleration is rad s−2. The dimensions are: MoLoT−2
4.2.6 Average Angular Acceleration
The ratio of the total change in angular velocity to the total interval of time is called average
angular acceleration.
Let ωi and ωf are the angular velocities at instants ti and tf , respectively. The average
angular acceleration during interval tf −ti is described as:
αavg =
ωf −ωi
tf −ti
=
∆ω
∆t
4.2.7 Instantaneous Angular Acceleration
The angular acceleration of the body at any instant of time is called instantaneous angular
acceleration.
If ∆ω is the angular velocity during the time interval t, as t approaches to zero, then
the instantaneous angular acceleration αins is described by the relation:
αins = lim
∆t→0
∆ω
∆t
4.2.8 Relation Between Angular Velocity and Linear Velocity
Consider a particle "P" in an object (in XY-plane) moving along a circular paths of radius
"r" about an axis through "O" , perpendicular to plane i.e. z-axis. Suppose the particles
moves through an angle ∆θ in time ∆t sec.
If ∆S is its distance for rotating through angle ∆θ then,

∆θ =
∆S
r
Dividing both sides by ∆t, we get
∆θ
∆t
=
∆S
r∆t
=⇒ r
∆θ
∆t
=
∆S
∆t
If time interval ∆t is very small ∆t → 0, then the angle through which the particle moves is
also very small and therefore the ratio ∆θ/∆t gives the instantaneous angular speed ωins.
lim
∆t→0
∆S
∆t
= r lim
∆t→0
∆θ
∆t
Now by deﬁnition:
v = lim
∆t→0
∆S
∆t
and ω = r lim
∆t→0
∆θ
∆t
Therefore
v = ωr
Tangential Velocity
If a particle "P" is moving in a circle of radius "r", then its linear velocity at any instant is
equal to tangential velocity which is :
vt = rω
Tangential Acceleration
Suppose an object rotating about a ﬁxed axis changes its angular velocity by ∆ω in time
∆t sec, then the change in tangential velocity ∆vt at the end of this interval will be:
∆vt = r∆ω
Change in velocity in unit time is given by:
∆vt
dt
=
r∆ω
dt
If ∆t approaches to zero then ∆vt/∆t will be instantaneous tangential acceleration and
∆ω/∆t will be instantaneous angular acceleration α:
at = rα

4.2.9 Centripetal Acceleration
When a body performs uniform circular motion its speed remains constant but velocity
continuously changes due to change of direction. Hence a body is continuously accelerated
and the acceleration experienced by the body is known as centripetal acceleration (that is
the acceleration directed towards the center). It is denoted by ac.
Consider a particle performing uniform circular motion with speed v. When the particle
changes its position from P1 to P2 its velocity changes from v1 to v2 due to change of
direction. The change in velocity from P1 to P2 is ∆v which is directed towards the center
of the circular path according to triangle law of subtraction of vectors.
From ﬁgure ∆OP1P2 and ∆ABC are similar, hence applying the condition of similarity:
BC
AB
=
P1P2
OP1
=⇒
∆v
v
=
∆S
r
∆v =
v∆S
r
Dividing both sides by ∆t, we get
∆v
∆t
=
v∆S
∆tr
But ∆v
∆t = a and ∆S
∆t = v, therefore:
ac =
v2
r
Putting v = rω,
ac = rω2
Since the change of velocity is directed towards the center of the circular path, the acceler-
ation responsible for the change in velocity is also directed towards center of circular path
and hence it is known as centripetal acceleration.
4.2.10 Centripetal Force
Force responsible for producing centripetal acceleration is known as centripetal force. Since
centripetal acceleration is directed towards the center of the circular path the centripetal
force is also directed towards the center of the circular path.
If a body is performing uniform circular motion with speed v and angular velocity ω
on a circular path of radius r, then centripetal Force is given by:
Fc =
mv2
r
= mrω2

Points to Note:
• If for the two angles of projection θ1 and θ2, the speeds are same then
ranges will be same. The condition is θ1 +θ2 = 90o.
• The weight of a body in projectile motion is zero as it is freely falling
body.
• Tangential acceleration (in circular motion) changes the magnitude of
the velocity of the particle.
• Regarding circular motion following possibilities will exist: [ar =
radial acceleration, at = tangential acceleration and a = a2
r +a2
t ].
1. If ar = 0 and at = 0, then a = 0 and motion is uniform translatory.
2. If ar = 0 and at = 0, then a = at and motion is accelerated trans-
latory.
3. If ar = 0 but at = 0, then a = ar and motion is uniform circular.
4. If ar = 0 and at = 0, then a = a2
r +a2
t and motion is non-
uniform circular.
• The maximum velocity of vehicle on a banked road is
√
rgtanθ.
• The weight that we feel is the normal force and not the actual weight.
• In the case of circular motion, centripetal force changes only the direc-
tion of velocity of the particle.
• Centrifugal force is equal and opposite to centripetal force.

5. STATICS
STATICS deals with the studies of bodies at rest or in motion under number of forces, the
equilibrium and the conditions of equilibrium.
Deﬁnition
5.0.1 Moment Arm
The perpendicular distance between the axis of rotation and the line of the action of force
is called the moment arm of the force.
5.0.2 Rigid body
A rigid body is deﬁned as that body which does not undergo any change in shape or
volume when external forces are applied on it. When forces are applied on a rigid body,
the distance between any two particles of the body will remain unchanged, however, large
the forces may be.
5.1 CENTER OF MASS
Centre of mass is an imaginary point in a body (object) where the total mass of the body
can be thought to be concentrated to make calculations easier.
Explanation
Let us consider a collection of N particles. Let the mass of the ith particle be mi and its
coordinates with reference to the chosen axes be xi, yi, zi . Write the product mi ×xi for
each of the particles and add them to get ∑
i
mixi . Similarly get ∑
i
miyi, and ∑
i
mizi. Then

64 Chapter 5. STATICS
the coordinates of the center of mass are X, Y and Z:
X =
1
M ∑
i
mixi , Y =
1
M ∑
i
miyi , Z =
1
M ∑
i
mizi
where M = ∑imi, is the total mass of the system. Locate the point with coordinates
(X,Y,Z). This point is called the centre of mass of the given collection of the particles. If
the position vector of the i th particle is ri, the centre of mass is defined to have the position
vector:
RCM =
1
M ∑imiri
5.1.1 EQUILIBRIUM
A body will be in equilibrium if the forces acting on it must be cancel the effect of each
other. In the other word we can also write that:
A body is said to be in equilibrium condition if there is no unbalance or net force acting on
it.
Static Equilibrium
When a body is at rest and all forces applied on the body cancel each other then it is said
to be in static equilibrium.
Dynamic Equilibrium
When a body is moving with uniform velocity and forces applied on the body cancel each
other then it is said to be in the dynamic equilibrium.
5.1.2 CONDITIONS OF EQUILIBRIUM
FIRST CONDITION OF EQUILIBRIUM
A body will be in first condition of equilibrium if sum of all forces along X-axis and sum
of all forces along Y-axis are are equal to zero, then the body is said to be in first condition
of equilibrium.
∑Fx = 0 and ∑Fy = 0
SECOND CONDITIONS OF EQUILIBRIUM
A body will be in second condition of equilibrium if sum of clockwise(Moment) torque
must be equal to the sum of anticlockwise torque(Moment), then the body is said to be in
second condition of equilibrium.
∑τ = 0
5.2 TORQUE
The turning effect of a force with respect to some axis, is called moment of force or torque
due to the force. Torque is measured as the product of the magnitude of the force and the
perpendicular distance of the line of action of the force from the axis of rotation. It is
denoted by Greek letter τ. Mathematically,
τ = r ×F

5.3 ANGULAR MOMENTUM 65
It is vector quantity. The magnitude of torque is give by:
τ = rF sinθ
SI unit of torque is Nm. The dimensions are : ML2T−2. The direction of torque is
perpendicular to the plane r ×F.
5.2.1 COUPLE OF FORCE
Two forces which are equal in magnitude but opposite in direction and not acting along the
same line constitute a couple.
Consider two equal and opposite forces F and −F acting oppositely along parallel lines
on two points A and B. Let r1 and r2 are their position vectors with respect to origin.
Torque due to F = r1 ×F
Torque due to -F = −r1 ×F
Total torque = τ1 +τ2 = r1 ×F −r2 ×F
Total torque = (r1 −r2)×F
But r = r1 −r2 is the displacement vector from B to A, therefore:
Total torque = r ×F
Magnitude of torque is given by: τ = rF sinθ, where θ is the angle between r and F.
rsinθ is the perpendicular distance between the line of action of the two forces. Let it is
denoted by d. Thus the magnitude of the torque of couple will be:
τ = Fd
Where d is called the moment arm of the couple.
Now the magnitude of the couple = Magnitude of any of the forces forming couple ×
moment arm of couple.
Examples
Examples of couple are
1. Forces applied to the handle of a screw press,
2. Opening or closing a water tap.
3. Turning the cap of a pen.
4. Steering a car.
5.3 ANGULAR MOMENTUM
The measure of the quantity of motion possessed by a body in rotational motion is called
ANGULAR MOMENTUM.
"The angular momentum of a body is equal to cross product of its linear momentum and
the vector distance from the axis of rotation."
If a body of mass m is moving in a circle or radius r with velocity v, the linear
momentum of body is P. The angular momentum of the body is given by:
L = r ×P
Putting the value of P, we get
L = r ×mv =⇒ L = m(r ×v)

Magnitude of angular momentum is given by:
L = mvrsinθ
Angular momentum is vector quantity and its S.I unit is Joule.second (J.s). The dimensions
of L are :[L2MT−1].
5.3.1 LAW OF CONSERVATION OF ANGULAR MOMENTUM
"When the net external torque acting on a system about a given axis is zero , the total
angular momentum of the system about that axis remains constant."
Mathematically,
If ∑τ = 0 then L = constant
Proof
According to the second law of motion net force acting on a body is equal to its rate of
change of linear momentum, i.e.,
F =
dP
dt
Taking vector product of r on both side of above expression:
r ×F = r ×
dP
dt
But r ×F is the torque τ acting on the body:
τ = r ×
dP
dt
−−−− > (1)
Angular momentum is deﬁned as:
L = r ×P

5.3 ANGULAR MOMENTUM 67
Differentiating both sides with respect to t:
dL
dt
=
d(r ×P)
dt
dL
dt
= r ×
dP
dt
+P×
dr
dt
dL
dt
= τ +P×
dr
dt
But
dr
dt
= v
dL
dt
= τ +P×v
Since P = mv
dL
dt
= τ +m(v×v)
dL
dt
= τ +m×0( v×v = 0)
dL
dt
= τ
This expression states that the torque acting on a particle is the time rate of change of its
angular momentum. If the net external torque on the particle is zero, then,
dL
dt
= 0 =⇒ L = 0
Integrating both sides:
dL = 0
L = constant
Thus the angular momentum of a particle is conserved if and only if the net external torque
acting on a particle is zero.

Points to Note:
1. The centre of mass need not to lie in the body.
2. Internal forces do not change the centre of mass.
3. When a cracker explodes in air, the centre of mass of fragments travel
along parabolic path.
4. The sum of moment of masses about its centre of mass is always zero.
5. The position of centre of mass does not depend upon the co-ordinate
system chosen.
6. Positive torque: If a body rotates about its axis in anti clockwise
direction, then the torque is taken positive .
7. Negative torque: If the body rotates in the clockwise direction, then the
torque is taken as negative .
8. The angular velocity of all points of a rigid body are same. But in the
case of non-rigid body, greater the distance of the point from the axis
of rotation, greater will be its angular displacement.
9. The angular velocity depends on the point about which rotation is
considered.
10. The sum of moment of masses about its centre of mass is always zero.
11. Moment of inertia depends upon the position of the axis of rotation.
12. If a number of torques acted on a system and the system is in rotational
equilibrium, then clockwise torque = anticlockwise torque.
13. If a body or system is in Complete equilibrium, then net force and net
torque on the body or system are zero.
14. In the case of couple, the sum of moment of all forces about any point
is the same.
(Class Review Notes for XI Physics)
By
Dr. Ram Chand, Government Girls Degree College, Jhudo
For video lectures please visit college fb page:
www.facebook.com/ggdcjhudo

6. GRAVITATION
6.1 KEPLER’S LAWS OF PLANETARY MOTION
Kepler’s ﬁrst law (law of elliptical orbit):-
A planet moves round the sun in an elliptical orbit with sun situated at one of its foci.
Kepler’s second law (law of areal velocities):-
A planet moves round the sun in such a way that its areal velocity is constant.
Kepler’s third law (law of time period):-
A planet moves round the sun in such a way that the square of its period is proportional to
the cube of semi major axis of its elliptical orbit.
T2
∝ R3
Here R is the radius of orbit.
T2
=
4π2
GM
R3
6.2 NEWTON’S LAW OF GRAVITATION
Newton proposed the theory that all objects in the universe attract each other with a force
known as gravitation. the gravitational attraction exists between all bodies. Hence, two
stones are not only attracted towards the earth, but also towards each other.

70 Chapter 6. GRAVITATION
It states that gravitational force of attraction acting between two point mass bodies of the
universe is directly proportional to the product of their masses and is inversely proportional
to the square of the distance between them.
Deﬁnition
Consider two bodies of masses m1 and m2 with their centres separated by a distance r. The
gravitational force between them is
F ∝ m1m2
F ∝ 1/r2
=⇒ F ∝
m1m2
r2
F = G
m1m2
r2
Where G = 6.67×10−11Nm2/kg2 is universal gravitational constant. In vector form,
it can be stated as:
F = −G
m1m2
r2
12
ˆr12
Minus shows that force is attractive. Unit vector ˆr12 shows that force acts along the line
joining the m1 and m2.
Gravitational constant (G)
It is equal to the force of attraction acting between two bodies each of unit mass, whose
centres are placed unit distance apart. Value of G is constant throughout the universe. It is
a scalar quantity. The dimensional formula is G = [M−1L3T−2].
Gravitational force is central force and conservative in nature. The value of G is determined
by Cavendish method in 1798. Gravitational force is always attractive in nature.
6.2.1 Gravity
It is the force of attraction exerted by earth towards its centre on a body lying on or near
the surface of earth. Gravity is the measure of weight of the body. The weight of a body of
mass m is equal to mass × acceleration due to gravity. The unit of weight of a body will
be the same as those of force.
Acceleration due to gravity (g)
It is deﬁned as the acceleration set up in a body while falling freely under the effect of
gravity alone. It is vector quantity. The value of g changes with height, depth and rotation
of earth. The value of g is zero at the centre of the earth. The value of g on the surface of
earth is 9.81m/s2. The acceleration due to gravity (g) is related with gravitational constant
(G) by the relation:
g =
GM
R2
where M = mass of earth, R = radius of earth.

6.2 NEWTON’S LAW OF GRAVITATION 71
Mass of the Earth
From the expression g = GM
R2 , the mass of the Earth can be calculated as follows:
M =
gR2
G
=
9.81×(6.38×106)2
6.67×10−11
M = 5.98×1024
kg
6.2.2 The variation of g
Variation of g with altitude (height)
Let P be a point on the surface of the Earth and Q be a point at an altitude h. Let the mass
of the Earth be M and radius of the Earth be R. Consider the Earth as a spherical shaped
body.
The acceleration due to gravity at P on the surface is:
g =
GM
R2
−−−− > (1)
Let the body be placed at Q at a height h from the surface of the Earth. The acceleration
due to gravity at Q is
gh =
GM
(R+h)2
−−−− > (2)
dividing (2) by (1):
gh
g
=
R2
(R+h)2
=⇒ gh = g(
R
R+h
)2
= g(1+
h
R
)−2
By simplifying and expanding using binomial theorem (supposing h << R):
gh = g(1−
2h
R
)
The value of acceleration due to gravity decreases with increase in height above the surface
of the Earth.
Variation of g with depth
Consider the Earth to be a homogeneous sphere with uniform density of radius R and mass
M. Let P be a point on the surface of the Earth and Q be a point at a depth d from the
surface.
The acceleration due to gravity at P on the surface is:
g =
GM
R2
−−−− > (1)
If ρ be the density, then, the mass of the Earth is:
M =
4
3
πR3
ρ
g =
4
3
πGRρ

Figure 6.1: Variation of g (left ﬁg for height and right ﬁg for depth)
The acceleration due to gravity at Q at a depth d from the surface of the Earth is:
gd =
GMd
(R−d)2
where Md is the mass of the inner sphere of the Earth of radius (R−d).
Md =
4
3
π(R−d)3
ρ
g =
4
3
πG(R−d)ρ −−−− > (2)
dividing (2) by (1),
gd
g
=
R−d
R
gd = g(1−
d
R
)
The value of acceleration due to gravity decreases with increase of depth.
• Due to rotation of earth, the value of g decreases as the speed of rotation of earth
increases. The value of acceleration due to gravity at a latitude φ is
gφ = g−Rω2
cos2
φ
At equator, φ = 0o and at the pole, φ = 90o
1. At the equator, gE = g−Rω2
2. At the pole, gpole = g

6.3 Gravitational field 73
6.3 Gravitational field
Two masses separated by a distance exert gravitational forces on one another. This is called
action at-a-distance. They interact even though they are not in contact. This interaction can
also be explained with the field concept. A particle or a body placed at a point modifies a
space around it which is called gravitational field. When another particle is brought in this
field, it experiences gravitational force of attraction.
The gravitational field is defined as the space around a mass in which it can exert gravita-
tional force on other mass.
Definition
6.3.1 Gravitational field intensity
Gravitational field intensity or strength at a point is defined as the force experienced by a
unit mass placed at that point. It is denoted by E. It is a vector quantity. Its unit is Nkg˘1.
Consider a body of mass M placed at a point Q and another body of mass m placed at
P at a distance r from Q.
The mass M develops a field E at P and this field exerts a force F = mE. The
gravitational force of attraction between the masses m and M is
F = G
Mm
R2
The gravitational field intensity at P is E = F
m:
E =
GM
r2
Gravitational field intensity is the measure of gravitational field.
Gravitational potential difference
Gravitational potential difference between two points is defined as the amount of work
done in moving unit mass from one point to another point against the gravitational force
of attraction.
Consider two points A and B separated by a distance ∆r in the gravitational field.
The work done in moving unit mass from A to B is ∆U = WA→B. Gravitational potential
difference is:
∆U = −E∆r
Here negative sign indicates that work is done against the gravitational field.
6.3.2 Gravitational Potential Energy
The gravitational potential energy of a point mass m placed in the gravitational field of a
point mass M can be found out by the work done in moving that point mass m from infinity
to the point at which gravitational potential energy is to be determined i.e.,

Gravitational potential at a point is defined as the amount of work done in moving unit
mass from the point to infinity against the gravitational field.
Definition
Mathematically,
U = mV = m(−
GM
r
) = −
GMm
r
It is a scalar quantity. Its unit is Nmkg−1.
6.3.3 Satellite
A satellite is a body which is revolving continuously in an orbit around a comparatively
much larger body. Orbital speed of satellite is the speed required to put the satellite into
given orbit around earth.
Orbital velocity
Artificial satellites are made to revolve in an orbit at a height of few hundred kilometres.
At this altitude, the friction due to air is negligible. The satellite is carried by a rocket to
the desired height and released horizontally with a high velocity, so that it remains moving
in a nearly circular orbit.
The horizontal velocity that has to be imparted to a satellite at the determined height so
that it makes a circular orbit around the planet is called orbital velocity.
Let us assume that a satellite of mass m moves around the Earth in a circular orbit of radius
r with uniform speed vo. Let the satellite be at a height h from the surface of the Earth.
Hence, r = R+h, where R is the radius of the Earth. The centripetal force required to keep
the satellite in circular orbit is:
F =
mv2
o
r
=
mv2
o
R+h
The gravitational force between the Earth and the satellite is:
F = G
Mm
r2
= G
mM
(R+h)2
For the stable orbital motion,
mv2
o
R+h
= G
mM
(R+h)2
Since the acceleration due to gravity on Earth’s surface is g = GM
R2 , therefore:
vo =
gR2
R+h
If the satellite is at a height of few hundred kilometres (say 200km), (R + h) could be
replaced by R:
Orbital velocity is vo = gR

Time period of Satellite(T)
It is the time taken by satellite to complete one revolution around the earth.
T =
circumference of the orbit
orbital velocity
T =
2πr
vo
=
2π(R+h)
vo
vo =
GM
R+h
so
T = 2π(R+h)
R+h
GM
= 2π
(R+h)3
GM
As GM = gR2, therefore:
T = 2π
(R+h)3
gR2
If the satellite orbits very close to the Earth, then h << R:
T = 2π
R
g
Escape Velocity
It is deﬁned as minimum speed of projection with which if a body is projected upwards,
then it does not return back to earth.
Mathematically, vesc =
2GM
R
= 2gR
Where M is the mass and R is the radius of the planet.
Geostationary satellite
A satellite which revolves around the earth with the same angular speed in the same
direction as is done by the earth around its axis is called geostationary or geosynchronous
satellite. The height of geostationary satellite is 36000km and its orbital velocity is
3.1kms−1.
6.3.4 Real Weight
The real weight of the object is the gravitational pull of the earth on the object.
6.3.5 Apparent Weight
The reading of weight on the scale of a spring balance is called apparent weight. Generally
the weight of the object is measured by spring balance. The force exerted by the object on
the scale is equal to weight of the object. This is not always true, so we call the reading of
the scale as apparent weight.

Apparent weight of a man inside a lift
(a) The lift possesses zero acceleration : W = mg
(b) The lift moving upward with an acceleration a:
W = mg+ma = mg+mg
W = 2mg
(c) The lift moving downward with an acceleration a:
W = mg−ma = mg−mg
W = 0
6.3.6 Weightlessness
It is a situation in which the effective weight of the body becomes zero.
6.3.7 Artificial Gravity
The weightlessness in satellite may affect the performance of astronaut in it. To overcome
this difficulty, an artificial gravity is created in the satellite. For this, the satellite is set into
rotation around its own axis.
Consider a satellite having outer radius R rotates around its own central axis with angular
speed ω, then the centripetal acceleration ac is
ac = Rω2
But ω = 2π
T where T is the period of the revolution of spaceship:
ac = R(
2π
T
)2
= 4R
π2
T2
As the frequency f = 1
T , therefore
ac = 4Rπ2
f2
=⇒ f2
=
ac
4Rπ2
=⇒ f =
1
2π
ac
R
The frequency f is increased to such an extent that ac equals to g. Therefore,
f =
1
2π
g
R
This is the expression of frequency for producing the artificial gravity in satellite equal to
that of earth.

Points to Note:
• If the earth stops spinning, then the value of g will increase slightly
( g).
• The earth has a bulge at the equator because of the spinning motion.
• Escape velocity is independent of the mass of projectile, but it depends
on the mass of planet.
• Escape velocity does not depend on angle of projection.
• If a particle of mass m is dropped from the end of tunnel along diameter
of earth, then the motion of the particle is S.H.M. having angular
frequency of
ω = g
R
• For earth, the value of escape speed is 11.2kms−1.
• For a point close to the earth’s surface , the escape speed and orbital
speed are related as vesc =
√
2vo.
• If a planet moves around sun, work done by gravitational force is zero.
So, total mechanical energy of planet remains constant.
• The total energy of a satellite in the orbit is always negative i.e., the
body is bound to the earth.
• Weightless" does not mean "no weight". "Weightless" means "free fall",
means the only force acting is gravity.
• Gravitational potential energy increases as height increases.
• Mechanical energy (PE +KE) does not change for a free falling mass
or a swinging pendulum (when ignoring air friction).
By

7. WORK, POWER AND ENERGY
7.1 WORK
The terms work and energy are quite familiar to us and we use them in various contexts.
In everyday life, the term work is used to refer to any form of activity that requires the
exertion of mental or muscular efforts.
In physics, work is said to be done by a force or against the direction of the force, when
the point of application of the force moves towards or against the direction of the force.
Definition
If no displacement takes place, no work is said to be done. Therefore for work to be done,
two essential conditions should be satisfied:
• a force must be exerted
• the force must cause a motion or displacement
If a particle is subjected to a force F and if the particle is displaced by an infinitesimal
displacement s , the work done W by the force is the scalar product given as:
W = F.s
W = |F||s|cosθ
where θ is the angle between F and s. Work is a scalar quantity.
Units:
In S.I system, the unit is Joule (J), in C.G.S, it is Erg and in F.P.S. system, the unit is ft.lb.

80 Chapter 7. WORK, POWER AND ENERGY
Conversions between Different Systems of Units
1Joule = 1N.m = 105
dyne = 107
erg
1watt = 1Joule/s = 107
erg/s
1kwh = 103
watt.hr = 3.6×106
Joule
1HP = 746watt
1MW = 106
watt
1cal = 4.2Joule
1eV = 1.6×10−19
Joule
(e = magnitude of charge on the electron in coulombs)
SIGN CONVENTION
(i) Positive work:
If force and displacement are in the same direction, work will be positive or if θ = 0 or
θ < 90o
(ii) Zero work:
If force and displacement are perpendicular to each other, work will be zero. i.e.,
cos90o = 0.
(iii) Negative work:
In force and displacement are in the opposite direction, work will be negative:
W = Fscos1800
= Fs×(−1) = −Fs.
NOTE:- A positive work can be deﬁned as the work done by a force and a negative
work as the work done against a force.
7.2 WORK DONE BY VARIABLE FORCE
Force varying with displacement
In this condition we consider the force to be constant for any elementary displacement
and work done in that elementary displacement is evaluated. Total work is obtained by
integrating the elementary work from initial to ﬁnal limits:
dW = F.ds
W =
s2
s1
F.s
Force varying with time
In this condition we consider the force to be constant for any elementary displacement and
work done in that elementary displacement is evaluated:
dW = F.ds
Multiplying and dividing by dt:
dW =
F.ds.dt
dt
dW = F.vdt( v = ds/dt)

7.3 ENERGY 81
Total work is obtained by integrating the elementary work from initial to final limits.
W =
t2
t1
F.vdt
7.3 ENERGY
Energy can be defined as the capacity to do work.
Definition
Energy can manifest itself in many forms like mechanical energy, thermal energy, electric
energy, chemical energy, light energy, nuclear energy, etc.
Mechanical Energy
The energy possessed by a body due to its position or due to its motion is called mechanical
energy.
Definition
The mechanical energy of a body consists of potential energy and kinetic energy.
7.3.1 Potential energy
The potential energy of a body is the energy stored in the body by virtue of its position or
the state of strain.
For example: water stored in a reservoir, a wound spring, compressed air, stretched rubber
chord, etc, possess potential energy. Potential energy is given by the amount of work done
by the force acting on the body, when the body moves from its given position to some
other position.
Expression for the potential energy
Let us consider a body of mass m, which is at rest at a height h above the ground. The
work done in raising the body from the ground to the height h is stored in the body as its
potential energy and when the body falls to the ground, the same amount of work can be
got back from it. Now, in order to lift the body vertically up, a force mg equal to the weight
of the body should be applied.
When the body is taken vertically up through a height h, then work done is:
W = Force×Displacement
W = mg×h
This work done is stored as potential energy in the body
P.E = mgh

7.3.2 Kinetic Energy
The kinetic energy of a body is the energy possessed by the body by virtue of its motion.
It is measured by the amount of work that the body can perform against the impressed
forces before it comes to rest. A falling body, a bullet fired from a rifle, a swinging
pendulum, etc. possess kinetic energy. A body is capable of doing work if it moves, but in
the process of doing work its velocity gradually decreases. The amount of work that can
be done depends both on the magnitude of the velocity and the mass of the body.
Expression for Kinetic energy
Consider a body of mass m starts moving from rest. After a time interval t its velocity
becomes v. If initial velocity of the body is vi = 0, final velocity vf = v and the displacement
of body is d. Then using equation of motion:
2aS = V2
f −V2
i
Putting the above mentioned values
2ad = v2
−0 =⇒ a =
v2
2d
Now force is given by
F = ma
Putting the value of acceleration
F = m(v2
/2d)
As we know that
Work done = Fd
Putting the value of F
Work done = (
v2
2d
)(d)
Work done =
mv2
2
=⇒ W = 1/2 ×mv2
Since the work done of motion is called “Kinetic Energy”:
KE =
1
2
mv2
7.3.3 Principle of work and energy (work – energy theorem)
The work done by a force acting on the body during its displacement is equal to the change
in the kinetic energy of the body during that displacement.
Definition

7.3 ENERGY 83
Consider a body of mass m is moving with velocity vi. A force F acting through a distance
d increases the velocity to vf , then from the 3rd equation of motion:
2ad = v2
f −v2
i
d =
v2
f −v2
i
2a
−−−− > (1)
From the second law of motion:
F = ma−−−− > (2)
Multiplying equation (1) and (2), we have:
Fd =
1
2
m(v2
f −v2
i )
Fd =
1
2
mv2
f −
1
2
mv2
i −−−− > (3)
Where the left hand side of the above equation gives the work done on the body and
the right hand side gives the change in kinetic energy of the body. This is the mathematical
form of work energy principle. It can also be written as:
work-done = ∆KE
7.3.4 Conservative forces and non-conservative forces
Conservative forces
If the work done by a force in moving a body between two positions is independent of the
path followed by the body, then such a force is called as a conservative force.
Examples : force due to gravity, spring force and elastic force. The work done by the
conservative forces depends only upon the initial and ﬁnal position of the body.
The work done by a conservative force around a closed path is zero.
Non-Conservative forces
Non-conservative force is the force, which can perform some resultant work along an
arbitrary closed path of its point of application. The work done by the non-conservative
force depends upon the path of the displacement of the body. For example: frictional force,
viscous force, etc.
7.3.5 Law of conservation of energy
The law states that, if a body or system of bodies is in motion under a conservative system
of forces, the sum of its kinetic energy and potential energy is constant. OR
Energy can neither be created nor it is destroyed, however energy can be converted from
one form energy to any other form of energy
Deﬁnition

Explanation
From the principle of work and energy:
Work done = change in the kinetic energy
W1→2 = KE2 −KE1 −−−− > (1)
If a body moves under the action of a conservative force, work done is stored as potential
energy:
W1→2 = −(PE2 −PE1)−−−− > (2)
Work done is equal to negative change of potential energy. Combining the equation (1)
and (2):
KE2 −KE1 = −(PE2 −PE1)
PE1 +KE1 = PE2 +KE2
which means that the sum of the potential energy and kinetic energy of a system of particles
remains constant during the motion under the action of the conservative forces.
7.3.6 Power
It is deﬁned as the rate at which work is done:
Power =
Work done
time
=⇒ P =
W
t
Power is scalar quantity and its SI unit is watt (W) and dimensional formula is ML2T˘3.
Power is said to be one watt, when one joule of work is said to be done in one second.
If ∆W is the work done during an interval of time t then:
P =
∆W
∆t
But W = (F cosθ)∆s, where θ is the angle between the direction of the force and
displacement. F cosθ is component of the force in the direction of the small displacement
∆s. Therefore:
P =
(F cosθ)∆s
∆t
= (F cosθ)
∆s
∆t
= (F cosθ)v( v =
∆s
∆t
)
Power = P = (F cosθ)v
If F and v are in the same direction, then power = Fvcosθ = Fv = Force×velocity. It is
also represented by the dot product of F and v:
P = F.v

7.3 ENERGY 85
Points to Note:
• Work depends upon the frame of reference.
• Work is used to convert energy from one form to another form.
• Work done by conservative force doesn’t depend upon path followed
by the object.
• Work done by constant force doesn’t depend upon path.
• Two bodies of mass m1 (heavy) and mass m2 (light) are moving with
same kinetic energy. If they are stopped by the same retarding force,
then
1. The bodies cover the same distance before coming to rest.
2. The time taken to come to rest is lesser for m2 and it has less
momentum i.e., t = P/F
3. The time taken to come to rest is more for m1 as it has greater
momentum.
• When a light and a heavy body have same kinetic energy, the heavy
body has greater momentum according to p =
√
2mKE.
• A body cannot have momentum without kinetic energy.
• Mechanical energy of a particle, object or system is deﬁned as the sum
KE and PE.
• Kinetic energy changes only if velocity changes.
• A body can have mechanical energy without having either kinetic or
potential energy.
• Mechanical energy of a body or a system can be negative and negative
mechanical energy implies that potential energy is negative and in
magnitude it is more than KE. Such a state is called bound state.
• The concept of potential energy exists only in the case of conservative
forces.
• If a body moves along a rough horizontal surface, with a velocity v,
then the power required is P = µmgv.
• If a block is pulled along the smooth inclined plane with constant
velocity v, the power spent is P = (mgsinθ)v.

8. WAVE MOTION AND SOUND
8.1 TYPES OF VIBRATORY MOTION
1. Periodic Motion
When a body or a moving particle repeats its motion along a definite path after regular
intervals of time, its motion is said to be Periodic Motion and interval of time is called
time or harmonic motion period (T). The path of periodic motion may be linear, circular,
elliptical or any other curve.
2. Oscillatory motion
To and Fro type of motion is called an Oscillatory Motion. It need not be periodic and
need not have fixed extreme positions. The force acting in oscillatory motion (directed
towards equilibrium point) is called restoring force.
3. Simple Harmonic Motion
Simple harmonic motion is the motion in which the restoring force is proportional to
displacement from the mean position and opposes its increase.
8.1.1 Simple harmonic motion (SHM)
A particle is said to move in SHM, if its acceleration is proportional to the displacement
and is always directed towards the mean position.
Explanation
Consider a particle P executing SHM along a straight line between A and B about the mean
position O. The acceleration of the particle is always directed towards a fixed point on
the line and its magnitude is proportional to the displacement of the particle from this point.
a ∝ x By definition a = −ω2
x

88 Chapter 8. WAVE MOTION AND SOUND
where ω is a constant known as angular frequency of the simple harmonic motion. The
negative sign indicates that the acceleration is opposite to the direction of displacement.
If m is the mass of the particle, restoring force that tends to bring back the particle to the
mean position is given by Hooke’s Law
F = −mω2
x =⇒ F = −kx
The constant k = mω2, is called force constant or spring constant. Its unit is Nm−1. The
restoring force is directed towards the mean position. From Newton’s 2nd Law of motion
F = ma, so
ma = −kx =⇒ a = −(k/m)x
where (k/m) is constant, so a ∝ −x. Thus, Simple harmonic motion is defined as oscillatory
motion about a fixed point in which the restoring force is always proportional to the
displacement and directed always towards that fixed point.
Condition for S.H.M
The conditions for simple Harmonic Motion are given below:
• Some resisting force must act upon the body.
• Acceleration must be directly proportional to the displacement.
• Acceleration should be directed towards mean position.
• System should be elastic.
• Motion under the influence of the type of force describe by the Hooke’s Law:
F = −kx
Examples
Following are the examples of S.H.M:
• Body attached to a spring horizontally on an ideal smooth surface.
• Motion of a simple and compound pendulum.
• Motion of a swing.
• Motion of the projection of a body in a circle with uniform circular motion.
8.1.2 Important terms in simple harmonic motion
Hooke’s Law
Springs extend in proportion to load, as long as they are under their proportional limit.
Limit of proportionality
Point at which load and extension are no longer proportional.
Elastic limit
Point at which the spring will not return to its original shape after being stretched.
Displacement (x)
It is the distance of a vibrating body at any instant from the equilibrium position. It is a
vector quantity. SI unit of displacement (x) is meter (m).

8.1 TYPES OF VIBRATORY MOTION 89
Amplitude (A)
The maximum distance of the body on either side of its equilibrium position is known as
amplitude. It is scalar quantity. SI unit of amplitude is meter (m).
Time Period (T)
The time required to complete vibration is known as time period. The SI unit of time
period is second (s).
Frequency
It is the number of vibrations executed by an oscillating body in one second. It is denoted
by f:
f = 1/T
SI unit of f is s−1.
Energy
E = KE +PE E = 1/2KA2
= 1/2kx2
+1/2mv2
Period of Mass Oscillating on a Spring
T = 2π
m
k
Wave Length
The distance between two consecutive crests and troughs is called wavelength. It is denoted
by Greek letter λ. SI unit of wave length is meter (m).
Velocity of wave
It is the distance λ travelled by the wave during the time (T), a particle completes one
vibration.
velocity of wave = (frequency) (wavelength)
v = fλ
Phase
The phase of a particle vibrating in SHM is the state of the particle as regards to its direction
of motion and position at any instant of time. In the equation y = Ao sin(ωt +φ) the term
(ωt +φ) is known as the phase of the vibrating particle.
Phase difference
If two vibrating particles executing SHM with same time period, cross their respective
mean positions at the same time in the same direction, they are said to be in phase.
If the two vibrating particles cross their respective mean position at the same time but
in the opposite direction, they are said to be out of phase (i.e they have a phase difference
of π).

8.2 Linear simple harmonic oscillator
The block-spring system is a linear simple harmonic oscillator. All oscillating systems like
diving board, violin string have some element of springiness, k (spring constant) and some
element of inertia, m.
8.2.1 Horizontal oscillations of spring
Consider a mass (m) attached to an end of a spiral spring (which obeys Hooke’s law)
whose other end is ﬁxed to a support as shown in ﬁgure. The body is placed on a smooth
horizontal surface. Let the body be displaced through a distance x towards right and
released. It will oscillate about its mean position. The restoring force acts in the opposite
direction and is proportional to the displacement.
Figure 8.1: Mass-spring system
Restoring force: F = −kx.
From Newton’s second law, we know that F = ma:
ma = −kx =⇒ a = −
k
m
x
Comparing with the equation of SHM a = −ω2x, we get
ω2
=
k
m
=⇒ ω =
k
m
But: T =
2π
ω
Time Period: T = 2π
m
k
Frequency: f =
1
2π
k
m
8.2.2 Vertical oscillations of a spring
When a mass m is attached to a light, elastic spiral spring suspended vertically from a rigid
support, the spring is extended by a length l such that the upward force F exerted by the
spring is equal to the weight mg.
The restoring force: F = kl and kl = mg −−−− > (1)

8.2 Linear simple harmonic oscillator 91
where k is spring constant. If we further extend the given spring by a small distance by
applying a small force by our ﬁnger, the spring oscillates up and down about its mean
position.
The resultant force is proportional to the displacement of the body from its equilibrium
position and the motion is simple harmonic. As the force acts in the opposite direction to
that of displacement, the restoring force is −ky and the motion is SHM.
F = −ky, and also, ma = −ky =⇒ a = −
k
m
y a = −ω2
y (expression for SHM)
Comparing the above equations, ω = k
m
But: T =
2π
ω
= 2π
m
k
From equation (1): mg = kl
m
k
=
l
g
Therefore time period: T =
l
g
Frequency: f =
1
2π
g
l
8.2.3 Relationship between Circular Motion and Simple Harmonic Motion
Consider a point P moves in a circle of radius x0, with uniform angular frequency ω = 2πT.
It can be visualized that when the point P moves along the circle of radius x0, its projection
(point N) execute simple harmonic motion on the diameter DE of the circle.
Thus the expression of displacement, velocity and acceleration for the object executing
SHM can be derived using the analogy between the uniform circular motion of point P
and SHM of point N on the diameter of the circle.
Displacement
It is the distance of projection of point N from the mean position O at any instant. According
to geometry:
∠O1OP = ∠NPO = θ
If x0 is the amplitude and x is the displacement of point N at any instant, then from triangle
OPN, we have
sinθ =
ON
OP
=
x
x0
x = x0 sinθ −−−− > (1)
This is the expression of instantaneous displacement for the object executing SHM.

Velocity
If the point P is moving in a circle of radius x0 with uniform angular velocity ω then the
tangential velocity of point P will be:
vp = x0ω
We want to ﬁnd out the expression of velocity for point N, which is executing SHM.
The velocity of N is actually the component of velocity vp in the direction parallel to the
diameter DE. Thus we can write the velocity v of point N as:
v = vp sin(900
−θ) = vp cosθ
v = x0ω cosθ −−−− > (2)
As from equation (1), we have: sinθ = x/x0, so
cosθ = 1−sin2
θ = 1−
x2
x2
0
=
x2
0 −x2
x2
0
=
x2
0 −x2
x0
Putting this value in equation (2), we get:
v = x0ω(
x2
0 −x2
x0
) = ω x2
0 −x2
This is the expression of velocity of the object executing simple harmonic motion.
Acceleration
When the point P moves in a circle of radius x0, then it will have an acceleration ap = x0ω2
that will be directed towards the center of the circle. We want to ﬁnd out the expression of
acceleration of point N that is executing SHM at the diameter of the circle. The acceleration
a of point N is the vertical component of acceleration ap along the diameter DE is:
a = ap sinθ = x0ω2
sinθ −−−− > (3)
As from equation (1), we have: sinθ = x/x0, therefore the equation (3) will become:
a = x0ω2
(
x
x0
)
a = ω2
x
Comparing the case of displacement and acceleration, it can be seen that the direction of
displacement and acceleration are opposite to each other. Considering the direction of x as
reference, the acceleration will be represented by:
a = −ω2
x
This expression shows that acceleration of SHM is proportional to displacement is directed
towards the mean position.

8.2 Linear simple harmonic oscillator 93
8.2.4 Simple Pendulum
Simple Pendulum consists of a heavy mass particle suspended by a light, ﬂexible and
in-extensible string. If mass is given small displacement, it will oscillate back and forth
around the mean position and execute SHM.
In order to prove this fact consider a simple pendulum having a bob of mass m and the
length of pendulum is l. Assuming that the mass of the string of pendulum is negligible.
When the pendulum is at rest at position A, the only force acting is its weight and tension
in the string. When it is displaced from its mean position to another new position say B
and released, it vibrates to and fro around its mean position.
Suppose that at this instant the bob is at point B as shown below:
Figure 8.2: Simple Pendulum
1. Weight of the bob (W) acting vertically downward.
2. Tension in the string (T) acting along the string.
The weight of the bob can be resolved into two rectangular components:
W cosθ along the string and W sinθ perpendicular to string.
Since there is no motion along the string, therefore, the component W cosθ must balance
the tension (T). This shows that only W sinθ is the net force which is responsible for the
acceleration in the bob of pendulum.
According to Newton’s second law of motion W sinθ will be equal to ma, i.e.,
W sinθ = ma
Since W sinθ is towards the mean position, therefore, it must have a negative sign:
ma = −W sinθ
ma = −mgsinθ( W = mg)
a = −gsinθ
In our assumption θ is very small because displacement is small, in this condition we can
take sinθ θ. Hence,
a = −gθ −−−− > (1)

If x be the linear displacement of the bob from its mean position, then from ﬁgure, the
length of arc AB is nearly equal to x. From elementary geometry we know that:
S = rθ =⇒ x = lθ OR θ =
x
l
Where S = x and r = l. Putting the value of θ in equation (1), we get:
a = −g
x
l
For a given pendulum g and l are constants, so
a = −(constant)x =⇒ a ∝ −x
As the acceleration of the bob of simple pendulum is directly proportional to displacement
and is directed towards the mean position, therefore the motion of the bob is simple
harmonic when it is given a small displacement.
Time period of Simple Pendulum
Time required to complete one vibration is called time period.
When the bob of the pendulum completes one vibration it travels 360o or 2π i.e. θ = 2π.
Using the relation:
T =
θ
ω
=
2π
ω
We know that a = −ω2x, and also for simple pendulum:
a = −
g
l
x
Comparing two equations, we get:
−ω2
x = −
g
l
=⇒ ω2
=
g
l
But ω =
2π
T
, so: (
2π
T
)2
=
g
l
=⇒
T2
4π2
=
l
g
T2
= 4π2 l
g
=⇒ T = 2π
l
g
This expression indicates that the time period of simple pendulum is independent of its
mass but it depends on the length of pendulum.
Second Pendulum
A seconds pendulum is a pendulum whose period is precisely two seconds; one second for
a swing in one direction and one second for the return swing, a frequency of 1/2Hz.
(a) T = 2 seconds
(b) length = 0.9925m

8.3 WAVE AND WAVE MOTION 95
8.3 WAVE AND WAVE MOTION
"A method of energy transfer without transferring mass and involving some form of
vibration is known as a WAVE."
WAVE MOTION is a form of disturbance, which travels through a medium due to periodic
motion of particles of the medium about their mean position.
Experiment
We see that if we dip a pencil into a tap of water and take it out a pronounced circular
ripple is set up on the water surface and travels towards the edges of the tub. However if
we dip the pencil and take it out many times, a number of ripples will be formed one after
the other.
Waves can also be produced on very long ropes. If one end of the rope is ﬁxed and the
other end is given sudden up and down jerk, a pulse-shaped wave is formed which travels
along the rope.
8.3.1 TYPES of WAVE
There are three types of wave:
Mechanical wave
he wave which need material medium for their propagation are mechanical wave. For eg:
the sound waves are mechanical wave and hence need a material medium for propagation.
Electromagnetic wave
The waves which do not need material medium for their propagation are called electro-
magnetic waves. For eg: Light waves are electromagnetic waves and hence do not need
material medium for propagation or propagates even in vacuum. The electromagnetic are
caused to propagate by two electromagnetic forces.
Matter wave
Also called de Broglie waves. These are associated with high speed moving mass.
According to the modes of vibration of the particles, in the medium, there are two
types of the wave.
8.3.2 Transverse Wave
"The wave in which amplitude is perpendicular to the direction of wave motion is known
as Transverse Wave."
In the transverse wave the medium particle or disturbance travels in the form of crest and
trough.
Examples
• Radio Waves
• Light Waves
• Micro Waves
• Waves in Water
• Waves in String
• Water wave

8.3.3 Longitudinal Wave
"The wave in which amplitude is parallel to wave motion is called longitudinal wave."
The longitudinal wave travels in the form compression and rarefaction.
Example
• Sound Waves
• Seismic Waves
8.3.4 PROGRESSIVE WAVE
Also called traveling wave. A wave in which the crest and trough or compression and
rarefaction travel toward is called progressive wave. In progressive wave, the crest and
trough or compression and rarefaction changes its position continuously and the velocity
of move equals to the velocity of wave.
Consider a wave travelling along positive X-axis with a velocity v. If Y be the displacement
of the particle along Y-axis then,
Y = Ao sinωt
where Ao is amplitude and ω is angular velocity of the wave. Consider a particle at p
which is at a distance x from point p. Since the particle at mean position O and p are not
in the same phase. Then displacement of the particle Y is given by,
Y = Ao sin(ωt −φ)
where φ is phase difference of the particle at O and p. Here φ = kx, then general form of
Progressive wave can be written as:
Y = Ao sin(ωt ±kx)
If the sign of f and x are opposite, wave is propagating along positive x-axis. If the sign
of f and x are same, then wave is propagating in negative x-direction. If phase of wave
ωt −kx is constant, then the shape of wave remains constant.
8.3.5 STANDING WAVE
Also called Stationary wave. When two progressive wave of the same frequency and
amplitude, travelling through a medium with the same speed but in opposite direction
superimpose on each other and they give rise to a wave called stationary wave. In stationary
wave, it does not seem to be moving and there is no net ﬂow of energy, along the wave.
When a stationary wave is formed due to the super position of the two waves of equal
frequency and amplitude travelling in opposite direction, the points of maximum and zero
amplitude are resulted alternatively. The points where amplitude is maximum are called
anti nodes (AN) and those with zero amplitude are called nodes (N).
Consider a progressive wave travelling in positive X-axis and another wave travelling in
X-axis the equation of the wave travelling along positive X-axis is given by,

RAM’S MIND MAP
Y1 = Ao sin(ωt −kx)
The equation of the wave travelling along negative X-axis is given by
Y2 = Ao sin(ωt +kx)
According to the super position principle, the displacement of resultant wave is given by,
Y = Y1 +Y2 = Ao sin(ωt −kx)+Ao sin(ωt +kx)
Y = Ao2sin(
ωt +kx+ωt −kx
2
)cos(
ωt +kx−ωt +kx
2
)
Y = 2Ao sinωt.coskx
This is the displacement of resultant wave and A = 2Ao coskx is the amplitude of resultant
wave. This implies that the amplitude of the stationary wave different at different points
i.e., amplitude become zero at some points and maximum at some other points.
Condition for maximum amplitude:
The amplitude of resultant wave is A = 2Ao coskx. For amplitude to be maximum,
coskx = 1 =⇒ kx = nπ; where n = 0,1,2,3,...
kx = 0π,1π,2π,.... But k = 2π/λ, so
x = 0,λ/2,λ,3λ/2,4λ/2,.....
Hence anti-nodes occur at the distance of x = 0,λ/2,λ,3λ/2,4λ/2,.....

Condition for minimum amplitude:
For amplitude to be maximum,
coskx = 0 =⇒ kx = (2n+1)π; where n = 0,1,2,3,...
Where k = 2π/λ, so
x = λ/4,3λ/4,5λ/4,.....
Hence nodes will be formed at the distance of x = λ/4,3λ/4,5λ/4,.....
Characteristics of stationary wave:
• The stationary waves are produced when two progressive waves of equal frequency
and amplitude travel in medium in opposite direction.
• In the stationary wave, the disturbance or energy is not transmitted from particle to
particle.
• At nods, the particles of the medium are permanently at rest.
• The particle at the anti-nodes vibrates with the maximum amplitude which is equal
to twice the amplitude of either waves.
• The period of vibration of stationary wave is equal to that of either of wave.
• The amplitude of particles on either side of an antinode gradually decreases to zero.
Difference between Standing waves and Progressive waves
Stationary Waves Progressive Waves
Stores energy Transmits energy
Have nodes & antinodes No nodes & antinodes
Amplitude increases from node to antinode Amplitude remains constant along length
of the wave
Phase change of π at node No phase change
8.3.6 Organ Pipe
Organ pipe is a pipe in which if the vibration is passed from the one end it is then reﬂected
from the other end and stationary wave is formed in the pipe. There are two types of Organ
pipe.
Closed organ pipe
The organ pipe in which one end is opened and another end is closed is called closed organ
pipe. Bottle, whistle, etc. are examples of closed organ pipe.
If the air is blown lightly at the open end of the closed organ pipe, then the air column
vibrates in the fundamental mode. There is a node at the closed end and an antinode at the
open end. If L is the length of the tube,
L =
λ1
4
or λ1 = 4L
If f1 is the fundamental frequency of the vibrations and v is the velocity of sound in air,
then
f1 =
v
λ1
=
v
4L

If air is blown strongly at the open end, frequencies higher than fundamental frequency
can be produced.
L =
3λ3
4
or λ3 =
4L
3
f3 =
v
λ3
=
3v
4L
= 3 f1( f1 =
v
4L
)
This is the ﬁrst overtone or third harmonic. Similarly:
f5 =
5v
4L
= 5 f1
This is called as second overtone or ﬁfth harmonic. Therefore the frequency of nth overtone
is (2n+1)f1 where f1 is the fundamental frequency. In a closed pipe only odd harmonics
are produced. The frequencies of harmonics are in the ratio of 1 : 3 : 5......
fn = (2n+1)f1 , where n = 1,2,3,....
Open organ pipe
The pipe in which the both of its ends are open is called open organ pipe. Flutes is the
example of open organ pipe.
When air is blown into the open organ pipe, the air column vibrates in the fundamental
mode. Antinodes are formed at the ends and a node is formed in the middle of the pipe. If
L is the length of the pipe, then
L =
λ1
2
or λ1 = 2L
The fundamental frequency f1 is
f1 =
v
2L
In the next mode of vibration additional nodes and antinodes are formed
f2 = 2
v
2L
= 2 f1
f3 = 3
v
2L
= 3 f1
f4 = 4
v
2L
= 4 f1
.
.
.
fn = nf1
where fn is the nth harmonic and n = 1,2,3,..... Hence, it is proved that all harmonics are
present in an open organ pipe.

8.3.7 Ripple Tank
RIPPLE TANK is an apparatus which is used to study the features or characteristics of
waves mechanics.
A ripple tank consists of a rectangular tray containing water. It is provided with a
transparent glass sheet at the bottom. A screen is placed well below the tray to observe the
characteristics of waves generated in water. A lamp is placed above the tray.
Working
When an observer dips a rod or his finger into the water of ripple tank, waves are generated.
There is also a mechanical way to generate pulses in water i.e. electric motor. The lamp
enlightens the waves which are focused on the bottom screen. The wave crests act as
converging lenses and tend to focus the light from the lamp. The wave troughs act as
diverging lenses and tend to spread it. This results that crests appear as bright bends and
troughs as dark bends on the screen.
• PRODUCTION OF STRAIGHT RIPPLES: Straight pulses are produced by dipping
a finger or a straight rod periodically in water.
• PRODUCTION OF CIRCULAR RIPPLES: Circular pulses are produced by dipping
the pointed end of a rod periodically in water.
If straight pulses are generated and a piece of paper is thrown on the surface of water, it is
found that the paper simply moves up and down as each of the waves passes across it. By
means of a stop watch time period of the rod and paper is measured. The two time periods
are found to be equal. This shows that the particles of medium execute simple harmonic
motion with the same time period as that of the body generating pulses.
8.3.8 Sonometer
A Sonometer is a device for demonstrating the relationship between the frequency of the
sound produced by a plucked string, and the tension, length and mass per unit length of
the string. These relationships are usually called Mersenne’s laws after Marin Mersenne
(1588-1648), who investigated and codified them.
The sonometer consists of a hollow sounding box about a metre long. One end of a
thin metallic wire of uniform cross-section is fixed to a hook and the other end is passed
over a pulley and attached to a weight hanger. The wire is stretched over two knife edges
P and Q by adding sufficient weights on the hanger. The distance between the two knife
edges can be adjusted to change the vibrating length of the wire.
A transverse stationary wave is set up in the wire. Since the ends are fixed, nodes are
formed at P and Q and antinode is formed in the middle.
The length of the vibrating segment is l = λ/2.
∴ λ = 2l. If f is the frequency of vibrating segment, then
f =
v
λ
=
v
2l
We know that v = T
µ , where T is the tension and µ is the mass per unit length of the
wire.
f =
1
2l
T
µ

This relationship shows that for small amplitude vibration, the frequency is proportional
to:
1. the square root of the tension of the string,
2. the reciprocal of the square root of the linear density (mass per unit length) of the
string,
3. the reciprocal of the length of the string.
8.3.9 Sound
"A vibration transmitted by air or other medium in the form of alternate compressions and
rarefactions of the medium is known as Sound."
Sound wave is longitudinal mechanical wave producing sensation of hearing on the ear.
On the basis of what range of frequency of longitudinal mechanical wave can be detected
by our ear. The longitudinal mechanical waves are divided into 3 types
1. Audible wave:
Range of frequency from 20Hz to 20kHz which can produce sensation of hearing in our
ears is called audible wave, which is in fact a sound wave. The sound wave can be produced
by vibration of tuning forks, air column human vocal cord etc.
2. Infrasonic wave:
The longitudinal wave whose frequency lies below lowest audible range are called infra-
sonic wave. The frequency of earthquakes lies in this range.
3. Ultrasonic wave:
Range lies above the 20kHz is ultra-sonic wave. The frequency given by vibration quartz
crystal is ultrasonic wave.
Production of Sound
Sound is produced by a vibrating body like a drum, bell, etc, when a body vibrates. due
to the to and fro motion of the drum, compressions and rarefactions are produced and
transmitted or propagated in air.
Properties of Sound
• Longitudinal in nature.
• It requires a material medium for its propagation.
• Sound waves can be reﬂected.
• Sound waves suffer refraction.
• Sound waves show the phenomenon of interference
• Sound waves shows diffraction
• Sound propagates with a velocity much smaller than that of light.
• Sound gets absorbed in the medium through which it passes.
8.3.10 Characteristics of musical sound
Musical Sound: A musical sound consists of a quick succession of regular and periodic
rarefactions and compressions without any sudden change in its amplitude.

Pitch
The pitch is the characteristics of a musical sound which depends upon the frequency. The
sound with low frequency is low pitch able sound and the sound with high frequency is
high pitch able sound.
Loudness
The loudness of musical sound is related to the intensity of the sound the higher is the
intensity, the higher will be the loudness.
Quality OR Timber
It measure the complexity of sound. Quality of sound depends upon the number and
intensity of harmonics present in the sound. A pure sound produces comparatively less
pleasing effect on ears then sound consisting of a number of harmonics. Usually a sounding
body produce a complex sound of frequency. The fo, 2 fo, 3fo, etc, where fo is called
fundamental frequency. The fo, 2 fo, 3fo etc are called first, 2nd, 3rd harmonics. In the
voice of different peoples different harmonics are present. Due to the different harmonics
present in the voices, we characteristics of sound is called Quality or Timber.
8.3.11 Intensity of sound
The intensity of sound at a point is defined as the amount of sound energy crossing the
point per unit area per second. Then the unit of intensity I of the sound is given by J/m2s
or Wm−2.
Sound intensity level β
Sound intensity levels are quoted in decibels (dB) much more often than sound intensities
in watts per meter squared. How our ears perceive sound can be more accurately described
by the logarithm of the intensity rather than directly to the intensity. The sound intensity
level β in decibels of a sound having an intensity I in watts per meter squared is defined to
be:
β(dB) = 10log10(
I
Io
)
where Io = 10−12W/m2 is a reference intensity. In particular, Io is the lowest or threshold
intensity of sound a person with normal hearing can perceive at a frequency of 1000Hz.
Sound intensity level is not the same as intensity. The units of decibels (dB) are used to
indicate this ratio is multiplied by 10 in its definition. The bel, upon which the decibel is
based, is named for Alexander Graham Bell, the inventor of the telephone.
The decibel level of a sound having the threshold intensity of 10−12W/m2 is β = 0dB,
because log101 = 0. That is, the threshold of hearing is 0 decibels.
Threshold of hearing:
The threshold of hearing is the lowest intensity of the sound that can be detected by our
ear within the range of audibility. The sensitivity of ears caries with the frequency i.e.
the sensitivity of ears is different range of frequencies. Also the threshold of hearing at a
frequency may very form ear to ear. Hence, the threshold of hearing has been defined for
a normal ear at a frequency of 1000Hz. The threshold of hearing is taken as 10−12 Watt
m−2 at the frequency of 1000Hz.

8.4 Velocity of sound 103
8.3.12 BEATS
When two sound waves of same amplitude and nearly equal frequency move in the same
direction then these two waves superimpose to each other giving rise to alternating hearing
of the sound. This phenomenon is called Beat.
The no of hearing produce in one second is beat frequency.
Consider two waves having frequency f1 and f2 then the displacement equation of these
two waves are given by:
y1 = Ao cos2π f1t and y2 = Ao cos2π f2t. If the two waves are sounded together, they will
interfere and the resulting displacement y according to the principle of super position is
given by,
y = y1 +y2 = Ao cos2π f1t +Ao cos2π f2t
Using a trigonometric identity, it can be shown that
y = 2Ao cos(2π fBt)cos(2π favgt)
where: fB = |f1 − f2|
is the beat frequency, and favg is the average of f1 and f2. These results mean that the
resultant wave has twice the amplitude and the average frequency of the two superimposed
waves, but it also fluctuates in overall amplitude at the beat frequency fB. The first cosine
term in the expression effectively causes the amplitude to go up and down. The second
cosine term is the wave with frequency favg. This result is valid for all types of waves.
However, if it is a sound wave, providing the two frequencies are similar, then what we
hear is an average frequency that gets louder and softer (or warbles) at the beat frequency.
Some important points regarding beats:
• The beats frequency = number of beats per second=|f1 − f2|.
• In the case of beats, the intensity at a point varies periodically.
• If beats frequency is fraction then round off is not allowed, e.g., if beats frequency is
5.2 Hz, then in five second 26 beats (not 25) are heard.
• Due to waxing or wanning to a tuning fork, frequency decreases.
• Due to filing a tuning fork, frequency increases.
• Human ear can hear fB = 7Hz.
8.4 Velocity of sound
• Velocity of sound in a medium is given by
v =
E
ρ
where E is the modulus of elasticity and ρ is the density of the medium.
• Velocity of sound is maximum in solids and minimum in gases since, solids are
more elastic.
• In a solid, elasticity E is replaced by Young’s modulus Y so that
v =
Y
ρ

• In a ﬂuid (liquid or gas) E is replaced by Bulk’s modulus B so that
v =
B
ρ
• In a gas,
v =
γRT
M
=
γP
ρ
Here, γ =
Cp
cv
= adiabatic constant
P = Normal pressure and ρ = density of gas.
8.4.1 Newton’s formula
On the basis of theoretical considerations, Newton proved that the velocity of sound in any
medium is given by:
v =
E
ρ
−−−− > (1)
Sound waves travels in gases in the form of compressions and rarefactions. Newton
assumed that when a sound wave travels through air, the temperature of the air during
compression and rarefaction remains constant. Such a process is called an isothermal
process.
Let V be the volume of a gas at pressure P, then Boyle’s Law for the isothermal process
is:
PV = constant −−−−− > (2)
If pressure increases from P to (P+∆P) at constant temperature, then its volume decreases
from V to (V −∆V). Now, according to Boyle’s Law:
(P+∆P)(V −∆V) = constant −−−−− > (3)
Comparing equations (2) and (3),
PV = (P+∆P)(V −∆V)
PV = PV −P∆V +V∆P−∆P∆V
0 = −P∆V +V∆P−∆P∆V
The product ∆P∆V is very small and can be neglected. So above equation becomes:
0 = −P∆V +V∆P
P∆V = V∆P
P =
V∆P
∆V
=⇒ P =
∆P
∆V/V
=
Stress
Volumetric Strain

8.4 Velocity of sound 105
P = E
Put this value in equation (1):
v =
E
ρ
=
P
ρ
Where atmospheric pressure P = 1.013×105N/m2 and density of air is 1.293kg/m3. The
speed of sound could be found as:
v =
1.013×105
1.293
= 280m/s
As, the experimental value of speed of sound in air is 332m/s and theoretical value comes
out to be 280m/s. This shows that Newton’s formula was not correct.
8.4.2 Laplace’s correction
The formula given by Newton is modified by Laplace assuming that propagation of sound
in air is an adiabatic process. In sound waves, the compressions and rarefactions occurs
so rapidly that heat produced in compressed regions does not have time to flow to the
neighboring rarefactions. This means that during compression, temperature rises and
during rarefaction, temperature falls. Hence, temperature of the air does not remain
constant.
Now, Boyle’s law for adiabatic process is:
PVγ
= constant −−−−− > (1)
Where γ = molar specific heat of gas at constant pressure
molar specific heat of gas at constant volume
γ =
Cp
Cv
If pressure of a given mass of a gas is changed from P to (P+∆P) and volume changes
from V to (V −∆V), then
(P+∆P)(V −∆V)γ
= constant −−−− > (2)
Comparing equations (1) and (2), we get:
PVγ
= (P+∆P)(V −∆V)γ
PVγ
= (P+∆P)[V(1−
∆V
V
)]γ
PVγ
= (P+∆P)Vγ
(1−
vV
V
)γ
P = (P+∆P)(1−
∆V
V
)γ
Applying Binomial Theorem:
(1+
∆V
V
)γ
= 1+γ(−
∆V
V
)+neglecting higher power terms
(1+
∆V
V
)γ
= 1−γ
∆V
V

Put this value in equation (3), we get:
P = (P+∆P)(1−γ
∆V
V
)
P = P−
γP∆V
V
+∆P−
γ∆P∆V
V
0 = −
γP∆V
V
+∆P−
γ∆P∆V
V
As ∆V << V, so (γ∆V∆P)/V can be neglected.
0 = −
γP∆V
V
+∆P
γP∆V
V
= ∆P
γP =
∆P
∆V/V
=
Stress
Volumetric Strain
γP = E
Hence, Laplace formula for speed of sound in a gas is:
v =
E
ρ
=
γP
ρ
Putting the value of atmospheric pressure P = 1.013×105N/m2, density ρ = 1.293kg/m3
and γ = 1.4 for air:
v =
1.4×1.013×105
1.293
= 333m/s
This value of speed of sound is very close to the experimental value. Hence Laplace’s
formula for speed of sound is correct.
If M is the mass and V is the volume of the air then: ρ = M/V, then Laplace’s equation
can be written as:
v =
γP
ρ
=
γPV
M
But PV = nRT, therefore:
v =
γnRT
M
In the above equation γ, n, R, and M all are constants:
v ∝
√
T
Thus, the velocity of sound in air is directly proportional to the square root of its absolute
temperature.

8.5 DOPPLER’S EFFECT 107
Some important points regarding velocity of sound in air or gaseous medium:
1. The speed of sound does not change due to variation of pressure.
2. Velocity of sound and temperature of the medium are related as:
v2
v1
=
T2
T1
3. Due to change of temperature by 1oC, the speed of sound is changed by 0.01m/s.
4. For small variation of temperature, vt = (vo +0.61t)m/s where, vo = speed of sound
at 0oC vt = speed of sound at toC.
5. The speed of sound increases due to increase of humidity.
6. The velocity of sound in air is measured by resonance tube.
7. The velocity of sound in gases is measured by Quinke’s tube.
8. Kundt’s tube is useful to measure the speed of sound in solid and gases.
8.5 DOPPLER’S EFFECT
The Doppler effect is an alteration in the observed frequency of a sound due to motion of
either the source or the observer. For example, if you ride a train past a stationary warning
bell, you will hear the bell’s frequency shift from high to low as you pass by.
The actual change in frequency due to relative motion of source and observer is called a
Doppler Shift. The phenomena is known as Doppler’s Effect.
Deﬁnition
The Doppler effect and Doppler shift are named for the Austrian physicist and mathe-
matician Christian Johann Doppler (1803–1853), who did experiments with both moving
sources and moving observers.
The apparent frequency due to Doppler effect for different cases can be deduced as
follows:
8.5.1 Both source and observer at rest
Suppose S and O are the positions of the source and the observer respectively. Let f be the
frequency of the sound and v be the velocity of sound. In one second, f waves produced
by the source travel a distance SO = v. The wavelength is:
λ =
v
f
8.5.2 Source is moving and observer is at rest
(1) When the source moves towards the stationary observer
If the source moves with a velocity vs towards the stationary observer, then after one
second, the source will reach S , such that SS = vs. Now f waves emitted by the source
will occupy a distance of (v−vs) only. Therefore the apparent wavelength of the sound is:
λ =
v−vs
f

The apparent frequency is:
f =
v
λ
= (
v
v−vs
)f
As f > f, the pitch of the sound appears to increase.
(ii) When the source moves away from the stationary observer
If the source moves away from the stationary observer with velocity vs, the apparent
frequency will be given by:
f = (
v
v−(−vs)
)f = (
v
v+vs
)f
As f < f, the pitch of the sound appears to decrease.
8.5.3 Source is at rest and observer in motion
(i) When the observer moves towards the stationary source
Suppose the observer is moving towards the stationary source with velocity vo. After one
second the observer will reach the point O such that OO = vo. The number of waves
crossing the observer will be f waves in the distance OA in addition to the number of
waves in the distance OO which is equal to vo/λ. Therefore, the apparent frequency of
sound is:
f = f +
vo
λ
= f +
vo
v
f
f = (
v+vo
v
)f
As f > f, the pitch of the sound appears to increase.
(ii) When the observer moves away from the stationary source
In this case velocity of observer is taken as negative:
f = (
v+(−vo)
v
)f = (
v−vo
v
)f
Asf < f, the pitch of sound appears to decrease.
Note: If the source and the observer move along the same direction, the equation for
apparent frequency is:
f = (
v−vo
v−vs
)f
General Equation
The apparent frequency f is given as:
f = (
v±vo
v vs
)f

8.5.4 Application of Doppler’s Effect
Doppler shifts and sonic booms are interesting sound phenomena that occur in all types
of waves. They can be of considerable use. For example, the Doppler shift in ultrasound
can be used to measure blood velocity, while police use the Doppler shift in radar (a
microwave) to measure car velocities. In meteorology, the Doppler shift is used to track
the motion of storm clouds; such “Doppler Radar” can give velocity and direction and
rain or snow potential of imposing weather fronts. In astronomy, we can examine the light
emitted from distant galaxies and determine their speed relative to ours.
As galaxies move away from us, their light is shifted to a lower frequency, and so to a
longer wavelength—the so-called red shift. Such information from galaxies far, far away
has allowed us to estimate the age of the universe (from the Big Bang) as about 14 billion
years.
RAM’S MIND MAP

Points to Note:
• In the case of S.H.M., total energy of the system remains constant at
every instant.
• In the case of S.H.M., particle is in stable equilibrium at the mean
position.
• At mean position, velocity is maximum and at extreme position, veloc-
ity is zero
• Acceleration is zero at mean position while maximum at extreme posi-
tion.
• If the spring is massless, then time period (T) is given by:
T = 2π
m
k
• If the spring is massive of mass ms, then
T = 2π
m+ ms
3
k
• In series combination of springs, the equivalent spring constant k can
be calculated as:
1
k
=
1
k1
+
1
k2
+
1
k3
+....
• For parallel combination of springs: k = k1 +k2 +k3 +....
• If a person sitting on an oscillating swing stands up, the time period of
the swing decreases.
• The time period of a simple pendulum having long length is:
T = 2π
lR
(l +R)g
where R is radius of the earth. If length is inﬁnite, then: T = 2π R
g
• If time period of one spring is T1 and that of second spring is T2 and
if they are connected in series, then Tseries = T2
1 +T2
2 . If they are
connected in parallel then:
Tseries =
T1T2
T2
1 +T2
2
• A surface wave is a mixture of transverse and longitudinal waves, and
a wave pulse is a single disturbance of a medium.
• Compression occur when the air particles are closer together and the
air pressure is higher than the surrounding pressure.

• Rarefaction occurs when the air particles are further apart and the air
pressure is lower than the surrounding pressure.
• A stationary wave is formed when two progressive waves of the same
frequency, amplitude and speed, travelling in opposite directions are
superposed.
• Node: region of destructive superposition where waves always meet
out of phase by π, =⇒ displacement = zero.
• Antinode: region of constructive superposition where waves meet in
phase; in this case particles vibrate with max amplitude.
• Neighboring nodes & antinodes separated by 1/2λ.
• When a string vibrates in one segment, the sound produced is called
fundamental note. The string is said to vibrate in fundamental mode.
• Harmonics are the integral multiples of the fundamental frequency.
If fo be the fundamental frequency, then nfo is the frequency of nth
harmonic.
• Overtones are the notes of frequency higher than the fundamental
frequency actually produced by the instrument.
• In the strings all harmonics are produced.
• In the open organ pipe all the harmonics are produced while in the
closed organ pipe only the odd harmonics are produced.
By

9. NATURE OF LIGHT
9.1 What is Light?
Light is form of energy and it is electromagnetic in nature. The speed of light is constant
which is denoted by c. The value of speed of light is 3 × 108m/s. Currently light is
considered to have dual nature. Following are theories of light.
9.1.1 Newton’s Corpuscular Theory of Light
This theory which was proposed by Newton is as follows:
1. Light is emitted from a luminous body in the form of tiny particles called corpuscles.
2. The corpuscles travel with the velocity of light.
3. When corpuscles strike the retina they make it sense light.
4. Medium is necessary for the propagation of light.
5. Velocity of light is greater in denser medium.
9.1.2 Wave Theory of Light
In 1676, Huygen proposed this theory. According to this theory:
1. Light propagates in space in the form of waves.
2. It can travel in space as well as in a medium.
3. Light does not travel in a straight line but in sine wave form.
4. Velocity of light is greater in rarer medium.
5. Medium is not necessary for propagation.
9.1.3 Quantum Theory of Light
Quantum Theory was proposed by Max Plank in 1901. According to this theory of Max
Plank:

114 Chapter 9. NATURE OF LIGHT
1. Light is emitted from a source discontinuously in the form of bundles of energy
called Photons or Quantum.
2. It travels in space as well as a medium.
3. Speed of light is greatest in space or vacuum.
9.1.4 Dual Nature of Light
Light has dual nature, it behaves not only as a particle (photon) but also as a wave. This is
called dual nature of light.
9.2 Wavefronts
The surface over which particles are vibrating in the same phase. The surface is normal to
rays in isotropic media.
Explanation
Consider a point source of light as S. Waves emitted from this source will propagate
outwards in all directions with speed c (c is the speed of light). After time t, they will reach
the surface of a sphere with center as S and radius ct. Every point on the surface of this
sphere will be set into vibration by the waves reaching there. As the distance of all these
points from the source is the same, so their state of vibration will be identical. In other
words we can say that all the points on the surface of the sphere will have the same phase.
Figure 9.1: Wavefront
Such a surface on which all the points have the same phase of vibration is known as
wavefronts.
Deﬁnition

9.3 Huygens Principle 115
Thus in case of a point source, the wavefront is spherical in shape. A line normal to the
wavefront including the direction of motion is called a ray of light.
With time, the wave moves farther giving rise to new wave fronts. All these wavefronts
will be concentric spheres of increasing radii. Thus the wave propagates in space by the
motion of the wavefronts is one wavelength. It can be seen that as we move away at greater
distance from the source, the wavefronts are parts of spheres of very large radii. A limited
region taken on such a wavefront can be regarded as a plane wavefront. For example, light
from the sun reaches the Earth in plane wavefronts.
9.3 Huygens Principle
The Dutch scientist Christiaan Huygens (1629–1695) developed a useful technique for
determining in detail how and where waves propagate. Huygens’s principle states that:
Every point on a wavefront is a source of wavelets that spread out in the forward direction
at the same speed as the wave itself. The new wavefront is a line tangent to all of the
wavelets.
Deﬁnition
Knowing the shape and location of a wavefront at any instant t, Huygen’s principle enables
us to determine the shape and location of the new wavefront at a later time t + t. This
principle consists of two parts:
1. Every point of a wavefront may be considered as a source of secondary wavelets
which spread out in forward direction with a speed equal to the speed of propagation
of the wave.
2. The new position of the wavefront after a certain interval of time can be found by
constructing a surface that touches all the secondary wavelets.
9.4 TERMS USED IN LIGHT
9.4.1 Coherent Sources
Coherent sources are the sources which either have no phase difference or have a constant
difference of phase between them.
9.4.2 Principle of superposition
It states that a number of waves travelling, simultaneously, in a medium behave independent
of each other and the net displacement of the particle, at any instant, is equal to the sum of
the individual displacements due to all the waves.
9.4.3 Interference
The modiﬁcation in the distribution of light energy obtained by the superposition of two or
more waves is called interference.

Conditions for interference:-
• The two sources should emit, continuously, waves of same wavelength or frequency.
• The amplitudes of the two waves should be either or nearly equal
• The two sources should be narrow.
• The sources should be close to each other.
• The two sources should be coherent one.
Condition for constructive interference
Path difference = (2n)λ/2
Phase difference = (2n)π
Condition for destructive interference
Path difference = (2n+1)λ/2
Phase difference = (2n+1)π
9.4.4 Fringe Width
It is the distance between two consecutive bright and dark fringes:
x =
λL
d
9.4.5 Maxima
A point having maximum intensity is called maxima.
x = 2n(λ/2)
A point will be a maxima if the two waves reaching there have a path difference of even
multiple of λ/2.
9.4.6 Minima
A point having minimum intensity is called a minima.
x = (2n+1)(λ/2)
A point will be a minima if the two waves reaching there have a path difference of odd
multiple of λ/2.
9.5 NEWTON’S RINGS
If monochromatic beam of light is allowed to fall normally on plano-convex lens placed
on a plane glass plate, and the film is viewed in reflected light, alternate bright and dark
concentric rings are seen around the point of contact. These rings were first discovered by
Newton, that’s why they are called NEWTON’S RINGS.

9.5 NEWTON’S RINGS 117
Explanation
When a Plano convex lens of long focal length is placed in contact on a plane glass plate, a
thin air film is enclosed between the upper surface of the glass plate and the lower surface
of the lens. The thickness of the air film is almost zero at the point of contact O and
gradually increases as one proceeds towards the periphery of the lens. Thus points where
the thickness of air film is constant, will lie on a circle with O as center.
Let us consider a system of plano-convex lens of radius of curvature R placed on flat glass
Figure 9.2: Newton’s Rings
plate it is exposed to monochromatic light of wavelength λ normally.
The incident light is partially reflected from the upper surface of air film between lens and
glass and light is partially refracted into the film which again reflects from lower surface
with phase change of 180 degree due to higher index of glass plate. Therefore the two parts
of light interfere constructively and destructively forming alternate dark and bright rings.
Now consider a ring of radius r due to thickness t of air film as shown in the figure given
below:
According to geometrical theorem, the product of intercepts of intersecting chord is equal
to the product of sections of diameter then,
DB×BE = AB×BC
But BD = BE = r, AB = t and BC = 2R−t:
r ×r = t(2R−t) =⇒ r2
= 2Rt −t2
Since "t" is very small as compared to "r", therefore, neglecting "t2"
r2
= 2Rt −−−− > (1)
In thin films, path difference for constructive interference (bright ring) is:
2nt cosθ = (m+1/2)λ

Where n is refractive index. for air the value of n = 1, therefore:
2t cosθ = (m+1/2)λ −−−− > (2)
For first bright ring m = 0, for second bright ring m = 1, for third bright ring m = 2.
Similarly, for Nth bright ring m = N −1. Putting the value of m in equation(2), we get:
2t cosθ = (N −1+1/2)λ =⇒ 2t cosθ = (N −1/2)λ
t = 1/2cosθ(N −1/2)λ −−−− > (3)
Now putting the value of t from equation (3) into equation (1), we get the radius of bright
ring as:
r2
= 2Rt =⇒ r2
= 2R×1/2cosθ(N −1/2)λ
rn =
Rλ(N −1/2)
cosθ
−−−− > Bright Ring
This is the expression for the radius of Nth bright ring where
rn = radius of Nth bright ring, N =Ring number, R =radius of curvature of lens, λ =Wave
length of light and θ is angle of reflection in air film.
For destructive interference OR dark ring, the path difference is:
2t cosθ = mλ
Putting in equation (1) and re-arranging, we get
r2
R
=
mλ
cosθ
rn =
mλR
cosθ
−−−− > Dark Ring
9.5.1 YOUNG’S DOUBLE SLIT EXPERIMENT
The first practical demonstration of optical interference was provided by THOMAS
YOUNG in 1801. His experiment gave a very strong support to the wave theory of
light.
Consider ’S’ is a slit, which receives light from a source of monochromatic light. As
’S’ is a narrow slit so it diffracts the light and it falls on slits A and B. After passing through
the two slits, interference between two waves takes place on the screen. The slits A and B
act as two coherent sources of light. Due to interference of waves alternate bright and dark
fringes are obtained on the screen.
Let the wave length of light = λ
Distance between slits A and B = d
Distance between slits and screen = L
Consider a point ’P’ on the screen where the light waves coming from slits A and B
interfere such that PC = y. The wave coming from A covers a distance AP = r1 and the

wave coming from B covers a distance BP = r2 such that PB is greater than PA.
Path difference = BP−AP = BD
S = r2 −r1 = BD
In right angled BAD
sinθ = BD/AB =⇒ sinθ = S/d
S = d sinθ −−−−−−−(1)
Since the value of d is very very small as compared to L, therefore, θ will also be very
small. In this condition we can assume that: sinθ = tanθ. Therefore, equation (1) will
become:
S = d tanθ −−−−−(2)
In right angled PEC:
tanθ = PC/EC = y/L
Putting the value of tanθ in eq. (2), w get
S = dy/L
Or
y =
SL
d
−−−−−(3)
Figure 9.3: Young’s Double Slit Experiment
FOR BRIGHT FRINGE
For bright fringe S = mλ. Therefore, the position of bright fringe is:
y =
mλL
d

FOR DARK FRINGE
For destructive interference (dark fringe), path difference between two waves is (m+1/2)λ.
Therefore, the position of dark fringe is:
y = (m+
1
2
)
λL
d
FRINGE SPACING
The distance between any two consecutive bright fringes or two consecutive dark fringes
is called fringe spacing. Fringe spacing or thickness of a dark fringe or a bright fringe is
equal. It is denoted by x.
Consider bright fringe: y =
mλL
d
For bright fringe m=1: y1 =
1×λL
d
For next order bright fringe m=2: y2 =
2×λL
d
fringe spacing = y2 −y1
x =
2×λL
d
−
1×λL
d
x =
λL
d
(2−1)
x =
λL
d
9.5.2 Interference in thin films
Thin films (e.g. soap bubbles,oil on water) often display brilliant coloration when reflecting
white light and show fringes when in monochromatic light.
Explanation
A thin film is a transparent medium whose thickness is comparable with the wavelength of
light. Brilliant and beautiful colors in soap bubbles and oil film on the surface of water are
due to interference of light reflected from the two surfaces of the film.
Consider a thin film of a reflecting medium. A beam AB of monochromatic light of
wavelength λ is incident on its upper surface. It is partly reflected along BC and partly
refracted into the medium along BD. At D it is again partly reflected inside the medium
along DE and then at E refracted along EF.
Reflected light has phase reversal of 180o (path difference of λ/2) as it is reflected
from a surface beyond which there is medium of higher refractive index (noil > nair). But
refracted ray has no phase change as it is reflected from a surface beyond which there
is a medium of lower index. Therefore the condition for constructive and destructive
interference are reversed then the Young’s double slit experiment. For nearly normal

Figure 9.4: Thin Film
incidence the path difference between the two interfering rays is twice the thickness of
the film i.e equal to 2t where t is the thickness of the film. If n is the refractive index of
medium of the film then:
Path difference = 2tn
Hence condition for the maxima or constructive interference is,
2nt = (m+
1
2
)λ ,(where m = 0,1,2,3....)
similarly condition for the minima or destructive interference is,
2nt = mλ ,(where m = 0,1,2,3....)
In case of varying thickness of film, there will be a pattern of alternate dark and bright
fringes.
9.5.3 DIFFRACTION OF LIGHT
The bending and spreading of light waves around sharp edges or corner or through small
openings is called Diffraction of Light.
Diffraction effect depends upon the size of obstacle. Diffraction of light takes place if the
size of obstacle is comparable to the wavelength of light. Light waves are very small in
wavelength, i.e. from 4×10−7m to 7×10−7m. If the size of opening or obstacle is near to
this limit, only then we can observe the phenomenon of diffraction.
Diffraction of light can be divided into two classes:
Fraunhoffer diffraction
In Fraunhoffer diffraction: -
• Source and the screen are far away from each other.
• Incident wave fronts on the diffracting obstacle are plane.
• Diffracting obstacle give rise to wave fronts which are also plane.
• Plane diffracting wave fronts are converged by means of a convex lens to produce
diffraction pattern

Fresnel diffraction
In Fresnel diffraction: -
• Source and screen are not far away from each other.
• Incident wave fronts are spherical.
• Wave fronts leaving the obstacles are also spherical.
• Convex lens is not needed to converge the spherical wave fronts.
9.5.4 DIFFRACTION GRATING
A diffraction grating is an optical device consists of a glass or polished metal surface over
which thousands of ﬁne, equidistant, closely spaced parallel lines are been ruled.
Figure 9.5: Diffraction Grating
Principle
Its working principle is based on the phenomenon of diffraction. The space between lines
act as slits and these slits diffract the light waves there by producing a large number of
beams which interfere in such away to produce spectra.
In diffraction grating, each ray travels a distance d sinθ different from that of its
neighbor, where d is the distance between slits. If this distance equals an integral number
of wavelengths, the rays all arrive in phase, and constructive interference (a maximum) is
obtained. Thus, the condition necessary to obtain constructive interference for a diffraction
grating is:
d sinθ = mλ , for m = 0, ± 1, ± 2, ± 3, ...(constructive)
where d is the distance between slits in the grating, λ is the wavelength of light, and m is
the order of the maximum. Note that this is exactly the same equation as for double slits
separated by d. However, the slits are usually closer in diffraction gratings than in double
slits, producing fewer maxima at larger angles.

Grating Element
Distance between two consecutive slits(lines) of a grating is called grating element. If "a"
is the separation between two slits and "b" is the width of a slit, then grating element "d" is
given by: -
d = a+b OR d =
length of grating
number of lines
=⇒ d =
L
N
Here, N is the total number of lines on the grating and L is the length of the grating.
9.5.5 Diffraction of X-Rays by Crystals
X-rays is a type of electromagnetic radiation of much shorter wavelength, about 10−10m.
In order to observe the effects of diffraction, the grating spacing must be of the order of
the wavelength of the radiation used. The regular array of the atoms in a crystal forms
a natural diffraction grating with spacing that is typically ≈ 10−10m. The scattering of
X-rays from the atoms in a crystalline lattice gives rise to diffraction effects very similar to
those observed with visible light incident on ordinary grating.
The study of atomic structure of crystals by X-rays was initiated in 1914 by W. H.
Bragg and W. L. Bragg with remarkable achievements. They found that a monochromatic
beam of X-rays was reflected from a crystal plane as if it acted like mirror. To understand
this effect, a series of atomic planes of constant inter planer spacing d parallel to a crystal
face are shown by lines PP , P1P1 , P2P2 and so on.
Figure 9.6: X-ray Diffraction
Suppose an X-rays beam is incident at an angle θ on one of the planes. The beam can
be reflected from both the upper and the lower planes of atoms. The beam reflected from
lower plane travels some extra distance as compared to the beam reflected from the upper
plane. The effective path difference between the two reflected beams is 2d sinθ, where d
is atom spacing. Therefore, for constructive interference, the path difference should be an
integral multiple of the wavelength. Thus:
2d sinθ = mλ
The value of m is referred to as the order of reflection. The above equation is known as the
Bragg equation. It can be used to determine inter planar spacing between similar parallel
planes of a crystal if X-rays of known wavelength are allowed to diffract from the crystal.

X-ray diffraction has been very useful in determining the structure of biologically
important molecules such as hemoglobin, which is an important constituent of blood, and
double helix structure of DNA.
9.5.6 Polarization
Light is an electromagnetic wave in which electric and magnetic field are varying in time
and space at right angle to the direction of the propagation of the wave. The process of
confining the vibration of these electric vectors of light waves to the one direction it is
called polarization of light. Since the polarization is the characteristics of the transverse
wave we can say light wave is transverse wave.
Unpolarized light
A beam of ordinary light consisting of large number of planes of vibrations, vibrates
in all directions in all possible directions perpendicular to the direction of propagation.
Such a beam is called unpolarized light. For example, the light emitted by an ordinary
incandescent bulb (and also by the sun) is unpolarized because its (electrical) vibrations
are randomly oriented in space.
Types of polarization
1. Plane polarized light
If the electric vector vibrates in the straight line perpendicular to the plane of direction of
polarization then the light is said to be plane polarized light.
2. Circularly polarized light
When the plane polarized light waves superimposed, the resultant light vector rotates in
constant magnitude in a plane perpendicular to the direction of polarization. The tip of
vectors traces the circular path and light is said to be circularly polarized.
3. Elliptically polarized light
If the magnitude of light vectors changes periodically during rotation,the tip of vectors
traces the ellipse and the light is said to be elliptically polarized light.
We can produce the polarized light by following ways:
1. Polarization by reflection
2. Polarization by selective absorption
3. Polarization of light by scattering
Polariod
Polaroids are the devices used to produce plane polarized light. It is made from the crystal
of iodosulpahate of quinine in thin sleet mounted between two thin sheets of glass of
cellulose. When the unpolarized light falls on the polariod, only the electric field vector
oscillating in the direction perpendicular to the alignment of molecules passes through
polariod so, the transmitted light has the electric field vector oscillating perpendicular
to the the direction of the alignment of the molecules. These transmitted light are plane
polarized.
The applications of the Polaroid are:
1. Polaroids are used in glass windows in train and aero planes to have desire intensity
of light.

2. They are used in three dimensional moving pictures.
3. They are used in headlight of vehicle to eliminate the dazzling light.
4. Polaroid are used in photo elasticity
5. They are used to produce and analyze the plane polarized light.
6. They are used as Polarized sun glass as they prevent the light from the shining
surface to reach the eye.
Points to Note:
• Wavefront is the peak of a transverse wave or the compression of a
longitudinal wave.
• The high points of each wave are the crests; the low points are the
troughs.
• The wave that strikes the boundary between the 2 media is the incident
wave
• The wave that continues in the new medium is the transmitted wave.
• A part of the wave moves back away from the boundary as a wave in
the old medium, this is the reflected wave.
• A soap bubble or oil film on water appears coloured in white light due
to interference of light reflected from upper and lower surfaces of soap
bubble or oil film.
• In interference fringe pattern central bright fringe is brightest and
widest, and remaining secondary maximas are of gradually decreasing
intensities.
• The difference between interference and diffraction is that the inter-
ference is the superposition between the wavelets coming from two
coherent sources while the diffraction is the superposition between the
wavelets coming from the single wavefront.

10. GEOMETRICAL OPTICS
10.1 LENS
A lens is one of the most familiar optical devices. A lens is made of a transparent material
bounded by two spherical surfaces. If the distance between the surfaces of a lens is very
small, then it is a thin lens. The word lens derives from the Latin word for a lentil bean,
the shape of which is similar to the convex lens.
As there are two spherical surfaces, there are two centres of curvature C1 and C2 and
correspondingly two radii of curvature R1 and R2. The line joining C1 and C2 is called the
principal axis of the lens. The centre O of the thin lens which lies on the principal axis is
called the optical centre.
A portion of refracting material bound between two spherical surfaces is called a lens.
Deﬁnition
There are two types of lens:
1. Convex OR Converging lens
A lens is said to be converging if the width of the beam decreases after refraction through
it. In another way:
The lens in which light rays that enter it parallel to its axis cross one another at a single
point on the opposite side with a converging effect is called converging lens.
It has three types:
1. Double Convex Lens
2. Plano Convex Lens
3. Concavo Convex Lens OR Converging Meniscus

128 Chapter 10. GEOMETRICAL OPTICS
Figure 10.1: Types of lenses
2. Concave OR Diverging lens
A lens is said to be diverging lens if the width of the beam increases after refraction through
it. In another way:
A lens that causes the light rays to bend away from its axis is called a diverging lens.
It has three types:
1. Double Concave Lens
2. Plano Concavo Lens
3. Convex Concave Lens OR Diverging Meniscus
Figure 10.2: Convex and Concave lenses
10.1.1 Useful terms:
Center of curvature
Center of curvature of a surface of a lens is deﬁned as the center of that sphere of which
that surface forms a part.
Radius of curvature
Radius of curvature of a surface of a lens is deﬁned as the radius of that sphere of which
the surface forms a part.
Focal point
The point at which the light rays cross is called the focal point F of the lens.

10.1 LENS 129
Principal focus
The point where rays parallel to the principal axis converge with a converging lens.
Focal length
Distance from the principle focus and the optical centre. It is denoted by f.
Principal axis
The line the goes through the optical centre, and the 2 foci.
Figure 10.3: Lens
10.1.2 Thin Lens
A thin lens is deﬁned to be one whose thickness allows rays to refract but does not allow
properties such as dispersion and aberrations.
An ideal thin lens has two refracting surfaces but the lens is thin enough to assume that
light rays bend only once. A thin symmetrical lens has two focal points, one on either side
and both at the same distance from the lens. Another important characteristic of a thin lens
is that light rays through its center are deﬂected by a negligible amount.
Ray Tracing
Ray tracing is the technique of determining or following (tracing) the paths that light rays
take. For rays passing through matter, the law of refraction is used to trace the paths.
Rules for Ray Tracing:
• A ray entering a converging lens parallel to its axis passes through the focal point F
of the lens on the other side.
• A ray entering a diverging lens parallel to its axis seems to come from the focal point
F.
• A ray passing through the center of either a converging or a diverging lens does not
change direction.
• A ray entering a converging lens through its focal point exits parallel to its axis.
• A ray that enters a diverging lens by heading toward the focal point on the opposite
side exits parallel to the axis.

Real Image
The image in which light rays from one point on the object actually cross at the location
of the image and can be projected onto a screen, a piece of ﬁlm, or the retina of an eye is
called a real image.
Virtual Image
An image that is on the same side of the lens as the object and cannot be projected on a
screen is called a virtual image.
10.1.3 Thin lens Formula
Let AB represents an object placed at right angles to the principal axis at a distance greater
than the focal length f of the convex lens. The image A B is formed beyond 2F2 and is
real and inverted.
Figure 10.4: Thin lens formula
OA = object distance = p
OA = image distance = q
OF2 = focal length = f
∆OAB and ∆OA B are similar, therefore:
A B
AB
=
OA
OA
−−−− > (1)
Similarly ∆OCF2 and ∆F2A B are similar:
A B
OC
=
F2A
OF2
But we know that OC = AB, therefore, above equation can be written as:
A B
OC
=
A B
AB
=
F2A
OF2
A B
AB
=
F2A
OF2
−−−− > (2)

10.1 LENS 131
From equation (1) and (2), we get:
OA
OA
=
F2A
OF2
From ﬁgure, OA = q, OA = p, OF2 = f and F2A = q− f, therefore:
q
p
=
q− f
f
qf = p(q− f) = pq− pf
Dividing whole equation by pqf, we get:
1
p
=
1
f
−
1
q
1
f
=
1
p
+
1
q
This is know as thin lens equation. It can be also used for concave lens by applying
appropriate sign convention.
10.1.4 Formation of Image by Convex Lens
The nature of images formed by a convex lens depends upon the distance of the object
from the Optical Center of the lens. Let us now see how the image is formed by a convex
lens for various positions of the object.
1. When the Object is Placed between F1 and O:
Figure 10.5: The object is placed between F1 and O
Here we consider two rays starting from the top of the object placed at F1 and optical
center. The ray parallel to the principal axis after refraction passes through the focus
(F2). The ray passing through the optical center goes through the lens undeviated. These
refracted rays appear to meet only when produced backwards. Thus, when an object is
placed between F1 and O of a convex lens, a virtual, erect and magniﬁed image of the
object is formed on the same side of the lens as the object.That is:-

• Formed on the same side of the lens
• Virtual
• Erected
• Magnified
2. When the Object is Placed at F1
Figure 10.6: The object is placed at F1
Consider two rays coming from the top of the object. One of the rays which is parallel
to the principal axis after refraction passes through F2 and the other ray which passes
through the optical center comes out without any deviation. These two refracted rays
are parallel to each other and parallel rays meet only at infinity. Thus, when an object is
placed at F1 of a convex lens, the image is formed at infinity and it is inverted, real and
magnified.That is:-
• Formed at infinity
• Real
• Inverted
• Magnified
3. When the Object is Placed between F1 and F2
Figure 10.7: The object is placed between F1 and F2

10.1 LENS 133
Let us consider two rays coming from the object. The ray which is parallel to the
principal axis after refraction passes through the lens and passes through F2 on the other
side of the lens. The ray passing through the optic center comes out of the lens without
any deviation. The two refracted rays intersect each other at a point beyond 2F2. So, when
an object is placed between F1 and 2F1 of a convex lens the image is formed beyond 2F2.
That is:
• Formed beyond 2F2
• Real
• Inverted
• Magniﬁed
4. When the Object is Placed at 2F1
Figure 10.8: The object is placed at 2F1
Here one of the rays starting from the top of the object placed at 2F1 passes through
the optic center without any deviation and the other ray which is parallel to the principal
axis after refraction passes through the focus. These two refracted rays meet at 2F2. Thus,
when an object is placed at 2F1 of a convex lens, inverted and real image of the same size
as the object is formed at 2F2 on the other side of the lens.
• Formed at 2F2
• Real
• Inverted
• Same size as the object
5. When the Object is Placed beyond 2F1
The ray parallel to the principal axis after refraction passes through F2 and the ray which
passes through the optical center comes out without any deviation. The refracted rays
intersect at a point between F2 and 2F2. The image is inverted, real and diminished. That
is:
• Formed between F2 and 2F2
• Real
• Inverted
• Diminished

Figure 10.9: The object is placed beyond 2F1
6. When the Object is Placed at Infinity
When the object is at infinity, the rays coming from it are parallel to each other. Let one of
the parallel rays pass through the focus F1 and the other ray pass through the optical center.
The ray which passes through F1 becomes parallel to the principal axis after refraction and
the ray which passes through the optical center does not suffer any deviation. That is:
• Formed at F2.
• Real
• Inverted
• Highly diminished
Figure 10.10: The object is placed at infinity
The table 10.5 gives at a glance the position, size and nature of the image formed by a
convex lens corresponding to the different positions of the object and also its application.
10.1.5 Formation of Image by Concave Lens
Because the rays always diverged by a concave lens, the emerging rays do not actually
intersect. But they deem to intersect on the incidence side by tracing backwards the
emerging rays. Hence concave lens images are always virtual images. Let us now draw
ray diagrams to show the position of the images when the object is placed at different
positions.
1. When the Object is at Infinity:
Nature of image is given below:

10.1 LENS 135
Figure 10.11: Object is at inﬁnity
• Formed at F1.
• Erected
• Virtual
• Diminished
2. When the Object is Placed between O and F:
Figure 10.12: Object is Placed between O and F
In this case the nature of image is given below:
• Formed between O and F1
• Erected
• Virtual
• Diminished

Figure 10.13: Object is placed at any position between O and infinity
3. When the Object is Placed at any Position between O and Infinity:
In this case the nature of image is given below:
• Formed between O and F1
• Erected
• Virtual
• Diminished
Concave Lens Examples
The ‘door eye’ is the most practical use of a concave lens. A door eye is a small concave
lens fitted in the entrance door of a house. Since a concave lens produces the image of
a real object much closer, you can identify the person who is knocking the door even if
he/she stands far from the door. Door eyes are extensively used as safety measure.
10.1.6 Magnification
Let us consider an object OO placed on the principal axis with its height perpendicular to
the principal axis. The ray OP passing through the optic centre will go undeviated. The
ray O A parallel to the principal axis must pass through the focus F2. The image is formed
where O PI and AF2I intersect. Draw a perpendicular from I to the principal axis. This
perpendicular II is the image of OO .
Figure 10.14: Magnification

10.1 LENS 137
The linear or transverse magnification is defined as the ratio of the size of the image to
that of the object.
RAM’S MIND MAP
Magnification m =
Size of the image
Size of the object
=
II
OO
=
h2
h1
where h1 is the height of the object and h2 is the height of the image.
From the similar right angled triangles OO P and II P, we have II PI
II
OO
=
PI
PO
Applying sign convention,
II = −h2;OO = +h1;PI = +q;PO = −p;
Substituting this in the above equation, we get magnification
m =
−h2
+h1
=
+q
−p
(10.1)
m = +
q
p
(10.2)
The magnification is negative for real image and positive for virtual image. In the case of a
concave lens, it is always positive.

Using lens formula the equation for magnification can also be obtained as
m =
h2
h1
=
q
p
=
f −q
f
=
f
f + p
This equation is valid for both convex and concave lenses and for real and virtual images.
10.1.7 Power of a lens
Power of a lens is a measure of the degree of convergence or divergence of light falling on
it. The power of a lens (P) is defined as the reciprocal of its focal length:
P =
1
f
The unit of power is dioptre (D) : 1D = 1m−1. The power of the lens is said to be 1 dioptre
if the focal length of the lens is 1 metre. P is positive for converging lens and negative for
diverging lens. Thus, when an optician prescribes a corrective lens of power +0.5D, the
required lens is a convex lens of focal length +2m. A power of −2.0D means a concave
lens of focal length −0.5m.
Table 10.1: Sign conventions for thin lens
Quantity Symbol In front In back
Object location p + −
Image location q − +
Lens radii R1, R2 − +
10.1.8 Combination of thin lenses in contact
Let us consider two lenses A and B of focal length f1 and f2 placed in contact with each
other. An object is placed at O beyond the focus of the first lens A on the common principal
axis. The lens A produces an image at I1. This image I1 acts as the object for the second
lens B. The final image is produced at I. Since the lenses are thin, a common optical centre
P is chosen.
Let PO = p, object distance for the first lens (A), PI = q, final image distance and
PI1 = q1, image distance for the first lens (A) and also object distance for second lens (B).
For the image I1 produced by the first lens A,
1
q1
+
1
p
=
1
f1
−−−− > (1)
For the final image I, produced by the second lens B,
1
q
−
1
q1
=
1
f2
−−−− > (2)

10.1 LENS 139
Figure 10.15: Combination of lenses
Adding equations (1) and (2),
1
q
+
1
p
=
1
f1
+
1
f2
−−−− > (3)
If the combination is replaced by a single lens of focal length F such that it forms the
image of O at the same position I, then
1
q
+
1
p
=
1
F
−−−−− > (4)
From equations (3) and (4)
1
F
=
1
f1
+
1
f2
−−−− > (5)
Here F is the focal length of the equivalent lens for the combination.
The derivation can be extended for several thin lenses of focal lengths f1, f2, f3... in
contact. The effective focal length of the combination is given by
1
F
=
1
f1
+
1
f2
+
1
f3
+... =⇒
1
F
=
n
∑
i=1
1
fi
−−−− > (6)
In terms of power, equation (6) can be written as
P = P1 +P2 +P3 +....
The power of a combination of lenses in contact is the algebraic sum of the powers of
individual lenses.
The combination of lenses is generally used in the design of objectives of microscopes,
cameras, telescopes and other optical instruments.

10.2 Aberrations
Lenses usually do not give a perfect image. Some causes are:
1. Chromatic aberration is caused by the fact that n = n(λ). This can be partially
corrected with a lens which is composed of more lenses with different functions
ni(λ). Using N lenses makes it possible to obtain the same f for N wavelengths.
2. Spherical aberration is caused by second-order effects which are usually ignored;
a spherical surface does not make a perfect lens. In coming rays far from the optical
axis will more bent.
3. Coma is caused by the fact that the principal planes of a lens are only flat near the
principal axis. Further away of the optical axis they are curved. This curvature can
be both positive or negative.
4. Astigmatism: from each point of an object not on the optical axis the image is an
ellipse because the thickness of the lens is not the same everywhere.
5. Field curvature can be corrected by the human eye.
6. Distortion gives aberrations near the edges of the image. This can be corrected with
a combination of positive and negative lenses.
RAM’S MIND MAP
10.3 Optical Instruments
10.3.1 Simple Microscope
Convex lens is called simple microscope as it is often used as a magnifier when an object is
brought within the focal length of convex lens. The magnified and virtual image is formed
at least distance of distinct vision d. A lens is placed in front of the eye in such a way that
a virtual image of the object is formed at a distance d from the eye. The size of the image
is now much larger than without the lens. If β and αare respective angles subtended by
the object when seen through lens (simple microscope) and when viewed directly, then
angular magnification M is defined as:
M =
β
α
−−−− > (1)

10.3 Optical Instruments 141
When angles are small, then they are nearly equal to their tangents:
α = tanα =
Size of object
distance of object
=
O
d
−−−− > (2)
Similarly:
β = tanβ =
Size of image
distance of image
=
I
q
−−−−− > (3)
Since the image is at the least distance of distinct vision, so q = d. Therefore, the equation
(3) becomes:
β =
I
q
=
I
d
Putting vales of β and α in equation (1), we have:
M =
I/d
O/d
=
I
O
As we already know that:
I
O
=
Size of image
size of object
=
distance of image
distance of object
=
q
p
Therefore:
M =
q
p
=
d
p
For virtual image, the lens formula if written as:
1
f
=
1
p
−
1
q
=⇒
1
f
=
1
p
−
1
d
Multiplying both sides of equation by d, we have:
d
f
=
d
p
−1 =⇒
d
p
= 1+
d
f
Since d/p = M, therefore:
M = 1+
d
f
It is seen that for a lens of high angular magnification, the focal length should be small.
10.3.2 Compound Microscope
Compound microscope is an optical instrument which is used to obtain high magnification.
It consists of two converging lenses:
Objective: The lens in front of object is called objective. Its focal length f1 = fo is taken
to be very small .The objective forms a real, inverted, and magnified image of the object
placed just beyond the focus of objective.
Eye piece The lens towards the observer’s eye is called piece. Focal length of eye piece is
greater than the focal length of objective. Eye piece works as a magnifying glass.
The objective is so adjusted that the object is very closed to its focus. The objective
forms a real, inverted and magnified image of the abject beyond 2 fo on the right hand side.
The eye piece is so adjusted that it forms a virtual image at the least distance of distinct
vision d. The final image is highly magnified.

Magnifying power
In order to determine the magnifying power of a compound microscope, we consider an
object OO placed in front of objective at a distance p1. Objective forms an inverted image
II at a distance of q1 from the objective. Magnification produced by the objective is given
by:
Mo =
size of image
size of object
=⇒ Mo =
q1
p1
−−−−−(1)
Eye piece works as a magnifying glass. It further magnifies the first image formed by
objective. Magnification produced by the eye piece is given by:
Me =
size of image
size of object
We know that the eye piece behaves as a magnifying glass therefore the final image will be
formed at least distance of distinct vision i.e at 25cm from the eye. Hence q2 = d
Me =
d
p2
−−−−−−−(2)
Using thin lens formula for eye piece :
1
f
=
1
p
+
1
q
=⇒
1
f2
=
1
q2
+
1
p2
Here f2 = fe, q2 = −d and p = p2:
1
fe
=
1
−d
+
1
p2
= −
1
d
+
1
p2

10.3 Optical Instruments 143
Multiplying both sides by d,
d
fe
= −1+
d
p2
d
p2
= 1+
d
fe
−−−− > (3)
Comparing equation (2) and (3):
Me = 1+
d
fe
−−−−−−−−(4)
Total magnification is equal to the product of the magnification produced by the objective
and the eye piece:
M = Mo ×Me
M = (
q1
p1
)(1+
d
fe
)
In order to get maximum magnification, we must decrease p1 and increase q1. Thus
maximum possible value of p1 is fo i.e p = fo and maximum possible value of q1 is the
length of microscope i.e q1 = L. Therefore, the magnification produced by a compound
microscope is given by:
M = (
L
fo
)(1+
d
fe
)
10.3.3 Astronomical Telescope
It is an optical instrument used to view heavenly bodies such as moon, stars, planets and
distant objects. Astronomical telescope consists of two convex lenses:
Objective:The objective is a convex lens of large focal length and large aperture. It usually
made of two convex lenses in contact with each other to reduce the chromatic and spherical
aberrations.
Eye piece: The eye piece is also a convex lens. Its focal length is smaller than that of
objective. It is also a combination of two lenses.
The objective is mounted on a wide metallic tube while the eye piece is mounted on
a small tube. The distance between the eye piece and the objective can be changed by
moving tubes.
Working
The rays coming from a distant object falls on objective as parallel beam at some angle say
α and these rays after refraction and passing through the objective converge at its focus
and make an inverted and real image AB. This image acts as an object for the eye piece.
The distance of the eye piece is so adjusted that the image lies within the focal length of
the eye piece. The eye piece forms the final image .The final image is magnified, virtual
and inverted with respect to object. The final image is formed at infinity.

Figure 10.16: Astronomical Telescope
Magnifying Power
The magnifying power (M) of astronomical telescope is given by:
M =
Angle suspended by final image @ eye
Angle suspended by object @ eye
=
Angle suspended by final image @ eye
Angle suspended by object @ object:
It is because the object is at infinite distance and hence the angle subtended by the object
at eye may be taken as the angle subtended by the object at objective: M = β
α . Since α and
β are small angles, therefore we can take: α = tanα and β = tanβ. Hence
M =
tanβ
tanα
In right angled triangles ∆ABO1 ∆ABO2:
M =
AB/AO2
AB/AO1
=
AO1
AO2
=⇒ M =
f0
fe
M =
focal length of objective
focal length of eye piece
This expression shows that in order to obtain high magnification, focal length of object
must be large and that of eye piece is small.
Length of Telescope
The distance b/w objective lens and the eye piece is equal to the length of the telescope.
From figure: O1O2 =length of telescope = L. But O1O2 = O1A + AO2. Where
O1A = fe and AO2 = fo. Therefore:
L = fo + fe
Or, L =focal length of objective + focal length of eye piece.
10.4 Spectrometer
The spectrometer is an optical instrument used to study the spectra of different sources of
light and to measure the refractive indices of materials. It consists of basically three parts.
They are collimator, prism table and Telescope.

10.5 Human eye 145
Collimator
The collimator is an arrangement to produce a parallel beam of light. It consists of a long
cylindrical tube with a convex lens at the inner end and a vertical slit at the outer end of
the tube. The distance between the slit and the lens can be adjusted such that the slit is at
the focus of the lens. The slit is kept facing the source of light. The width of the slit can be
adjusted. The collimator is rigidly fixed to the base of the instrument.
Turn table
The turn table is used for mounting the prism, grating etc. It consists of two circular metal
discs provided with three levelling screws. It can be rotated about a vertical axis passing
through its centre and its position can be read with verniers V1 and V2. The prism table can
be raised or lowered and can be fixed at any desired height.
Telescope
The telescope is an astronomical type. It consists of an eyepiece provided with cross
wires at one end of the tube and an objective lens at its other end co-axially. The distance
between the objective lens and the eyepiece can be adjusted so that the telescope forms a
clear image at the cross wires, when a parallel beam from the collimator is incident on it.
The telescope is attached to an arm which is capable of rotation about the same vertical
axis as the prism table. A circular scale graduated in half degree is attached to it.
10.5 Human eye
Eyes are organ of the sight approximately 2.5cm in diameter. It consists of one convex
lens. It is composed of three layers. They are: sclera, choroid and retina. It consists of
two types of muscles called irish and ciliary muscles. Irish helps to regulate the amount
of light entering the eyeball whereas ciliary muscles helps to focus light at the retina by
contraction and relaxation of the ligaments attached to the ciliary muscles. Inverted image
is formed on retina.
Defects
For a normal eye, far point is infinity and near point is 25cm from the eye. When an eye
cannot focus the light at the retina, the object cannot be seen clearly. Such defects of eyes
vision observed in human eye are as Myopia or Hypermetroypia.
• Myopia: Also called nearsightedness, is common name for impaired vision in which
a person sees near objects clearly while distant objects appear blurred. In such
a defective eye, the image of a distant object is formed in front of the retina and
not at the retina itself. Consequently, a nearsighted person cannot focus clearly on
an object farther away than the far point for the defective eye. This defect can be
corrected by using a concave (diverging) lens. A concave lens of appropriate power
or focal length is able to bring the image of the object back on the retina itself.
• Hypermetroypia: Also called farsightedness, common name for a defect in vision
in which a person sees near objects with blurred vision, while distant objects appear
in sharp focus. In this case, the image is formed behind the retina. This defect can
be corrected by using a convex (converging) lens of appropriate focal length.

Table 10.2: Sign conventions for thin lens
Quantity Positive when ... Negative when...
Object location
(p)
Object is in the front of lens.
(Real object)
Object is in the back of lens. (vir-
tual object)
Image location
(q)
Image is in the back of lens.
(Real image)
Image is in the front of lens. (Vir-
tual image)
Image height
(h )
Image is upright Image is inverted
R1 and R2 Center of curvature is in the back
of lens
Center of curvature is in the front
of lens
Focal length (f) Converging lens Diverging lens
Points to Note:
• Light waves slow down when they pass from a less to a more dense
material and vice versa.
• When a wave is slowed down, it is refracted towards the normal and
when a wave is sped up, it is refracted away from the normal.
• A converging lens is one which has a positive focal length. It is also
called a positive lens.
• A diverging lens is defined to be a lens which has a negative focal
length. It is also called a negative lens.
• Lens formula is only applicable for thin lens.
• Magnification formula is only applicable when object is perpendicular
to optical axis.
• Lens formula and the magnification formula is only applicable when
medium on both sides of lenses are same.
• Thin lens formula is applicable for converging as well diverging lens.
• If a lens is cut along the diameter, focal length does not change.
• If lens is cut by a vertical, it converts into two lenses of different focal
lengths.
• The minimum distance between real object and real image in the case
of thin lens is 4f.
• If a number of lenses are in contact, then:
1
f
=
1
f1
+
1
f2
+
1
f3
+...
• Real images are always inverted and Virtual images are always upright.
• Diverging lens (concave) produce only small virtual images.
• The focal length of a converging lens (convex) is shorter with a higher
index (n) value lens or if blue light replaces red.

10.5 Human eye 147
Table 10.3: Formation of Image by Convex Lens
Position of
the object
Position of
the image
Nature
of the
image
Size of the
image
Application
Between O
and F1
on the same
side of the
lens
Erected
and
virtual
Magnified Magnifying lens (simple micro-
scope), eye piece of many instru-
ments
At 2F1 At 2F2 Inverted
and real
Same size Photocopying camera
Between F
and 2F1
Beyond At
2F2
Inverted
and real
Magnified Projectors, objectives of microscope
At F1 At infinity Inverted
and real
Magnified Theater spot lights
Beyond At
2F1
Between F2
and 2F2
Inverted
and real
Diminished Photocopying (reduction camera)
At infinity At F2 Inverted
and real
Diminished Objective of a telescope

APPENDIX
10.6 Useful Formula
Pythagoras’s Theorem: C = A2 +B2 (10.3)
Quadratic Formula: x =
−b±
√
b2 −4ac
2a
(10.4)
circumference: C = 2πr (10.5)
Area of circle: A = πr2
(10.6)
Voulme of sphere: V =
4
3
πr3
(10.7)
Volume of cylinder: V = πr2
h (10.8)
Binomial Theorem: (a+b)n
=
n
∑
k=0
n
k
an−k
bk
(10.9)
where
n
k
=
n!
k!(n−k)!

Physical Constants
Name Symbol Value Unit
Number π π 3.14159265358979323846
Number e e 2.71828182845904523536
Elementary charge e 1.60217733×10−19 C
Gravitational constant G 6.67259×10−11 m3kg−1s−2
Speed of light in vacuum c 2.99792458×108 m/s (def)
Permittivity of the vacuum ε0 8.854187×10−12 F/m
Permeability of the vacuum µ0 4π ·10−7 H/m
(4πε0)−1 8.9876·109 Nm2C−2
Planck’s constant h 6.6260755×10−34 Js
Bohr magneton µB = e¯h/2me 9.2741·10−24 Am2
Bohr radius a0 0.52918 Å
Rydberg’s constant Ry 13.595 eV
Stefan-Boltzmann’s constant σ 5.67032·10−8 Wm−2K−4
Wien’s constant kW 2.8978·10−3 mK
Molar gas constant R 8.31441 J×mol−1·K−1
Avogadro’s constant NA 6.0221367×1023 mol−1
Boltzmann’s constant k = R/NA 1.380658×10−23 J/K
Electron mass me 9.1093897·10−31 kg
Proton mass mp 1.6726231×10−27 kg
Neutron mass mn 1.674954×10−27 kg
Elementary mass unit mu = 1
12m(12
6 C) 1.6605656·10−27 kg
Nuclear magneton µN 5.0508·10−27 J/T
Mass of the moon MM 7.36×1022 kg
Radius of the moon RM 1.74×106 m
Mean earth-moon distance dEM 3.84×108 m
Diameter of the Sun D 1392×106 m
Mass of the Sun M 1.989×1030 kg
Rotational period of the Sun T 25.38 days
Radius of Earth RA 6.378×106 m
Mass of Earth MA 5.976×1024 kg
Escape speed from the earth vesc 11.2 km/s
Escape speed from the moon vesc 2.38 km/s
Rotational period of Earth TA 23.96 hours
Earth orbital period Tropical year 365.24219879 days
Astronomical unit AU 1.4959787066×1011 m
Light year lj 9.4605×1015 m
Parsec pc 3.0857×1016 m
Hubble constant H ≈ (75±25) km·s−1×Mpc−1

10.6 Useful Formula 151
Trigonometric Identities
sin2
θ +cos2
θ = 1 (10.10)
sin(A+B) = sinAcosB+cosAsinB (10.11)
cos(A+B) = cosAcosB−sinAsinB (10.12)
tan2
θ +1 = sec2
θ (10.13)
1+cot2
θ = csc2
θ (10.14)
cos(−θ) = cosθ (10.15)
sin(−θ) = −sinθ (10.16)
cos(A−B) = cosAcosB+sinAsinB (10.17)
sinAcosB =
1
2
sin(A−B)+
1
2
sin(A+B) (10.18)
cosAcosB =
1
2
cos(A−B)+
1
2
cos(A+B) (10.19)
sinAsinB =
1
2
cos(A−B)−
1
2
cos(A+B) (10.20)
sin2A = 2sinAcosA (10.21)
cos2A = cos2
A−sin2
A (10.22)
cos2
A =
1
2
(1+cos2A) (10.23)
sin2
A =
1
2
(1−cos2A) (10.24)
tan(A+B) =
tanA+tanB
1−tanAtanB
(10.25)
tan(A−B) =
tanA−tanB
1+tanAtanB
(10.26)
By
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Physics Notes First Year Class

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