Download Physics , subject code 3110011 and more Study notes Physics in PDF only on Docsity!
Module 1 - Properties of Matter
1.1 INTRODUCTION
Elasticity is a branch of physics which deals with the elastic property of materials. Elasticity is the
property by virtue of which bodies regain original shape and size after removal of the deforming force.
when a deforming force acts on a body, there is a change in its length or shape or volume. Then, the
body is said to strained. In this chapter, elastic properties of materials and their applications for
different uses are discussed.
1.2 LOAD
A force which is applied to the structure is called a load.
Load causes stress, deformation and displacement in the structures.
1.2.1 CLASSIFICATION OF LOADS-BASED ON TIME
Static load: The load which are relatively constant for an extended time is called static load.
Dynamic load: The load which varies with respect to time is called dynamic load.
Normal load: The load which acts perpendicular to cross sectional area is called normal load.
Shear load: The load which tends to produce a sliding failure of a body along a plane parallel to the
direction of load or force is called shear load.
Concentrated load: The load which is concentrated at a point is called concentrated load.
Distributed load: The load which is distributed over the length or area of the structure is called
distributed load.
1.3 STRESS
Restoring force per unit area is called stress.
σ = F/A
There are three types Longitudinal stress, volume stress and tangential (shearing) stress.
1.3.1 LONGITUDINAL STRESS
Longitudinal stress can be compressive or tensile.
In compressive length will decrease and in tensile length will increase.
l
= F/A
Module 1 - Properties of Matter
1.3.2 Volume (Bulk) stress
Volume stress is nothing but pressure because to change the volume force must be applied normal to
the surface.
σ
V
F
A
= P
1.3.3 Tangential stress
In tangential stress shape will change.
σ
T
= F/A
Module 1 - Properties of Matter
Portion AB: In this portion, Hook’s law is not obeyed, although the material may still be elastic. The
point B indicates the elastic limit.
Portion BC: In this portion, the metal shows a strain even without increase in stress and the strain is
not fully return when load is removed.
Portion CD: Yielding start in this portion and there is a drop of stress at the point D directly after yielding
begins at C. The point D is termed as lower yield point and C is called upper yield point.
Portion DE: After yielding has taken place at D, further straining takes place at this portion by increasing
the stress and the stress–strain curve continues to rise up to the point E. Strain in this portion is about
100 times that of portion O-A. At the point E, the bar begins to form a local neck. The point E is termed
as ultimate tensile stress point.
Portion EF: In this portion, the load is falling off from the maximum and fracture at F takes place. The
point F is termed as fracture or breaking point and the identical stress is called breaking stress.
1.5.1. Ductility
It is the property due to which material can be drawn into wires.
Higher the strain more is the ductility.
Ductility increases with temperature.
Gold is the most ductile material.
1.5.2. BRITTLENESS
It is the property due to which material do not deform under load but suddenly breaks.
It is opposite to ductility.
In this material no or negligible strain.
1.5.3. PLASTICITY
It is the property due to which there is permanent deformation without breaking.
Module 1 - Properties of Matter
1.6 Hook’s law
Stress is directly proportional to strain for small stress applied.
Ratio of stress to strain is constant and is known as coefficient of elasticity or modulus of elasticity.
stress ∝ strain
stress = E × strain
E =
stress
strain
1.7 Types of elastic moduli
1.7.1. Young’s modulus
Y =
longitudinal stress
longitudinal strain
If load applied to a wire of length l and cross-section area A = πr 2 , is mg and extension in wire due to
this is ∆l. Then
Y =
F/A
∆l/l
Y =
mgl
πr 2 ∆l
Unit of Young’s modulus is N/m
2
or Pascal.
1.7.2. Bulk modulus
B =
volume stress
volume strain
If load applied normal to substance, then only change in volume will be there. Thus volume stress is
nothing but pressure. Due to normal force if change in volume is ∆V then,
B =
P
∆V/V
B =
PV
∆V
Unit of Bulk modulus is N/m
2
or Pascal.
1.7.3. Modulus of rigidity
η =
shearing stress
shearing strain
If load applied tangential to surface of the substance shape of the substance will change.
η =
F/A
∆l/l
η =
Fl
A∆l
Unit of Modulus of rigidity is N/m
2
or Pascal.
Module 1 - Properties of Matter
ϵ
x
∆l
l
∆b
b
∆h
h
ϵ
x
∆l
l
− μ
∆l
l
− μ
∆l
l
(∵ μ =
∆d
d
∆l
l
ϵx =
∆l
l
1 − 2μ
Using equation 1
ϵ
x
σ
Y
1 − 2μ
= ϵy = ϵz − − − − − − − 2
Volume is V = lbh
Volume strain is
∆V
V
∆l
l
∆b
b
∆h
h
∴ ϵ
v
= ϵ
x
y
z
Using equation 2 Volume strain is
ϵ
v
3σ
Y
1 − 2μ
B =
σ
ϵ
v
⇒ ϵ
v
σ
B
Using equation 4 in 3
σ
B
3σ
Y
1 − 2μ
⇒ Y = 3B
1 − 2μ
1.10 Relation between Y and η
Longitudinal strain is,
Module 1 - Properties of Matter
ϵ =
∆l
l
Also from, Y =
σt
ϵ
⇒ ϵ =
σt
Y
Comparing both
∆l
l
σt
Y
Net strain in AC
ϵ
AC
AC
′
− AC
AC
BD
′
− BD
BD
∆l
l
∆b
b
ϵ
AC
∆l
l
∆l
l
∆l
l
( 1 + μ)
Using equation 1
ϵ
AC
σt
Y
( 1 + μ) − − − − − 2
Net strain in AC
ϵ
AC
AC
′
− AC
AC
BD
′
− BD
BD
∆l
l
∆b
b
ϵ
AC
∆l
l
∆l
l
∆l
l
( 1 + μ)
Using equation 1
ϵ
AC
σt
Y
1 + μ
Strain
ϵ =
AC
′
− AC
AC
ϵ =
AC
′
− AN
AC
NC′
AC
CC
′
cos 45
BC√ 2
ϵ =
CC
′
2BC
tan θ
θ
ϵ =
θ
Using equation 2 in 3
σt
Y
( 1 + μ) =
θ
σt
Y
1 + μ
σt
2η
− − − − − (∵ η =
σt
θ
Y = 2η( 1 + μ)
1.11 Relation between Y, B and η
Module 1 - Properties of Matter
1.12.4 Effect of annealing (heating or cooling)
Operations like annealing (i.e., heating and then cooling gradually) tends to produce a uniform pattern
of orientation of the constituent crystals, by orienting them all in one particular direction and thus
forming larger crystal grains, resulting in a decrease in their elastic properties or an increase in softness
or plasticity of the material. Effect of impurities
Effect of impurities
It is well known that sometimes suitable impurities are deliberately added to metals to help bind their
crystal grains better, without affecting their orientation. For example, carbon and potassium are added
in minute quantities to molten iron and gold respectively for this purpose. Such impurities naturally
affect the elastic properties of the metal to which they are added, enhancing or impairing them. In
either case, the elastic properties are considerably strengthened.
1.13 Working stress
Working stress is defined as the stress that is developed in the body during working condition.
While considering working stress, other stresses are also considered in addition to its loading stress.
Working stress =
actual applied load
cross section area under loading
1.14 Factor of safety
It is defined as the ratio of maximum stress to the working stress.
It is always greater than 1.
For ductile material, it is ratio of yield stress to working stress.
For brittle material, it is ratio of ultimate stress to working stress.
Some factors are also considered for factor of safety.
Type of materials
Behaviour of material under different loading conditions.
Type of loading conditions.
Effect of temperature, weather, chemical, radiation etc.
Safety of users.
Module 1 - Properties of Matter
1.15 TWISTING COUPLE ON A WIRE
Torsion is the twisting of an object due to an applied torque. Torque is a rotating force capable of
turning a body. A stress is an internal resistance offered by a body per unit area of the cross section.
For studying torsional stress, we may simply define it in terms of shearing stress which is produced
when we apply the twisting moment to the end of a shaft about its axis. For example, when we turn a
screw driver to produce torsion, our hand applies a torque ‘T’ to the handle and twists the shank of
the screw driver.
Deformation means change in the shape or dimensions of a body as a result of stress and strain on a
material. Let us understand this by taking an example of a shaft attached to the wall and rotating it, as
shown.
A circular shaft remains undistorted because its axis is symmetric about the centre. A non-circular
shaft, on the other hand, when subjected to torsion, will be distorted, because it is not having an axis
that is symmetrical about its centre. For any type of circular shafts, whether it is a solid material or a
hollow material, a circular shaft will remain undistorted due to torsion.
1.15.1 TWISTING COUPLE ON A WIRE
The upper end is clamped through a fixed support. The wire consists of a number of cylindrical tubes
i.e., coaxial tubes. The radius of tubes ranges from zero to a. Consider such a cylindrical shell of radius
r and thickness dr.
In elementary tube, the line AC is parallel to axis of the tube. Consider that a couple is applied at the
bottom of wire. As a result, the wire is twisted through an angle θ. Thus, in this twisted state, the
position of AC takes a new position as AG which is shown.
From figure, angle COG = θ and CG = r θ
It is known that the displacement is maximum for the points lying on the rim while it decreases as one
move towards the centre of wire O. This results a maximum shearing strain at the rim and a minimum
shearing strain at the axis or centre.
Consider the line AC which is flattened out from the hollow cylinder. Therefore, rectangle ACDB and
AC1D1B are obtained before and after twisting. Hence, the shear and shearing strain are obtained i.e.
CAG = φ
Or CG = l φ.
Comparing above two equations, l φ = r θ
Module 1 - Properties of Matter
- The radial lines before and after twist remain same.
Let dx be the thickness of an elementary thin tube and r be its radius. Consider that the shaft is twisted
through an angle θ. The cross-sectional area of the elementary tube is
𝐴 = π r
2
𝑑𝐴 = 2π r dr
And
The torque carried out by the tube
3
dr
The total torque or moment of the couple, carried out by solid shaft is the sum of all the elementary
torques, i.e.,
3
dr
𝑑/ 2
0
We know that the radial lines before and after twist remains same and hence, angle θ of the twist is
constant. Therefore, the rigidity modulus η and the length of the shaft remain constant. Thus equation
of torque can be written as
3
dr
𝑑/ 2
0
[ 2 𝜋 (
4
]
Or torque T can be obtained as
[
4
]
where d is the diameter of the shaft. The couple per twist is
4
4
The above equation gives a couple per angular twist. Further, it is clear that the angle of twist is directly
proportional to the length of the shaft.
1.15.3 TORSIONAL PENDULUM
Torsional pendulum is a pendulum which performs torsional oscillations. It is used to determine the
moment of inertia and torsional rigidity of a given body. The compound and torsional pendulum differ
from each other in terms of oscillations. The oscillations are twisted or torsional in a horizontal plane
in torsional pendulum, while they are linear for a compound pendulum.
A torsional pendulum consists of a circular disc D as shown. The metal rod or wire is used to clamp the
disc. The metal rod is suspended symmetrically by clamping at the torsion head H1. The length of
torsion wire is adjusted employing the head H2. In order to measure the time period, a pointer is
marked at the disc. The length of pendulum, i.e., the distance between head H2 and head H1, is
adjusted by using the head H2. The head H1 is not disturbed.
Module 1 - Properties of Matter
The disc D is rotated through a small angle and left free. As a result, the elements in the wire undergo
shearing strain due to the twisting of wire when it is rotated. Therefore, restoring couple which is
acting on the wire brings back to the normal original position. Therefore, the restoring couple = - Cθ,
where C is the couple per unit twist.
Therefore, angular acceleration in the wire is due to the restoring couple
2
2
If I is the moment of the wire about its axis,
2
2
Above equation indicates that the angular acceleration
𝑑
2
𝜃
𝑑𝑡
2
is directly proportional to angular
displacement θ. The negative sign indicates a decrease in twist on the wire with couple and directed
towards the mean position. Therefore, torsional oscillations made by torsional pendulum are simple
harmonic motion and the period of oscillation is controlled by the momentum of inertia of the
suspended mass about the axis of suspension and a couple per unit twist produced in the wire.
Therefore, time period of oscillator is
The above equation gives the time period of torsional oscillation of torsional pendulum. From the
above equation, it can be derived that if the torsional pendulum is allowed to oscillate, the angular
velocity and the time period only depends on moment of inertia of wire as well on torsional rigidity.
1.16 BENDING OF BEAM
A beam is defined as a rod or a bar made of homogenous and isotropic elastic material with uniform
cross-section, either circular or rectangular. The length of a beam is very large compared to its other
dimensions, viz breadth and thickness. Thus, the shearing stress at any point of the beam is very small
and negligible. Generally, beams are used to carry heavy loads in applications like roofs, bridges, etc.
In order to study flexural rigidity of the beam for heavy loading applications, a simple theory is
developed to study the bending of a beam. The following are the important points to be considered
- The weight of the beam should be low.
- The beam should have small curvature.
- The minimum deflection of the beam is small compared to its length.
- There are no shearing forces.
- The cross-section of the beam remains unaltered during bending. Hence, the geometrical
moment of inertia of the beam remains constant.
One end of the beam is fixed while a load is applied at the other end. As a result, longitudinal filament
is extended at the convex side, while a contraction is obtained at the concave side. However, there is
no change in the filament at the centre i.e., in between the two sides. The central filament is in the
neutral stage and hence, it is known as neutral filament. The corresponding central axis is known as
neutral axis.
Module 1 - Properties of Matter
1.16.2 CANTILEVER
A cantilever is a beam fixed horizontally at one end and loaded at the other end. If a load ‘W’ is applied
at the free end, a couple is created between two forces i.e., (a) force (W) applied at the free end
towards downward direction and (b) reaction (R) acting in the upward direction at the supporting end
as shown
Due to the load applied at the free end of the cantilever, an external couple is created between the
load W at A and the force of reaction Q. Here the arm of the couple (distance between the two equal
and opposite forces) is l – x.
The external bending moment = W (l – x )
We know that the internal bending moment under equilibrium condition =
𝑌
𝑅
𝑔
Under the situation where External bending moment = Internal bending moment
W (l – x ) =
𝑌
𝑅
𝑔
𝑌 𝑎 𝑘
2
𝑅
We know that the moment of a couple increases as one moves from U towards the fixed end P. The
radius of curvature is different at different points and it decreases as one approaches the point P.
Consider a point V at a distance dx from U. The radius of curvature at V is same as that of U, since the
point V is very close to the point U. Thus, the equation
Thus, the above equation gives
2
And
2
The observed depression dy of V below U is
2
2
The total depression of the beam can be calculated taking the limit from zero to l.
2
2
𝑙
0
2
[𝑙
2
2
3
]
0
𝑙
2
3
𝑔
3
Module 1 - Properties of Matter
Above equation gives the depression value at the free end of cantilever.
1.16.2.1 CANTILEVER-YOUNG’S MODULUS
Experimental arrangements used to determine the Young’s modulus of a rod employing the cantilever.
The given rod or bar whose Young’s modulus is to be determined is clamped
UNIFORM BENDING – YOUNG’S MODULUS
NON – UNIFORM BENDING – YOUNG’S MODULUS
I – SHAPED GIRDERS
The girders with upper and lower sections broadened and middle section tapered, so that it can
withstand heavy loads over it is called as I-shaped girders. Since the girder looks like the letter I, they
are known as I-shaped girders.
We know that the depression in the case of a rectangular section is given as,
3
3
Depression y can be minimized by either decreasing the load (W) or the length of the girder (l) or by
increasing the Young’s modulus or the breadth (b) or the thickness (d) of the girder. Since length l is a
fixed quantity, it cannot be decreased. Therefore, breadth and thickness may be adjusted by increasing
the depth and decreasing the breadth (since thickness increases by d
3
). Thus, volume of girder is
increased and hence depression produced is reduced. Depression can also be reduced by properly
choosing materials of high Young’s modulus.
Applications of I-shaped Girders
- They are used in the construction of bridges over rivers.
- They are very much useful in the production of iron nails which are used in railway tracks.
- They are used in supporting beams for ceilings in the construction of buildings.
- They are used in construction of dams.
Advantages
- More stability
- Stronger
- High durability
1.17 VISCOSITY
Viscosity is a property of the fluid which describes the frictional resistance to flow of the fluid and it
measures the resistance of fluid to deform under shear stress. If the viscosity of the fluid is high, it
implies that particles have more affinity with each other and vice versa.
Consider a fluid is flowing over a fixed horizontal surface with streamline flow. The layer which is in
contact with fixed horizontal surface remains stationary and the velocity of the other layer increase
gradually with the distance from the fixed surface. So, the upper most layer has highest velocity and
gradually bottom layers have decreasing velocity which develops velocity gradient.
Module 2 – Waves, Motion and Acoustics
Introduction:
2.1.1 WAVES
The wave is a form of disturbance which transmits energy from one place to another without the actual
flow of matter as a whole. Wave motion is a form of disturbance which is due to the repeated periodic
vibrations of the particles of the medium about their mean positions and the motion is handed over
from one particle to the another without any net transport of the medium. Water waves or sound
waves are called mechanical or elastic waves as they require a material medium for their propagation
which possess elasticity as well as inertia. Light waves are electromagnetic waves or non-mechanical
waves which can propagate through vacuum i.e. don’t require any material medium.
Matter waves are associated with moving electrons, protons, neutrons and other fundamental
particles and even atoms and molecules and all these constitute the matter, so are called matter
waves. Matter waves arise in quantum mechanical description of nature. Thus, waves are of three
types: Mechanical waves, electromagnetic waves and matter waves.
Mechanical waves are of two types; transverse waves and longitudinal waves.
A wave is said to be progressive or travelling wave if it travels from one point of the medium to another.
Waves on the surface of water are of two types; capillary waves and gravity waves. The restoring force
that produces capillary waves is the surface tension of water. The restoring force that produces gravity
waves is the pull of gravity which tends to keep the water surface at its lowest level. The oscillations
of the particles in the gravity waves are not confined to the surface only, but extend with diminishing
amplitude to the very bottom.
The particle motion in water waves involves a complicated motion, they not only move up and down
but also back and forth. The waves in an ocean are a combination of both longitudinal waves travel
with different speeds in the same medium. Speed of transverse waves in a string is determined by two
factors; linear mass density and tension in the string.
2.1.2 Definitions:
- Phase:
Phase is the position of a point in time on a waveform cycle.
- Wavelength:
It is equal to the distance travelled by a wave during the time in which any one particle of the
medium completes one vibration about its mean position. Distance between two consecutive
crests or troughs.
- Wave number:
Wave number k and wavelength are related as 𝑘 =
2 𝜋
𝜆
. Its unit is m - 1
- Wave velocity:
Velocity = distance travelled / time taken
Velocity of wave = length of pulse / time period for pulse
- Frequency:
The number of waves passing through a given point during the interval of one second. The
higher the frequency, the shorter the wavelength. Its unit is Hz
- Time period:
Module 2 – Waves, Motion and Acoustics
The time required for completing one cycle measured is termed as time period. Its unit is
second.
- Transverse waves:
A wave in which the particles of medium vibrates up and down in a direction perpendicular to
the direction of propagation of wave motion are defined as transverse waves. It travels in the
form of crests and troughs.
Movement of string of a sitar or violin, movement of the membrane of tabla or Dholak,
movement of kink on a rope, wave set up on the surface of water.
- Longitudinal waves
If the particles of a medium vibrate in the direction of wave propagation, the wave is called
longitudinal waves. It travels in the form of compression and rarefaction.
Sound waves are example of longitudinal waves.
MOTION
For simple harmonic motion, the force is proportional to the mean position. We can write
But, acceleration for SHM can be written as
2
So, on comparing two equations, we get,
2
2
Pendulum and spring under force due to suspended mass are the examples of SHM. In this type of
motion, rate of acceleration is directly proportional to its displacement from mean position. Further,
SHM can be represented by means of vector of magnitude A rotating at constant angular velocity as
shown. This type of motion is represented by a sinusoidal curve as shown in figure and hence the
instantaneous displacement at any time can be expressed by following equation
𝑦 = 𝐴 sin(𝜔𝑡 + 𝜃)
where A is the displacement from the mean position at time ‘t’.
If phase angle is assumed to be zero, then the equation becomes
𝑦 = 𝐴 sin(𝜔𝑡)
Therefore, velocity at any instant ‘t’ is given by
= 𝐴𝜔 cos(𝜔𝑡)
Further, acceleration at any instant 't' is given by
2
2
2
sin
