Table of
Contents
* Work, Energy, and Power
* Impulse and Momentum
* Kinematics
Physics
introduction
to
Kinematics
* Dynamics and Rotational Motion
* Elastic properties of Solids and
Liquids
* Elastic properties of Solids and
Liquids 2
* Vibratory Motion
* Fluids
* Heat Transfer
* Waves
* Electrostatics
* Magnetism
* Optics
Work, Energy, and Power
1.1 Work
A. Definition
- Work is the measure of energy transfer that occurs when an object
is
moved over a distance by an external force at least part of which is
applied in
the direction of the displacement. If the force is constant, work may
be
computed by multiplying the length of the path by the component of
the force
acting along the path. B. Calculation formula
- To express this concept mathematically, the work W is equal to the
force f times the distance d, or W = fd. If the force is being exerted
at an
angle θ to the displacement, the work done is W = fd cos θ. C.
Equivalent Units
- The SI unit of work is joule (J). Joule is defined as the work done
by a force
of one newton causing a displacement of one meter. Sometimes,
newton- metre (N-m) is also used for measuring work. However, as
this unit is also
used for torque it can get quite confusing. Thus, SI authority does
not
encourage anyone to use this unit. Following is the table of units
and dimensional formula:
SI unit N.m Joule
CGS unit dyne-cm Erg
Dimensional formula ML2T-2 –
1.2 Energy
A. Definition
- Energy is defined as the “ability to do work, which is the ability to
exert a
force causing displacement of an object.” Despite this confusing
definition, its
meaning is very simple: energy is just the force that causes things
to move. B. Calculation formula
- K.E.=12×m×v2
- P.E.=m×g×h
C. Equivalent Units
- As power doesn’t have any direction, it is a scalar quantityT.he SI
unit of
power is Joules per Second (J/s), which is termed as Watt. Watt can
be
defined as the power taken to do one joule of work in one second.
The unit
Watt is dedicated in honour of So James Watt, the developer of the
steam
engine. A garage hoist lifts a truck up 2 meters above the ground in
15 seconds. Find
the power delivered to the truck. [Given: 1000 kg as the mass of the
truck]
First we need to calculate the work done, which requires the force
necessary
to lift the truck against gravity:
F = mg = 1000 x 9.81 = 9810 N. W = Fd = 9810N x 2m = 19620 Nm =
19620 J. The power is P = W/t = 19620J / 15s = 1308 J/s
Equivalent Units
SI unit watt (W)
In SI base units (kg)(m^2)(s^-3)
2.0 Potential and Kinetic Energy
2.1 Definition
- The best way to think about them is that potential energy occurs
before an
action, and kinetic energy happens during an action. Imagine you
are holding
your physics textbook up in the air. It has the potential to drop, just
because of
its high position. If you let the textbook drop, the potential energy is
converted
into kinetic energy – the energy in the movement itself. 2.2
Calculation
- Kinetic Energy: The energy exists due to the motion of an object is
known as
Kinetic Energy. For example, a moving van, flowing water, etc.
K.E.=12×m×v2
Where, K.E. Kinetic Energy
m Mass of the object
v The velocity of the object
Potential Energy: This is the energy stored in an object due to its
position and
height. It is measured by the amount of work done. For example, a
book on a
table, water stored in a lake, etc. P.E.=m×g×h
Where, P.E. Potential Energy
m Mass of the object
g Acceleration due to gravity
h Height
3.0 Friction
Definition and Formula
Friction, force that resists the sliding or rolling of one solid object
over
another. Frictional forces, such as the traction needed to walk
without slipping, maybe beneficial , but they also present a great
measure of opposition
to motion . This constant ratio is called the coefficient of friction
and is usually symbolized
by the Greek letter mu (μ). Mathematically, μ = F/L. Because both
friction and
load are measured in units of force (such as pounds or newtons),
the coefficient of friction is dimensionless. 3.1 Sliding Friction
- Sliding friction is the resistance created by any two objects when
sliding
against each other. This friction is also known as kinetic friction and
is defined
as the force that is needed to keep a surface sliding along another
surface. It
depends on two variables- one is material and the other is the
weight of the
object. Any change in the surface area in contact does not change
the sliding
friction. In most of the materials, sliding friction is less than static
friction. There are exceptions that include metals having static and
sliding friction
coefficients and are essentially the same with small surfaces where
molecular
attraction forces take over. 3.2 Static Friction
- Acts between surfaces at rest with respect to each other. The
value of static
friction varies between zero and the smallest force needed to start
motion. This smallest force required to start motion, or to overcome
static friction, is
always greater than the force required to continue the motion, or to
overcome
kinetic friction.
3.3 Rolling Friction
- For a moving solid body, there are two principal types of friction
that act
surface is known
as rolling friction or rolling resistance. Rolling of ball or wheel on the
ground is an example of Rolling frictio
is sliding friction. In this type of friction, there
is a restriction on the body’s movement as only one side of the body
is
in contact with the surface. Pushing a box across the table is an
example of Sliding friction. Rolling friction is considerably weaker
than sliding friction. Laws of Rolling Friction
There are three laws of rolling friction: - With the increase in
smoothness, the force of rolling friction decreases. - Rolling friction
is expressed as a product of load and constant to the
fractional power. F = kLn
- Rolling friction force is directly proportional to load and inversely
proportional
to the radius of curvature. F=μ×Wr
3.4 Fluid Friction
- Fluid friction describes the friction between layers of a viscous
fluid that are
moving relative to each other. 3.5 Stopping Distance
- When the body is moving with a certain velocity and suddenly one
applies
brakes. You will observe that the body stops entirely after covering
a certain
distance. This is stopping distance. The stopping distance is the
distance covered between the time when the body
decides to stop a moving vehicle and the time when the vehicle
stops entirely. The stopping distance relates to factors containing
road surface, and reflexes of
the car’s driver and it is denoted by d. The SI unit for stopping
distance meters. The Formula for Stopping Distance:
Stopping Distance formula is given by,
d= v22μg
Where
v velocity
μ friction coefficient
g acceleration due to gravity
d distance
The stopping distance formula is also given by, d= kv2
Where, k a constant of
proportionality
v speed
d distance
3.6 Coefficient of Kinetic Friction
- The coefficient of friction is a dimensionless scalar value. It is a
ratio of the
force of friction between two bodies and the force pressing them
together. 3.7 Limiting angle (angle repose)
- The term has a related usage in mechanics, where it refers to the
maximum
angle at which an object can rest on an inclined plane without
sliding down. This angle is equal to the arctangent of the coefficient
of static friction μs
between the surfaces. 4. Conservation of Energy
- Conservation of energy, principle of physics according to which the
energy
of interacting bodies or particles in a closed system remains
constant. ... When the pendulum swings back down, the potential
energy is converted
back into kinetic energy. At all times, the sum of potential and
kinetic energy is
constant.
5. Transformations of Kinetic and Potential Energy. - You can
transfer energy to an object by doing work on that object. ... When
you drop a book, gravitational potential energy is transformed into
kinetic
energy. Your car transforms the chemical potential energy stored in
gasoline
into the kinetic energy of the car's motion
6. Actual Mechanical Advantage
- The actual mechanical advantage (AMA) is the mechanical
advantage
determined by physical measurement of the input and output forces.
Actual
mechanical advantage takes into account energy loss due to
deflection,
friction, and wear. 7. Ideal Mechanical Advantage
- The ideal mechanical advantage (IMA) of an inclined plane is the
length of
the incline divided by the vertical rise, the so-called run-to-rise ratio.
The
mechanical advantage increases as the slope of the incline
decreases, but
then the load will have to be moved a greater distance. 8. Efficiency
- Efficiency is a comparison of the energy output to the energy input
in a given
system. It is defined as the percentage ratio of the output energy to
the input
energy, given by the equation: This equation is commonly used in
order to
represent energy in the form of heat or power.
Impulse and Momentum
1.Impulse
Impulse is a term that quantifies the overall effect of a force acting
over time. It
is conventionally given the symbol J, and expressed in Newtonseconds.
For a constant force, J=F⋅ Δt
When we calculate impulse, we are multiplying force by time. This is
equivalent
to finding the area under a force-time curve. This is useful because
the area
can just as easily be found for a complicated shape—variable force—
as for a
simple rectangle—constant force. It is only the overall net impulse
that matters
for understanding the motion of an object following an impulse.
The concept of impulse that is both external and internal to a
system is also
fundamental to understanding conservation of momentum.
2. Momentum
Momentum is a word that we hear used colloquially in everyday life.
We are
often told that sports teams and political candidates have "a lot of
momentum".
In this context, the speaker usually means to imply that the team or
candidate
has had a lot of recent success and that it would be difficult for an
opponent to
change their trajectory. This is also the essence of the meaning in
physics,
though in physics we need to be much more precise.
Momentum is a measurement of mass in motion: how much mass is
in how
much motion. It is usually given the symbol of P.
Formula: p=m⋅ v where m is the mass and v is the velocity. The
standard units
for momentum are kg•m/s and momentum is always a vector
quantity.
3. Law of Conservation of Momentum
The law of conservation of momentum states as for a collision
occurring
between object 1 and object 2 in an Isolated system, the total
momentum of
the two objects before the collision is equal to the total momentum
of the two
objects after the collision.
The total momentum of a collection of objects is conserved – that is,
the total
amount of momentum is a constant or unchanging value. p = mv.
This is where
p is the momentum of an object, measured in kilogram meters per
second; m
is the mass of that object, measured in kilograms; and v is the
velocity of the
object, measured in meters per second. The law of conservation of
momentum
is generously confirmed by experiment and can even be
mathematically
deduced on the reasonable presumption that space is uniform.
Conservation of momentum is a major law of physics which states
that the
momentum of a system is constant if no external forces are acting
on the
system. It is embodied in Newton’s First Law or The Law of Inertia.
4. Coefficient of Restitution
The ratio of the relative velocity after impact to the relative velocity
before the
impact of two colliding bodies, equal to 1 for an elastic collision and
0 for an
inelastic collision.
A parameter associated with the behaviour of two bodies during a
collision.
Suppose that two billiard balls are travelling in the same straight
line and have
velocities u 1 and u 2 before the collision, and velocities v 1 and v 2
after the
collision. If the coefficient of restitution is e, then
v 2−v 1=−e(u 2−u 1).
This formula is Newton's law of restitution. The coefficient of
restitution always
satisfies 0≤e≤1. When e=0, the balls remain in contact after the
collision. When
e=1, the collision is elastic: there is no loss of kinetic energy.
5. Moment of Momentum
In a similar way, if a particle at position r has linear momentum
p=mv, its
moment of momentum with respect to the origin is the vector l
defined by
l=r×p(3.3.1) and its components are the moments of momentum with
respect to
the axes. Moment of momentum plays a role in rotational motion
analogous to
the role played by linear momentum in linear motion, and is also
called angular
momentum. The dimensions of angular momentum are ML2T−1.
Several
choices for expressing angular momentum in SI units are possible;
the usual
choice is J s (joule seconds).
6.Elastic Collision
An elastic collision is defined as one in which both conservation of
momentum
and conservation of kinetic energy are observed. This implies that
there is no
dissipative force acting during the collision and that all of the
kinetic energy of
the objects before the collision is still in the form of kinetic energy
afterward.
7. Conservation of Angular Momentum
The law of conservation of angular momentum states that when no
external
torque acts on an object, no change of angular momentum will
occur.Angular
Momentum- A vector quantity describing an object in circular motion;
its
magnitude is equal to the momentum of the particle, and the
direction is
perpendicular to the plane of its circular motion.
8. Second Law of Motion
The second law of motion states that the rate of change of
momentum of a
body over time is directly proportional to the force applied, and
occurs in the
same direction as the applied force. F=dp/dt where p is the
momentum of the
body.
According to Newton’s second law, i.e, Fnet = m. An acceleration of
a body is
directly proportional to the net force that acts on the body and
inversely
proportional to the mass. If combined with the acceleration (a = ?v /
t), the
resultant equation is described as:
F = m •a
or
F = m • ?v / t
Constant Mass
For objects and systems with constant mass, the second law can be
re-stated
in terms of an object's acceleration.
F=d(mv)/dt= m(dv/dt)=ma, where F is the net force applied, m is the
mass of
the body, and a is the body's acceleration. Thus, the net force
applied to a
body produces a proportional acceleration.
Variable-mass systems
Variable-mass systems, like a rocket burning fuel and ejecting spent
gases,
are not closed and cannot be directly treated by making mass a
function of
time in the second law; The equation of motion for a body whose
mass m
varies with time by either ejecting or accreting mass is obtained by
applying
the second law to the entire, constant-mass system consisting of
the body and
its ejected or accreted mass; the result is
F+u(dm/dt)=m(dv/dt)
9.Precession
Precession, phenomenon associated with the action of a gyroscope
or a
spinning top and consisting of a comparatively slow rotation of the
axis of
rotation of a spinning body about a line intersecting the spin axis.
The smooth,
slow circling of a spinning top is precession, the uneven wobbling is
nutation.
10.Gyroscopes
Gyroscope, device containing a rapidly spinning wheel or circulating
beam of
light that is used to detect the deviation of an object from its
desired orientation.
Gyroscopes are used in compasses and automatic pilots on ships
and aircraft,
in the steering mechanisms of torpedoes, and in the inertial
guidance systems
installed in space launch vehicles, ballistic missiles, and orbiting
satellites
KINEMATICS
Physics introduction to Kinematics.
1. Quantities
- A physical quantity is any phenomenon that can be measured with
an instrument or be calculated for. A physical quantity can be
expressed
as a value, which is the algebraic multiplication of a numerical value
and
a unit . For example, the physical quantity mass can be quantified
as n kg ,where n is the numerical value and kg is the unit. A physical
quantity
possesses at least two characteristics in common, one is the
numerical
magnitude and the other is the unit in which it is measured.
2. Units
Units used in Kinematics
• length unit – meter (1 m),
• time – second (1 s)
• weight – kilogram (1 kg),
• the amount of substance is mol (1 mol),
• temperature – kelvin (1 K),
• electric current – ampere (1 A)
• Reference: luminous intensity – candela (1 cd, is not actually used
when
solving
school problems).
3. Vectors
- In mathematics, physics, and engineering, a vector is a geometric
object that
has a magnitude
(or length) and direction and can be added to other vectors
according to vector
algebra. The
direction of a vector in one-dimensional motion is given simply by a
plus (+) or
minus (−) sign.
4. Displacement
• Displacement is the change in position of an object relative to its
reference
frame. For
example, if a car moves from a house to a grocery store, its
displacement is
the relative
distance of the grocery store to the reference frame, or the house.
The word
“displacement” implies that an object has moved or has been
displaced.
Displacement is
the change in position of an object and can be represented
mathematically as
follows:
Δx=xf−x0Δx=xf−x0
where Δx is displacement, xf is the final position, and x0 is the
initial position.
5. Velocity
• There are a variety of quantities associated with the motion of
objects displacement (and
distance), velocity (and speed), acceleration, and time. Knowledge
of each of
these
quantities provides descriptive information about an object's motion.
For
example, if a
car is known to move with a constant velocity of 22.0 m/s, North for
12.0
seconds for a
northward displacement of 264 meters, then the motion of the car is
fully
described. And
if a second car is known to accelerate from a rest position with an
eastward
acceleration
of 3.0 m/s2 for a time of 8.0 seconds, providing a final velocity of 24
m/s, East
and an
eastward displacement of 96 meters, then the motion of this car is
fully
described. These
two statements provide a complete description of the motion of an
object.
However, such
completeness is not always known. It is often the case that only a
few
parameters of an
object's motion are known, while the rest are unknown. For example
as you
approach the
stoplight, you might know that your car has a velocity of 22 m/s,
East and is
capable of a
skidding acceleration of 8.0 m/s2, West. However you do not know
the
displacement that
your car would experience if you were to slam on your brakes and
skid to a
stop; and you
do not know the time required to skid to a stop. In such an instance
as this, the
unknown
parameters can be determined using physics principles and
mathematical
equations (the
kinematic equations).
6. Acceleration
- is a vector quantity that is defined as the rate at which an object
changes its
velocity.
An object is accelerating if it is changing its velocity.
Acceleration units are expressed as length per time divided by time
such as
meters/second/second or in abbreviated form as m/s 2 .
Calculating the Average Acceleration
The average acceleration (a) of any object over a given interval of
time (t) can
be
calculated using the equation
7. Graphing Motion
- First note that graphs in this text have perpendicular axes, one
horizontal and
the other
vertical. When two physical quantities are plotted against one
another in such
a graph,
the horizontal axis is usually considered to be an independent
variable and the
vertical
axis a dependent variable. If we call the horizontal axis the x-axis
and the
vertical axis
the y-axis, as in Figure 1, a straight-line graph has the general form
y=mx+b
DYNAMICS AND ROTATIONAL MOTION
I Concept of Dynamics
1. Newton's First and Second Laws of Motion.
Newton’s first law states that, if a body is at rest or moving at a
constant speed
in a straight line, it will remain at rest or keep moving in a straight
line at
constant speed unless it is acted upon by a force. This postulate is
known as
the law of inertia. The law of inertia was first formulated by Galileo
Galilei for
horizontal motion on Earth and was later generalized by René
Descartes.
Before Galileo it had been thought that all horizontal motion
required a direct
cause, but Galileo deduced from his experiments that a body in
motion would
remain in motion unless a force (such as friction) caused it to come
to rest.
Newton's second law of motion pertains to the behavior of objects
for which all
existing forces are not balanced. The second law states that the
acceleration
of an object is dependent upon two variables - the net force acting
upon the
object and the mass of the object. The acceleration of an object
depends
directly upon the net force acting upon the object, and inversely
upon the mass
of the object. As the force acting upon an object is increased, the
acceleration
of the object is increased. As the mass of an object is increased, the
acceleration of the object is decreased.
2. Newton's Third Law of Motion and Tension Forces.
The third law states that for every action (force) in nature there is
an equal and
opposite reaction. If object A exerts a force on object B, object B
also exerts an
equal and opposite force on object A. In other words, forces result
from
interactions.
The tension force is the force that is transmitted through a string,
rope, cable or
wire when it is pulled tight by forces acting from opposite ends. The
tension
force is directed along the length of the wire and pulls equally on
the objects on
the opposite ends of the wire.
3. Applications of Newton's Laws of Motion.
The final common application of Newton's Laws deals with tension.
Tension
usually arises in the use of ropes or cables to transmit a force.
Consider a
block being pulled by a rope. The person doing the pulling at one end
of the
rope is not in contact with the block, and cannot exert a direct force
on the
block. Rather a force is exerted on the rope, which transmits that
force to the
block. The force experienced by the block from the rope is called
the tension
force.
Almost all situations you will be presented with in classical
mechanics deal
with massless ropes or cables. If a rope is massless, it perfectly
transmits the
force from one end to the other: if a man pulls on a massless rope
with a force
of 10 N the block will also experience a force of 10 N. An important
property of
massless ropes is that the total force on the rope must be zero at all
times. To
prove this, we go back to Newton's Second Law. If a net force acts
upon a
massless rope, it would cause infinite acceleration, as a = F/m, and
the mass
of a massless rope is 0. Such a situation is physically impossible
and,
consequently, a massless rope can never experience a net force.
Thus all
massless ropes always experience two equal and opposite tension
forces. In
the case of a man pulling a block with a rope, the rope experiences
a tension in
one direction from the pull of the man, and a tension in the other
direction from
the reactive force of the block:
Newton’s three laws of motion, the foundation of classical
mechanics, can be
stated very simply, as we have seen. But applying these laws to
situations
such as a locomotive, a suspension bridge, a car rounding a banked
curve, or
a toboggan sliding down a hill requires specific problem-solving
skills. Although
we will not introduce any new principles in this chapter, we will help
you
develop the skills you will need in order to solve problems with
Newton’s laws
of motion. We begin with equilibrium problems, concentrating on
systems at
rest. Then we generalize our problem-solving techniques to include
systems
that are not in equilibrium, for which we need to deal precisely with
the
relationships between forces and motion. We’ll learn how to
describe and
analyze the contact force that acts on an object when it rests or
slides on a
surface, as well as the elastic forces that are present when a solid
object is
deformed. Finally, we take a brief look at the fundamental nature of
force and
the kinds of forces found in the physical universe.
4. Friction
The resistance to motion of one object moving relative to another. It
is not a
fundamental force, like gravity or electromagnetism. Instead,
scientists believe
it is the result of the electromagnetic attraction between charged
particles in
two touching surfaces.
5.Springs
Elastic object that stores mechanical energy. Springs are typically
made of
spring steel. There are many spring designs. In everyday use, the
term often
refers to coil springs. When a conventional spring, without stiffness
variability
features, is compressed or stretched from its resting position, it
exerts an
opposing force approximately proportional to its change in length
(this
approximation breaks down for larger deflections). The rate or
spring constant
of a spring is the change in the force it exerts, divided by the change
in
deflection of the spring. That is, it is the gradient of the force versus
deflection
curve. An extension or compression spring's rate is expressed in
units of force
divided by distance, for example or N/m or lbf/in. A torsion spring is a
spring
that works by twisting; when it is twisted about its axis by an angle,
it produces
a torque proportional to the angle. A torsion spring's rate is in units
of torque
divided by angle, such as N·m/rad or ft·lbf/degree. The inverse of
spring rate is
compliance, that is: if a spring has a rate of 10 N/mm, it has a
compliance of 0.1
mm/N. The stiffness (or rate) of springs in parallel is additive, as is
the
compliance of springs in series.
6. Simple Pendulum
A simple pendulum consists of a relatively massive object hung by a
string
from a fixed support. It typically hangs vertically in its equilibrium
position. The
massive object is affectionately referred to as the pendulum bob.
II. Concept of Rotational Motion
1. Rotational Kinematics
Kinematics is the description of motion. The kinematics of
rotational motion
describes the relationships among rotation angle, angular velocity,
angular
acceleration, and time.
Rotational motion Examples:
• Motion of wheel, gears, motors, etc is rotational motion.
• Motion of the blades of the helicopter is also rotatory motion.
• A door, swiveling on its hinges as you open or close it.
• A spinning top, motion of a Ferris Wheel in an amusement park.
2. Torque
Torque is a measure of the force that can cause an object to rotate
about an
axis. Just as force is what causes an object to accelerate in linear
kinematics,
torque is what causes an object to acquire angular acceleration.
Torque is a
vector quantity.
Torque is defined as Γ = r × F = r Fsin(θ)
In other words, torque is the cross product between the distance
vector (the
distance from the pivot point to the point where force is applied) and
the force
vector, 'a' being the angle between r and F.
ELASTIC PROPERTIES OF SOLIDS AND LIQUIDS
I Concepts of Elastic Properties of Solids and Liquids.
- In physics and materials science, elasticity is the ability of a body
to resist a distorting
influence and to return to its original size and shape when that
influence or force is
removed. Solid objects will deform when adequate loads are applied
to them; if the
material is elastic, the object will return to its initial shape and size
after removal.
This is in contrast to plasticity, in which the object fails to do so and
instead remains
in its deformed state. The physical reasons for elastic behavior can
be quite different
for different materials. In metals, the atomic lattice changes size
and shape when
forces are applied (energy is added to the system). When forces are
removed, the
lattice goes back to the original lower energy state. For rubbers and
other polymers,
elasticity is caused by the stretching of polymer chains when forces
are applied. In
engineering, the elasticity of a material is quantified by the elastic
modulus such as
the Young’s modulus, bulk modulus or shear modulus which
measure the amount of
stress needed to achieve a unit of strain; a higher modulus indicates
that the material
is harder to deform. The SI unit of this modulus is the pascal (Pa).
The material’s
elastic limit or yield strength is the maximum stress that can arise
before the onset of
plastic deformation. Its SI unit is also the pascal (Pa).
II Elasticity
- Elasticity is the property of solid materials to return to their
original shape and
size after the forces deforming them have been removed. You would
have noticed
that when an external force is applied on an object, its shape or size
(or both) change,
i.e. deformation takes place. The extent of deformation depends on
the material and
shape of the body and the external force. When the deforming forces
are withdrawn,
the body tries to regain its original shape and size.
A body which regains its original state completely on removal of the
deforming
force is called perfectly elastic. On the other hand, if it completely
retains its modified
form even on removing the deforming force, i.e. shows no tendency
to recover the
deformation, it is said to be perfectly plastic. However, in practice
the behavior of all
bodies is in between these two limits. There exists no perfectly
elastic or perfectly
plastic body in nature. The nearest approach to a perfectly elastic
body is quartz fiber
and to the perfectly plastic is ordinary putty. Here it can be added
that the object
which opposes the deformation more is more elastic. No doubt
elastic deformations
are very important in science and technology, but plastic
deformations are also
important in mechanical processes. You might have seen the
processes such as
stamping, bending and hammering of metal pieces. These are
possible only due to
plastic deformations. Due to inter-atomic forces, solid takes such a
shape that each
atom remains in a stable equilibrium. When the body is deformed,
the atoms are
displaced from their original positions and the inter-atomic
distances change. If in
deformation, the separation increases beyond their equilibrium
separation
(i.e., R > R0), strong attractive forces are developed. However, if
inter–atomic
separation decreases (i.e. R < R0), strong repulsive forces develop.
These forces,
called restoring forces, drive atoms to their original positions. The
behavior of atoms
in a solid can be compared to a system in which balls are connected
with springs.
III Hooke's Law
- Hooke’s Law is a principle of physics that states that the that the
force needed to extend
or compress a
spring by some distance is proportional to that distance. The law is
named after 17th
century British
physicist Robert Hooke, who sought to demonstrate the relationship
between the forces
applied to a spring and its elasticity. He first stated the law in 1660
as a Latin anagram, and
then published the solution in 1678 as ut tensio, sic vis – which
translated, means “as the
extension, so the force” or “the extension is proportional to the
force”).
This can be expressed mathematically as F= -kX, where F is the
force applied to the
spring (either in the form of strain or stress); X is the displacement
of the spring, with a
negative value demonstrating that the displacement of the spring
once it is stretched; and
k is the spring constant and details just how stiff it is.
Hooke’s law is the first classical example of an explanation of
elasticity – which is the
property of an object or material which causes it to be restored to
its original shape after
distortion. This ability to return to a normal shape after experiencing
distortion can be
referred to as a “restoring force”. Understood in terms of Hooke’s
Law, this restoring force
is generally proportional to the amount of “stretch” experienced.
SI Unit: N/m or kg/s2
IV Stress and Strain
- In mechanics, stress is defined as a force applied per unit area. It
is given by the
formula
σ=FA
where,
σ is the stress applied
F is the force applied
A is the area of force application
The unit of stress is N/m2
Stress applied to a material can be of two types. They are:
● Tensile Stress: It is the force applied per unit area which results in
the increase in
length (or area) of a body. Objects under tensile stress become
thinner and longer.
● Compressive Stress: It is the force applied per unit area which
results in the
decrease in length (or area) of a body. The object under compressive
stress becomes
thicker and shorter.
What is Strain?
According to the strain definition, it is defined as the amount of
deformation experienced
by the body in the direction of force applied, divided by initial
dimensions of the body.
The relation for deformation in terms of length of a solid is given
below.
ε=δlL
where,
ε is the strain due to stress applied
δl is the change in length
L is the original length of the material.
The strain is a dimensionless quantity as it just defines the relative
change in shape.
Depending on stress application, strain experienced in a body can
be of two types. They
are:
● Tensile Strain: It is the change in length (or area) of a body due to
the application
of tensile stress.
● Compressive Strain: It is the change in length (or area) of a body
due to the
application of compressive strain
When we study solids and their mechanical properties, information
regarding
their elastic properties is most important. These can be obtained by
studying the stressstrain relationships, under different loads, in these materials.
V Young's Modulus
- Young’s modulus, numerical constant, named for the 18th-century
English physician and
physicist Thomas Young, that describes the elastic properties of a
solid undergoing
tension or compression in only one direction, as in the case of a
metal rod that after being
stretched or compressed lengthwise returns to its original length.
Young’s modulus is a
measure of the ability of a material to withstand changes in length
when under lengthwise
tension or compression. Sometimes referred to as the modulus of
elasticity, Young’s
modulus is equal to the longitudinal stress divided by the strain.
Stress and strain may be
described as follows in the case of a metal bar under tension.
If a metal bar of cross-sectional area A is pulled by a force F at each
end, the bar stretches
from its original length L0 to a new length Ln. (Simultaneously the
cross section
decreases.) The stress is the quotient of the tensile force divided by
the cross-sectional
area, or F/A. The strain or relative deformation is the change in
length, Ln – L0, divided by
the original length, or (Ln – L0)/L0. (Strain is dimensionless.) Thus
Young’s modulus may
be expressed mathematically as
Young’s modulus = stress/strain = (FL0)/A(Ln – L0).
SI unit : Pascal
VI Elasticity: Bulk Modulus
Bulk modulus, numerical constant that describes the elastic
properties of
a solid or fluid when it is under pressure on all surfaces. The applied
pressure
reduces
the volume of a material, which returns to its original volume when
the pressure is
removed. Sometimes referred to as the incompressibility, the bulk
modulus is a
measure of the ability of a substance to withstand changes in
volume when under
compression on all sides. It is equal to the quotient of the applied
pressure divided by
the relative deformation.
In this case, the relative deformation, commonly called strain , is
the change in
volume divided by the original volume. Thus, if the original volume V
o of a material
is reduced by an applied pressure p to a new volume V n , the strain
may be expressed
as the change in volume, V o − V n , divided by the original volume,
or (V o − V n )/V o .
When the bulk modulus is constant (independent of pressure), this is
a specific
form of Hooke’s law of elasticity .
Because the denominator, strain, is a ratio without dimensions, the
dimensions of the
bulk modulus are those of pressure, force per unit area. In the
English system the bulk
modulus may be expressed in units of pounds per square inch
(usually abbreviated to
psi), and in the metric system , newtons per square metre (N/m 2 ),
or pascals.
The value of the bulk modulus for steel is about 2.3 × 10 7 psi, or 1.6
×
10 11 pascals, three times the value for glass . Thus, only one-third
the pressure is
needed to reduce a glass sphere the same amount as a steel sphere
of the same initial
size. Under equal pressure, the proportional decrease in volume of
glass is three times
that of steel. One may also say that glass is three times more
compressible than steel.
In fact, compressibility is defined as the reciprocal of the bulk
modulus. A substance
that is difficult to compress has a large bulk modulus but a small
compressibility. A
substance that is easy to compress has a high compressibility but a
low bulk modulus.
VII Elasticity of Shear
Shear modulus, numerical constant that describes the elastic
properties of
a solid under the application of transverse internal forces such as
arise, for
example, in torsion, as in twisting a metal pipe about its lengthwise
axis. Within
such a material any small cubic volume is slightly distorted in such
a way that two
of its faces slide parallel to each other a small distance and two
other faces change
from squares to diamond shapes. The shear modulus is a measure of
the ability of
a material to resist transverse deformations and is a valid index of
elastic behavior
only for small deformations, after which the material is able to
return to its
original configuration. Large shearing forces lead to flow and
permanent
deformation or fracture. The shear modulus is also known as the
rigidity.
• Mathematically the shear modulus is equal to the quotient of the
shear
stress divided by the shear strain. The shear stress, in turn, is equal
to the shearing
force F divided by the area A parallel to and in which it is applied, or
F/A. The
shear strain or relative deformation is a measure of the change in
geometry and in
this case is expressed by the trigonometric function, tangent (tan)
of the
angle θ (theta), which denotes the amount of change in the 90°, or
right, angles of
the minute representative cubic volume of the unstrained material.
Mathematically, shear strain is expressed as tan θ or its equivalent,
by
definition, x/y. The shear modulus itself may be expressed
mathematically as
shear modulus = (shear stress)/(shear strain) = (F/A)/(x/y)
This equation is a specific form of Hooke’s law of elasticity.
Because the
denominator is a ratio and thus dimensionless, the dimensions of
the shear
modulus are those of force per unit area. In the English system the
shear modulus
may be expressed in units of pounds per square inch (usually
abbreviated to psi);
the common SI units are newtons per square metre (N/m2). The
value of the shear
modulus for aluminum is about 3.5 × 10^6 psi, or 2.4 × 10^10 N/m^2.
By comparison, steel
under shear stress is more than three times as rigid as aluminum.
ELASTIC PROPERTIES OF SOLIDS AND LIQUIDS
I. Types of Stress-Strain Relations
Tensile Stress-Strain Relationship in Materials:
Tensile stresses can be originated from axial forces. So in
standardized
conditions, materials tensile characteristics can be explored. You
can learn
these conditions, tests and tensile stress-strain curves in that
article.
Compression Stress-Strain Relationship in Materials:
Also there can be compression of materials in engineering
applications. So in
literature, compressive strengths are defined for materials. You can
learn the
compression stress-strain in materials and curves of that, and the
compression
tests on standard material specimens in this article.
Bending(Flexure) Test of Material Specimens:
In bending, there are two of stress types exist. These stresses are
compression and tension. You can learn about the materials’ flexure
characteristics, bending stress-strain curves from this article.
Shear Properties of Materials; Shear Stress-Strain:
Shear stresses can occur in many engineering applications. Torque
generally
leads to the phenomenon called as shear stress on materials. In this
article,
what is shear stress on materials and how is the shear stress test is
applied on
test specimens.
II. Rolling Friction
Rolling friction occurs when a wheel, ball, or cylinder rolls freely
over a surface,
as in ball and roller bearings. The main source of friction in rolling
appears to
be dissipation of energy involved in deformation of the objects. If a
hard ball is
rolling on a level surface, the ball is somewhat flattened and the
level surface
somewhat indented in the regions in contact. The elastic
deformation or
compression produced at the leading section of the area in contact
is a
hindrance to motion that is not fully compensated as the substances
spring
back to normal shape at the trailing section. The internal losses in
the two
substances are similar to those that keep a ball from bouncing back
to the level
from which it is dropped. Coefficients of sliding friction are generally
100 to
1,000 times greater than coefficients of rolling friction for
corresponding
materials. This advantage was realized historically with the
transition from
sledge to wheel.
❖ Laws of Rolling Friction
There are three laws of rolling friction:
With the increase in smoothness, the force of rolling friction
decreases. Rolling
friction is expressed as a product of load and constant to the
fractional power.
F = kL n
Rolling friction force is directly proportional to load and inversely
proportional to
the radius of curvature.
F = μ×W/r
III. Thermal Stress
Thermal stress is mechanical stress created by any change in
temperature of
a material. These stresses can lead to fracturing or plastic
deformation
depending on the other variables of heating, which include material
types and
constraints.[1] Temperature gradients, thermal expansion or
contraction and
thermal shocks are things that can lead to thermal stress. This type
of stress is
highly dependent on the thermal expansion coefficient which varies
from
material to material. In general, the greater the temperature change,
the higher
the level of stress that can occur. Thermal shock can result from a
rapid
change in temperature, resulting in cracking or shattering.
Thermal Stress Formula
Consider a thermal conducting rod, on heating, the rod expands. The
change
in length will be directly proportional to the amount of heat supplied
and the
coefficient of thermal expansion. Thus, we can mathematically write
Thermal
stress as:
δT=Lα(Tf−Ti) δT=LαΔT
Where,
• Listhelengthinm
• Ti is the initial temperature in∘ C
• Tf is the final temperature in ∘ C
• ΔT=(Tf−Ti) is the change in temperature in ∘ C • Α is coefficient of
thermal
expansion in m/m∘ C
IV. Relation Among Elastic Constants
Young’s modulus, bulk modulus and Rigidity modulus of an elastic
solid are
together called Elastic constants. When a deforming force is acting
on a solid,
it results in the change in its original dimension. In such cases, we
can use the
relation between elastic constants to understand the magnitude of
deformation.
Elastic constant formula E=9KGG+3K
Where,
K is the Bulk modulus
G is shear modulus or modulus of rigidity.
E is Young’s modulus or modulus of Elasticity.
Derivation of relation between elastic constants
We can derive the elastic constant’s relation by combining the
mathematical
expressions relating terms individually.
Young modulus can be expressed using Bulk modulus and Poisson’s
ratio as
E=3K(1−2μ)
Similarly, Young’s modulus can also be expressed using rigidity
modulus and
Poisson’s ratio as
E=2G(1+2μ)
Combining the above two-equation and solving them to eliminate
Poisson’s
ratio we can get a relation between Young’s modulus and bulk
modulus k and
modulus of rigidity as
E=9KGG+3K
V. Elastic Behavior and Atomic Structure
The physical reasons for elastic behavior can be quite different for
different
materials. In metals, the atomic lattice changes size and shape
when forces
are applied (energy is added to the system). When forces are
removed, the
lattice goes back to the original lower energy state. For rubbers and
other
polymers, elasticity is caused by the stretching of polymer chains
when forces
are applied.
Hooke's law states that the force required to deform elastic objects
should be
directly proportional to the distance of deformation, regardless of
how large
that distance becomes. This is known as perfect elasticity, in which
a given
object will return to its original shape no matter how strongly it is
deformed.
This is an ideal concept only; most materials which possess
elasticity in
practice remain purely elastic only up to very small deformations,
after which
plastic (permanent) deformation occurs.
In engineering, the elasticity of a material is quantified by the
elastic modulus
such as the Young's modulus, bulk modulus or shear modulus which
measure
the amount of stress needed to achieve a unit of strain; a higher
modulus
indicates that the material is harder to deform. The SI unit of this
modulus is
the pascal (Pa). The material's elastic limit or yield strength is the
maximum
stress that can arise before the onset of plastic deformation. Its SI
unit is also
the pascal (Pa).
An atom is a complex arrangement of negatively charged electrons
arranged
in defined shells about a positively charged nucleus. This nucleus
contains
most of the atom's mass and is composed of protons and neutrons
(except for
common hydrogen which has only one proton). All atoms are roughly
the same
size. A convenient unit of length for measuring atomic sizes is the
angstrom (Å),
which is defined as 1 x 10-10 meters. The diameter of an atom is
approximately
2-3 Å.
In 1897, J. J. Thomson discovered the existence of the electron,
marking the
beginning of modern atomic physics. The negatively charged
electrons follow
a random pattern within defined energy shells around the nucleus.
Most
properties of atoms are based on the number and arrangement of
their
electrons. The mass of an electron is 9.1 x 10-31 kilograms.
One of the two types of particles found in the nucleus is the proton.
The
existence of a positively charged particle, a proton, in the nucleus
was proved
by Sir Ernest Rutherford in 1919. The proton's charge is equal but
opposite to
the negative charge of the electron. The number of protons in the
nucleus of
an atom determines what kind of chemical element it is. A proton
has a mass of
1.67 x 10- 27 kilograms. The neutron is the other type of particle
found in the
nucleus. It was discovered by a British physicist, Sir James
Chadwick. The
neutron carries no electrical charge and has the same mass as the
proton.
With a lack of electrical charge, the neutron is not repelled by the
cloud of
electrons or by the nucleus, making it a useful tool for probing the
structure of
the atom. Even the individual protons and neutrons have internal
structure,
called quarks. Six types of quarks exist. These subatomic particles
cannot be
freed and studied in isolation. Current research continues into the
structure of
the atom.
VI. Some Further Properties of Matter
Physical properties are properties that can be measured or observed
without
changing the chemical nature of the substance. Some examples of
physical
properties are:
▪ color (intensive)
▪ density (intensive)
▪ volume (extensive)
▪ mass (extensive)
▪ boiling point (intensive): the temperature at which a substance
boils
▪ melting point (intensive): the temperature at which a substance
melts
Matter has mass and volume, as demonstrated by this concrete
block. You can
observe its mass by feeling how heavy it is when you try to pick it
up; you can
observe its volume by looking at it and noticing its size. Mass and
volume are
both examples of extensive physical properties.
VIBRATORY MOTION
Concepts of Vibratory Motion and Oscillations.
Vibration is a mechanical phenomenon whereby oscillations occur
about an equilibrium point. The
oscillations may be periodic, such as the motion of a pendulum—or
random, such as the movement
of a tire on a gravel road.
Vibration can be desirable: for example, the motion of a tuning fork,
the reed in a woodwind
instrument or harmonica, a mobile phone, or the cone of a
loudspeaker.
In many cases, however, vibration is undesirable, wasting energy
and creating unwanted sound. For
example, the vibrational motions of engines, electric motors, or any
mechanical device in operation
are typically unwanted. Such vibrations could be caused by
imbalances in the rotating parts,
uneven friction, or the meshing of gear teeth. Careful designs
usually minimize unwanted
vibrations.
The studies of sound and vibration are closely related. Sound, or
pressure waves, are generated by
vibrating structures (e.g. vocal cords); these pressure waves can
also induce the vibration of
structures (e.g. ear drum). Hence, attempts to reduce noise are
often related to issues of vibration.
Oscillation is the repetitive variation, typically in time, of some
measure about a central value
(often a point of equilibrium) or between two or more different
states. The term vibration is
precisely used to describe mechanical oscillation. Familiar
examples of oscillation include a
swinging pendulum and alternating current.
Oscillations occur not only in mechanical systems but also in
dynamic systems in virtually every
area of science: for example the beating of the human heart (for
circulation), business
cycles in economics, predator–prey population cycles in ecology,
geothermal geysers in geology,
vibration of strings in guitar and other string instruments, periodic
firing of nerve cells in the brain,
and the periodic swelling of Cepheid variable stars in astronomy.
The Three (3) Types of Motion.
Translational motion
Motion that results in a change of location is said to be translational.
This category may seem
ridiculous at first as motion implies a change in location, but an
object can be moving and yet not
go anywhere. I get up in the morning and go to work (an obvious
change in location), but by
evening I'm back at home — back in the very same bed where I
started the day. Is this
translational motion? Well, it depends. If the problem at hand is to
determine how far I travel in a
day, then there are two possible answers: either I've gone to work
and back (22 km each
way for a total of 44 km) or I've gone nowhere (22 km each way for a
total of 0 km). The first
answer invokes translational motion while the second invokes
oscillatory motion.
Oscillatory motion
Motion that is repetitive and fluctuates between two locations is
said to be oscillatory. In the
previous example of going from home to work to home to work I am
moving, but in the end I
haven't gone anywhere. This second type of motion is seen in
pendulums (like those found in
grandfather clocks or Big Ben), vibrating strings (a guitar string
moves but goes nowhere), and
drawers (open, close, open, close — all that motion and nothing to
show for it). Oscillatory
motion is interesting in that it often takes a fixed amount of time for
an oscillation to occur. This
kind of motion is said to be periodic and the time for one complete
oscillation (or one cycle) is
called a period. Periodic motion is important in the study of sound,
light, and other waves. Large
chunks of physics are devoted to this kind repetitive motion. Doing
the same thing over and over
and going nowhere is pretty important. Which brings us to our next
type of motion.
Rotational motion
Motion that occurs when an object spins is said to be rotational. The
Earth is in a constant state of
motion, but where does that motion take it? Every twenty-four hours
it makes one complete rotation
about its axis. (Actually, it's a bit less than that, but let's not get
bogged down in details.) The sun
does the same thing, but in about twenty-four days. So do all the
planets, asteroids, and comets;
each with its own period. (Note that rotational motion too is often
periodic.) On a more
mundane level, boccie balls, phonograph records, and wheels also
rotate. That should be enough
examples to keep us busy for a while.
Periodic Motion
Periodic motion, in physics, motion repeated in equal intervals of
time. Periodic motion is
performed, for example, by a rocking chair, a bouncing ball, a
vibrating tuning fork, a swing in
motion, the Earth in its orbit around the Sun, and a water wave. In
each case the interval of time for
a repetition, or cycle, of the motion is called a period, while the
number of periods per unit time is
called the frequency. Thus, the period of the Earth’s orbit is one year,
and its frequency is one orbit
per year. A tuning fork might have a frequency of 1,000 cycles per
second and a period of 1
millisecond (1 thousandth of a second).
Simple harmonic motion is a special case of periodic motion. In the
examples given above, the
rocking chair, the tuning fork, the swing, and the water wave
execute simple harmonic motion, but
the bouncing ball and the Earth in its orbit do not.
Waves that can be represented by sine curves are periodic. If the
wave is propagated with a velocity
v and has a wavelength λ, then the period T is equal to wavelength
divided by velocity, or T= λ/v.
The frequency f is the reciprocal of the period; thus, f = 1/T = v/λ.
Simple Harmonic Motion
Simple harmonic motion, in physics, repetitive movement back and
forth through an equilibrium,
or central, position, so that the maximum displacement on one side
of this position is equal to the
maximum displacement on the other side. The time interval of each
complete vibration is the same.
The force responsible for the motion is always directed toward the
equilibrium position and is
directly proportional to the distance from it. That is, F = −kx, where
F is the force, x is the
displacement, and k is a constant. This relation is called Hooke’s
law.
A specific example of a simple harmonic oscillator is the vibration of
a mass attached to a
vertical spring, the other end of which is fixed in a ceiling. At the
maximum displacement −x, the
spring is under its greatest tension, which forces the mass upward.
At the maximum displacement
+x, the spring reaches its greatest compression, which forces the
mass back downward again. At
either position of maximum displacement, the force is greatest and
is directed toward
the equilibrium position, the velocity (v) of the mass is zero, its
acceleration is at a maximum, and
the mass changes direction. At the equilibrium position, the velocity
is at its maximum and the
acceleration (a) has fallen to zero. Simple harmonic motion is
characterized by this changing
acceleration that always is directed toward the equilibrium position
and is proportional to the
displacement from the equilibrium position. Furthermore, the
interval of time for each complete
vibration is constant and does not depend on the size of the
maximum displacement. In some form,
therefore, simple harmonic motion is at the heart of timekeeping.
To express how the displacement of the mass changes with time,
one can use Newton’s second
law, F = ma, and set ma = −kx. The acceleration a is the second
derivative of x with respect to
time t, and one can solve the resulting differential equation with x =
A cos ωt, where A is the
maximum displacement and ω is the angular frequency in radians
per second. The time it takes
the mass to move from A to −A and back again is the time it takes
for ωt to advance by 2π.
Therefore, the period T it takes for the mass to move from A to −A
and back again is ωT = 2π,
or T = 2π/ω. The frequency of the vibration in cycles per second is
1/T or ω/2π.
Many physical systems exhibit simple harmonic motion (assuming
no energy loss): an oscillating
pendulum, the electrons in a wire carrying alternating current, the
vibrating particles of the medium
in a sound wave, and other assemblages involving relatively small
oscillations about a position of
stable equilibrium.
Period, Frequency, and Amplitude.
Period
A period T is the time required for one complete cycle of vibration to
pass a given point. As the
frequency of a wave increases, the period of the wave decreases.
Frequency and Period are in
reciprocal relationships and can be expressed mathematically as:
Period equals the Total time
divided by the Number of cycles.
Frequency
Frequency, in physics, the number of waves that pass a fixed point
in unit time; also, the number of
cycles or vibrations undergone during one unit of time by a body in
periodic motion. A body in
periodic motion is said to have undergone one cycle or one vibration
after passing through a series
of events or positions and returning to its original state. See also
angular velocity; simple harmonic
motion. If the period, or time interval, required to complete one
cycle or vibration is 1/2
second, the frequency is 2 per second; if the period is 1/100 of an
hour, the frequency is 100 per hour.
In general, the frequency is the reciprocal of the period, or time
interval; i.e., frequency = 1/period
= 1/(time interval). The frequency with which the Moon revolves
around Earth is slightly more than
12 cycles per year. The frequency of the A string of a violin is 440
vibrations or cycles per second.
The symbols most often used for frequency are f and the Greek
letters nu (ν) and omega (ω). Nu is
used more often when specifying electromagnetic waves, such as
light, X-rays, and gamma rays.
Omega is usually used to describe the angular frequency—that is,
how much an object rotates or
revolves in radians per unit time. Usually, frequency is expressed in
the hertz unit, named in honour
of the 19th-century German physicist Heinrich Rudolf Hertz, one
hertz being equal to one cycle per
second, abbreviated Hz; one kilohertz (kHz) is 1,000 Hz, and one
megahertz (MHz) is 1,000,000 Hz.
In spectroscopy another unit of frequency, the wavenumber, the
number of waves in a unit of
distance, is sometimes used.
Amplitude
Amplitude, in physics, the maximum displacement or distance
moved by a point on a vibrating
body or wave measured from its equilibrium position. It is equal to
one-half the length of the
vibration path. The amplitude of a pendulum is thus one-half the
distance that the bob traverses in
moving from one side to the other. Waves are generated by vibrating
sources, their amplitude being
proportional to the amplitude of the source.
For a transverse wave, such as the wave on a plucked string,
amplitude is measured by the
maximum displacement of any point on the string from its position
when the string is at rest. For a
longitudinal wave, such as a sound wave, amplitude is measured by
the maximum displacement of
a particle from its position of equilibrium. When the amplitude of a
wave steadily decreases
because its energy is being lost, it is said to be damped.
The Circle Reference
The gauge's output can be displayed as a polar profile or graph,
which, although providing a simple
graphical representation, can be time consuming and subjective to
deduce real values from. As a
result, we'll need a way to analyze the data in order to get reliable
and consistent results. Because
we're seeking to analyze deviations from genuine circularity and
need a point of reference, it's
logical to try to fit a circle to our profile and base all of our
calculations on it.
There are a number of ways of assessing out of roundness using a
number of types of reference
circle. All reference circles are used to establish the centre of the
component. Roundness is then
established as the radial deviations from the component center.
Acceleration and Speed in Simple Harmonic Motion.
To study the energy of a simple harmonic oscillator, we need to
consider all the forms of energy.
Consider the example of a block attached to a spring, placed on a
frictionless surface, oscillating in
SHM. The potential energy stored in the deformation of the spring is
U=1/2kx2.
In a simple harmonic oscillator, the energy oscillates between
kinetic energy of the mass K=1/2mv2
and potential energy U=1/2kx2 stored in the spring. In the SHM of
the mass and spring system,
there are no dissipative forces, so the total energy is the sum of the
potential energy and kinetic
energy. In this section, we consider the conservation of energy of
the system. The concepts
examined are valid for all simple harmonic oscillators, including
those where the gravitational force
plays a role.
Period and Speed in Simple Harmonic Motion.
The oscillations of a system in which the net force can be described
by Hooke’s law are of
special importance, because they are very common. They are also
the simplest oscillatory systems.
Simple Harmonic Motion (SHM) is the name given to oscillatory
motion for a system where the net
force can be described by Hooke’s law, and such a system is called
a simple harmonic oscillator. If
the net force can be described by Hooke ’s law and there is no
damping (by friction or other
non-conservative forces), then a simple harmonic oscillator will
oscillate with equal displacement
on either side of the equilibrium position, as shown for an object on
a spring in Figure 1. The
maximum displacement from equilibrium is called the amplitude X.
The units for amplitude and
displacement are the same, but depend on the type of oscillation.
For the object on the spring, the
units of amplitude and displacement are meters; whereas for sound
oscillations, they have
units of pressure (and other types of oscillations have yet other
units). Because amplitude is the
maximum displacement, it is related to the energy in the oscillation.
Energy in Simple Harmonic Motion.
In the SHM of the mass and spring system, there are no dissipative
forces, so the total energy is the
sum of the potential energy and kinetic energy. In this section, we
consider the conservation of
energy of the system. The concepts examined are valid for all
simple harmonic oscillators,
including those where the gravitational force plays a role.
The Simple Pendulum, the Compound (Physical) Pendulum, and
Simple
Angular Harmonic Motion.
A simple pendulum is defined to have a point mass, also known as
the pendulum bob, which is
suspended from a string of length L with negligible mass. Here, the
only forces acting on the bob
are the force of gravity (i.e., the weight of the bob) and tension from
the string. The mass of the
string is assumed to be negligible as compared to the mass of the
bob.
A physical pendulum is any object whose oscillations is similar to
those of the simple pendulum,
but cannot be modeled as a point mass on a string, and the mass
distribution must be included into
the equation of motion. As for the simple pendulum, the restoring
force of the physical pendulum is
the force of gravity. Each pendulum has a back panel that fixes
separately to the Test Frame. The
back panel of each pendulum has an accurate scale and indicator,
referenced to pendulum pivot or
centre of mass points. This improves measurement accuracy essential for good results.
The simple pendulum has a choice of two spheres suspended by a
cord. An adjustable indicator
also acts as the pendulum pivot point - allowing you to adjust the
cord length in seconds. This can
also provide a quick visual demonstration of the effect of cord
length on period. Each sphere has a
different mass for comparison and an internal spring retainer so you
can easily swap them between
experiments.
Angular Simple Harmonic Motion
A body free to rotate about an axis can make angular oscillations.
For example, a photo frame or a
calendar suspended from a nail on the wall. If it is slightly pushed
from its mean position and
released, it makes angular oscillations.
Resonance
Resonance is a phenomenon in which an oscillator responds most
strongly to a driving force that
matches its own natural frequency of vibration. For example,
suppose a child is on a playground
swing with a natural frequency of 1 Hz. That is, if you pull the child
away from equilibrium, release
her, and then stop doing anything for a while, she&#39;ll oscillate at
1 Hz. If there was no friction,
as we assumed in section 2.5, then the sum of her gravitational and
kinetic energy would remain
constant, and the amplitude would be exactly the same from one
oscillation to the next. However,
friction is going to convert these forms of energy into heat, so her
oscillations would gradually die
out. To keep this from happening, you might give her a push once
per cycle, i.e., the frequency of
your pushes would be 1 Hz, which is the same as the swing&#39;s
natural frequency. As long as
you stay in rhythm, the swing responds quite well. If you start the
swing from rest, and then give
pushes at 1 Hz, the swing&#39;s amplitude rapidly builds up, as in
figure a, until after a while it
reaches a steady state in which friction removes just as much
energy as you put in over the course
of one cycle
The Physiological Effects of Vibration.
We can feel vibrations and know that people might be exposed to it.
But we cannot determine if
what we feel is going to be harmful. For that, we must measure
vibration exposure.
Vibration is the mechanical oscillations of an object about an
equilibrium point. The oscillations
may be regular such as the motion of a pendulum or random such as
the movement of a tire on a
gravel road. The study of health effects of vibration require
measures of the overall "pressure
waves" (vibration energy) generated by the vibrating equipment or
structure.
Vibration enters the body from the part of the body or organ in
contact with vibrating equipment.
When a worker operates hand-held equipment such as a chain saw
or jackhammer, vibration affects
hands and arms. Such an exposure is called hand-arm vibration
exposure. When a worker sits or
stands on a vibrating floor or seat, the vibration exposure affects
almost the entire body and is
called whole-body vibration exposure.
The risk of vibration induced injury depends on the average daily
exposure. An evaluation of
the risk takes into account the intensity and frequency of the
vibration, the duration (years) of
exposure and the part of the body which receives the vibration
energy.
Hand-arm vibration causes damage to hands and fingers. It appears
as damage to blood vessels,
nerves and joints in the fingers. The resulting condition is known as
white finger disease,
Raynaud's phenomenon or hand-arm vibration syndrome (HAVS). One
of the symptoms is that
affected fingers may turn white, especially when exposed to cold.
Vibration-induced white finger
disease also causes a loss of grip force and loss of sensitivity to
touch.
The health effect of whole-body vibration (WBV) is poorly understood.
Studies of drivers of
heavy vehicles have revealed an increased incidence of the
disorders of bowel and the circulatory,
musculoskeletal and neurological systems.
However, disorders of the nervous, circulatory and digestive
systems are not specific to
whole-body vibration exposure only. These disorders can be caused
by a combination of various
other working conditions and life style factors rather than by one
physical factor alone. More
information is available in the OSH. Answers document Vibration Health Effects that describes
the effects of hand-arm vibration and whole body vibration.
FLUIDS
A. Fluids at Rest
1. Concepts of Fluids
A fluid is a state of matter that yields to sideways or shearing forces.
Liquids
and gases are both fluids. Fluid statics is the physics of stationary
fluids. A fluid
is a liquid, gas, or other substance that deforms (flows) under an
applied shear
stress, or external force, in physics. They have a zero shear modulus,
or, to put
it another way, they are substances that cannot withstand any
shear stress.
Mechanics is the oldest physical science that deals with both
stationery and
moving boundaries under the influence of forces. The branch of the
mechanics
that deals with bodies at rest is called statics while the branch that
deals with
bodies in motion is called dynamics.
Fluid Mechanics is the science that deals with behavior of fluids at
rest (fluid
statics) or in motion (fluid dynamics) and the interaction of fluids
with solids or
other fluids at the boundaries.
2. Density, Specific Gravity/Relative Density and Weight Density
Density
A mass per unit volume. Density is measured simultaneously at an
infinite
number of points in the fluid, we would obtain an expression for the
density
distribution as a function of the space coordinates, at the given
instant. The
density at a point may also vary with time (as a result of work done
on or by the
fluid and/or heat transfer to the fluid).
Specific gravity is density of a substance (solid or fluid) compared to
an
accepted reference value, typically the maximum density of water.
Specific weight is defined as the weight of a substance per unit
volume.
3. Hydrostatic Pressure, Pressure Due to the Weight of a Liquid, A
Liquid
Seeks to its Own Level, Pressure in Liquids at Rest and Pressure in
Gases
Hydrostatic Pressure
Hydrostatic pressure is defined as the pressure exerted by a fluid at
equilibrium at any point of time due to the force of gravity.
Hydrostatic pressure is proportional to the depth measured from the
surface as
the weight of the fluid increases when a downward force is applied.
The fluid pressure can be caused by gravity, acceleration or forces
when in a
closed container. Consider a layer of water from the top of the bottle.
There is a
pressure exerted by the layer of water acting on the sides of the
bottle. As we
move down from the top of the bottle to the
4. Pascal's Law, Boyle's Law and Archimedes' Principle.
Pascal’s principle, also called Pascal’s law, in fluid (gas or liquid)
mechanics,
statement that, in a fluid at rest in a closed container, a pressure
change in one
part is transmitted without loss to every portion of the fluid and to
the walls of
the container. The principle was first enunciated by the French
scientist Blaise
Pascal.
Pressure is equal to the force divided by the area on which it acts.
According to
Pascal’s principle, in a hydraulic system a pressure exerted on a
piston
produces an equal increase in pressure on another piston in the
system. If the
second piston has an area 10 times that of the first, the force on the
second
piston is 10 times greater, though the pressure is the same as that
on the first
piston. This effect is exemplified by the hydraulic press, based on
Pascal’s
principle, which is used in such applications as hydraulic brakes.
Pascal also found that the pressure at a resting point in a fluid is the
same in all
directions; the pressure would be the same in all planes passing
through that
point. Pascal's principle, or Pascal's law, is the name given to this
fact.
Boyle's law, also known as Mariotte's law, is a relationship that
describes
how a gas compresses and expands at a constant temperature. This
empirical
relationship, proposed by scientist Robert Boyle in 1662, says that
under
constant temperature, the pressure (p) of a given quantity of gas
changes
inversely with its volume (v); i.e., pv = k, a constant.
Archimedes' principle, or the physical law of buoyancy, states that
anybody
totally or partially immersed in a fluid (gas or liquid) at rest is acted
upon by an
upward, or buoyant, force, the magnitude of which is equal to the
weight of the
fluid displaced by the body. The volume of displaced fluid is equal to
the
volume of an item completely submerged in a liquid or a percentage
of the
volume below the surface of an object partially submerged in a
liquid. The
amount of the buoyant force is equal to the weight of the displaced
part of the
fluid. The buoyant force on a body floating in a liquid or gas is equal
in size to
the floating item's weight and acts in the opposite direction; the
object does not
rise or sink. A ship thrown into the ocean, for example, sinks until
the weight of
the water it displaces is just equal to its own weight. As the ship is
loaded, it
sinks deeper, displacing more water, and so the magnitude of the
buoyant
force continuously matches the weight of the ship and its cargo.
5. Volume Elasticity of Gases
Volume Elasticity of Gases is which the Gases and liquids also
possess elastic
properties since their volume changes under the action of pressure.
For small
volume changes, the bulk modulus, κ, of a gas, liquid, or solid is
defined by the
equation where P is the pressure that reduces the volume V0 of a
fixed mass
of material to V, because gases may be compressed considerably
more readily
than liquids or solids, the value of is substantially lower for a gas
than for a
liquid or solid. Fluids, unlike solids, cannot withstand shearing
forces and have
no Young's modulus.
6. Surface Phenomena, Surface Tension, Pressure Due to Surface
Tension and Capillarity.
Surface Phenomena
The special properties of surface layers, or thin layers of a material
at the
border of contiguous bodies, media, or phases, are known as surface
phenomenon. These qualities are due to the surface layer's surplus
free
energy as well as the layer's unique structure and composition.
Surface Tension
Due to the cohesive structure of the water molecules, surface
tension may be
described as a characteristic of a liquid's surface that permits it to
resist an
external force.
Pressure Due to Surface Tension and Capillarity
The surface tension acts to hold the surface intact. Capillary action
occurs
when the adhesion to the surface material is stronger than the
cohesive forces
between the water molecules. The height to which capillary action
will take
water is limited by surface tension and gravity.
Capillarity is the result of cohesion of water molecules and adhesion
of those
molecules to a solid material. In the case of a glass tube inserted in
water with
openings at both ends, as the edges of the tube are brought closer
together,
such as in a very narrow tube, the liquid will be drawn upward in the
tube. The
more narrow the tube, the greater the rise of the liquid. Greater
surface tension
and increased ratio of adhesion to cohesion also result in greater
rise.
If one takes a small capillary tube an inserts it in water and the tube
does not
have a vacuum like a barometer but is open at top, water will start
to rise up.
Water wants to stick to the glass and surface tension will push the
water up,
until the force of gravity prevents further rise.
Capillary action is due to the pressure of cohesion and adhesion
which cause
the liquid to work against gravity.
B. Fluids in Motion
1. Fluid Flow
Fluid Flow is a part of fluid mechanics and deals with fluid dynamics.
It involves
the motion of a fluid subjected to unbalanced forces. This motion
continues as
long as unbalanced forces are applied.
For example, if you are pouring water from a mug, the velocity of
water is very
high over the lip, moderately high approaching the lip, and very low
at the
bottom of the mug. The unbalanced force is gravity, and the flow
continues as
long as the water is available and the mug is tilted.
Types of Fluids
• Ideal fluid
A fluid is said to be ideal when it cannot be compressed and the
viscosity
doesn’t fall in the category of an ideal fluid. It is an imaginary fluid
which
doesn’t exist in reality.
• Real fluid
All the fluids are real as all the fluid possess viscosity.
• Newtonian fluid
When the fluid obeys Newton’s law of viscosity, it is known as a
Newtonian
fluid.
• Non-Newtonian fluid
When the fluid doesn’t obey Newton’s law of viscosity, it is known as
Non-Newtonian fluid.
• Ideal plastic fluid
When the shear stress is proportional to the velocity gradient and
shear stress
is more than the yield value, it is known as ideal plastic fluid.
• Incompressible fluid
When the density of the fluid doesn’t change with the application of
external
force, it is known as an incompressible fluid.
• Compressible fluid
When the density of the fluid changes with the application of
external force, it is
known as compressible fluid.
Fluid Flow Equation
Mass flow rate is the rate of movement of a massive fluid through a
unit area.
In simple words it is the movement of mass per unit time. The
formula for mass
flow rate is given as follows:
Mass flowrate=ρAV
To calculate the total mass of fluid flowing through the tube, we use
Massflowrate=ρAV
Substituting the values in the above equation,
we get Massflowrate=1.5×15×0.4=9g/s
2. Streamlines, Tubes of Flow and Flow Through a Constriction
Streamlines
Streamline, In fluid mechanics, the path of imaginary particles
suspended in
the fluid and carried along with it. In steady flow, the fluid is in
motion but the
streamlines are fixed. Where streamlines crowd together, the fluid
speed is
relatively high; where they open out, the fluid is relatively still.
Streamlines are
a family of curves that are instantaneously tangent to the velocity
vector of the
flow. These show the direction in which a massless fluid element
will travel at
any point in time. By definition, different streamlines at the same
instant in a
flow do not intersect, because a fluid particle cannot have two
different
velocities at the same point.
Tubes of Flow
Laminar flow is often encountered in common hydraulic systems,
such as
where fluid flow is through an enclosed, rigid pipe; the fluid is
incompressible,
has constant viscosity, and the Reynolds number is below this lower
critical
threshold value. It is characterized by the flow of a fluid in parallel
layers, in
which there is no disruption or interaction between the different
layers, and in
which each layer flows at a different velocity along the same
direction. The
variation in velocity between adjacent parallel layers is due to the
viscosity of
the fluid and resulting shear forces.
Flow Through a Constriction
The Venturi effect is the reduction in fluid pressure that results
when a fluid
flows through a constricted section (or choke) of a pipe. The Venturi
effect is
named after its discoverer, Giovanni Battista Venturi.
3. Bernoulli’s Theorem, Effect of Friction on Flow
Bernoulli’s Theorem
Bernoulli’s principle states that an increase in the speed of a fluid
occurs
simultaneously with a decrease in static pressure or a decrease in
the fluid’s
potential energy. Although Bernoulli deduced that pressure
decreases when
the flow speed increases, it was Leonhard Euler who derived
Bernoulli’s
equation in its usual form in 1752. The principle is only applicable
for isentropic
flows: when the effects of irreversible processes (like turbulence)
and
non-adiabatic processes (e.g. heat radiation) are small and can be
neglected.
Bernoulli’s principle can be applied to various types of fluid flow,
resulting in
various forms of Bernoulli’s equation. The simple form of Bernoulli’s
equation is
valid for incompressible flows (e.g. most liquid flows and gases
moving at low
Mach number). More advanced forms may be applied to
compressible flows at
higher Mach numbers (see the derivations of the Bernoulli equation).
Fluid particles are subject only to pressure and their own weight. If
a fluid is
flowing horizontally and along a section of a streamline, where the
speed
increases it can only be because the fluid on that section has moved
from a
region of higher pressure to a region of lower pressure; and if its
speed
decreases, it can only be because it has moved from a region of
lower
pressure to a region of higher pressure. Consequently, within a fluid
flowing
horizontally, the highest speed occurs where the pressure is lowest,
and the
lowest speed occurs where the pressure is highest.
Where,
p = fluid density
g = acceleration
P1 = pressure at elevation 1 v1 = velocity at elevation 1 h1 = height
at
elevation 1 P2 = pressure at elevation 2 v2 = velocity at elevation 2
h2 =
height at elevation 2
Effect of Friction on Flow
Frictional effects, or pressure drop, are of primary importance in
production
facilities, flow lines, and pipeline design. As a fluid travels down a
pipe, flow is
retarded by frictional shear stresses with the pipe walls. The
pressure levels
decrease downstream as energy is used to overcome the frictional
effects. The
only exception occurs in downwardly inclined sections of pipe
where elevation
effects may overcome the pressure-decreasing effects of friction.
The faster
the fluid travels in the pipe, the greater the frictional stresses and
the greater
the pressure gradient.
Flows inside ducts, channels and pipes are very important because
they occur
in many practical applications (water pipes, air conditioning ducts,
gas lines,
ventilation shafts, heat exchanger tubes, etc.). Friction is usually
important in
these flows because there is a resistance to relative motion: when
one layer of
fluid is moving with respect to an adjacent layer, there exists
friction between
the layers. The amount of friction depends on the fluid viscosity and
the
velocity gradient (that is, the relative velocity between fluid layers).
The velocity
gradients are set up by the no-slip condition at the wall. When a fluid
is in
contact with a solid surface, there can be no relative motion
between the fluid
in contact with the solid surface and the surface itself: if the wall
has zero
velocity, then the fluid in contact with the wall has zero velocity
also.
4. Viscosity, Nature of Viscosity, and pressure and Speed
Viscosity
Viscosity is a measure of a fluid’s resistance to flow. It describes
the internal
friction of a moving fluid. A fluidwith large viscosity resists motion
because its
molecular makeup gives it a lot of internal friction. A fluid with low
viscosity
flows easily because its molecular makeup results in very little
friction when it
is in motion.
Gases also have viscosity, although it is a little harder to notice it in
ordinary
circumstances.
A fluid that has no resistance to shear stress is known as an ideal or
inviscid
fluid. Zero viscosity is observed only at very low temperatures in
superfluids.
Otherwise, the second law of thermodynamic requires all fluids to
have
positive viscosity;[2][3] such fluids are technically said to be
viscous or viscid. A
fluid with a high viscosity, such as pitch, may appear to be a solid.
Nature of Viscosity
Viscosity, resistance of a fluid (liquid or gas) to a change in shape,
or
movement of neighbouring portions relative to one another.
Viscosity denotes
opposition to flow. The reciprocal of the viscosity is called the
fluidity, a
measure of the ease of flow. Molasses, for example, has a greater
viscosity
than water. Because part of a fluid that is forced to move carries
along to some
extent adjacent parts, viscosity may be thought of as internal
friction between
the molecules; such friction opposes the development of velocity
differences
within a fluid. Viscosity is a major factor in determining the forces
that must be
overcome when fluids are used in lubrication and transported in
pipelines. It
controls the liquid flow in such processes as spraying, injection
molding, and
surface coating.
For many fluids the tangential, or shearing, stress that causes flow
is directly
proportional to the rate of shear strain, or rate of deformation, that
results. In
other words, the shear stress divided by the rate of shear strain is
constant for
a given fluid at a fixed temperature. This constant is called the
dynamic, or
absolute, viscosity and often simply the viscosity. Fluids that
behave in this
way are called Newtonian fluids in honour of Sir Isaac Newton, who
first
formulated this mathematical description of viscosity. Wider/slower
side P1 has
to be larger than the pressure on the narrow/faster side P2.
This inverse relationship between the pressure and speed at a point
in a fluid
is called Bernoulli’s principle.
Bernoulli’s principle: At points along a horizontal streamline, higher
pressure
regions have lower fluid speed and lower pressure regions have
higher fluid
speed.
It might be conceptually simplest to think of Bernoulli’s principle as
the fact that
a fluid flowing from a high pressure region to a low pressure region
will
accelerate due to the net force along the direction of motion.
The idea that regions where the fluid is moving fast will have lower
pressure
can seem strange. Surely, a fast moving fluid that strikes you must
apply more
pressure to your body than a slow moving fluid, right? Yes, that is
right. But
we’re talking about two different pressures now. The pressure that
Bernoulli’s
principle is referring to is the internal fluid pressure that would be
exerted in all
directions during the flow, including on the sides of the pipe. This is
different
from the pressure a fluid will exert on you if you get in the way of it
and stop its
motion.
Note that Bernoulli’s principle does not say that a fast moving fluid
can’t have
significantly high pressures. It just says that the pressure in a
slower region of
that same flowing system must have even larger pressure than the
faster
moving region.
5. Airfoils, Discharge from an orifice and Turbines
Airfoils
In the field of fluid dynamics, an area of significant practical
importance is the
study of airfoils. An airfoil refers to the cross sectional shape of an
object
designed to generate lift when moving through a fluid.
A solid body moving through a fluid produces an aerodynamic force.
The
component of this force perpendicular to the relative freestream
velocity is
called lift. The component parallel to the relative freestream
velocity is called
drag. An airfoil is a streamlined shape that is capable of generating
significantly more lift than drag.[1] Subsonic flight airfoils have a
characteristic
shape with a rounded leading edge, followed by a sharp trailing edge,
often
with a symmetric curvature of upper and lower surfaces. Foils of
similar
function designed with water as the working fluid are called
hydrofoils.
Orifice
An orifice is a small aperture through which the fluid passes. The
thickness of
an orifice in the direction of flow is very small in comparison to its
other
dimensions.
An orifice may be defined as an opening provided in the side or
bottom of a
tank, for the purpose of discharging the liquid contained in the tank.
It should
be noted that the opening will be considered as an orifice only when
the liquid
surface in the tank is above the upper edge of the opening. Orifices
may be
classified based on their size, shape, sharpness and discharge
conditions.
Based on their size orifices are classified into small and large
orifices. In a
small orifice, the size of the orifice is so small compared with the
head over it,
the velocity at the level of the centre of the orifice may be taken as
the mean
velocity through the orifice. In a large orifice however, this is not
correct.
An orifice may be circular, rectangular or square though often,
circular orifices
are adopted.
An orifice may be sharp edged or bell mouthed depending on the
shape of the
entrance edge. In the case of a sharp edged orifice the inner edge
(i.e., at
entrance) is made sharp and is tapered to a slightly larger diameter
at the
outer edge. The liquid discharged through the orifice will touch only
the sharp
edge at entrance.
In the case of a bell mouthed orifice, a rounded passage is provided
in the
orifice and the discharge liquid will be in contact with the entire
inner surface of
the orifice. Due to decreased friction a bell mouthed orifice has a
greater
discharging capacity.
The orifices mentioned above may discharge a liquid either from a
tank into the
atmosphere or from one tank into another. If the liquid surfaces on
the two
sides of an orifice are above the upper edge of the orifice, then the
orifice is
called a submerged or drowned orifice. If an orifice discharges a
liquid to the
atmosphere then the discharge is said to be free.
Coefficient of Discharge (Cd):
The ratio of the actual discharge to the theoretical discharge of the
orifice is
called the coefficient of discharge.
Turbine
Turbine is a machine which uses the kinetic energy of fluids and
transforms
into mechanical energy through a designed mechanism.The turbine
consists of
several blades which are connected to an axle which is used
generally to
drives a generator. The basic functioning of the turbine is that the
blades of
turbine are rotated by the moving fluid(e.g steam, water, gas etc.)
which in turn
rotates the axle which is connected to a device which uses this
rotational
energy for the required use.
The turbine is of two types based on the type of its working as
follows:
1. Impulse turbines: Its blades are rotated by the impulse created
the steam
on the blades. It consists of blades in form of buckets and nozzles
which
directs the steam to the curved
bucket shaped blades.
2. Reaction turbines: its blades are rotated by the torque which is
generated
by the virtue of fluid’s pressure or mass. It consists of blades of
aeroplane
wings. Two rows of movable blades are separated by a row of fixed
blades
attached to the
casings acting as nozzles.
HEAT TRANSFER
What is Heat Flow?
Heat flow is the transfer of heat energy or enthalpy. Heat energy is
transferred through
conduction (physical contact of surfaces), convection (movement of
fluids),and radiation
(emitted electromagnetic energy). As heat energy is transferred
from one source to
another, the kinetic energy of the receiving source increases. An
increase in kinetic energy
results in an increase in sensible heat or temperature. Sensible heat
will continue to
increase until the source has reached a phase change temperature.
At this point, the
source begins to change from a solid to a liquid or a liquid to a gas.
At this point, the
source begins storing latent heat until it completely changes phase
and cycles back to
storing sensible heat.
The heat flow is the measurement of the energy transfer, which is
caused by a
temperature difference and leads to the temperature balance
between substances. In this
context, the energy is called heat.
The amount of heat that passes from one substance to another per
unit of time, is the heat
flow with the unit of measure Joule per second [J/s]. This is the unit
of measure Watts [W]
that is commonly used to indicate power.
Describe the Methods of Heat Transfer.
Equally as interesting as the effects of heat transfer on a system are
the methods by
which this occurs. Whenever there is a temperature difference, heat
transfer occurs. Heat
transfer may occur rapidly, such as through a cooking pan, or slowly,
such as through the
walls of a picnic ice chest. We can control rates of heat transfer by
choosing materials
(such as thick wool clothing for the winter), controlling air
movement (such as the use of
weather stripping around doors), or by choice of color (such as a
white roof to reflect
summer sunlight). So many processes involve heat transfer, so that
it is hard to imagine a
situation where no heat transfer occurs. Yet every process involving
heat transfer takes
place by only three methods:
➢ Conduction
Conduction is the transfer of heat through stationary matter by
physical contact. (The
matter is stationary on a macroscopic scale—we know there is
thermal motion of the
atoms and molecules at any temperature above absolute zero.) Heat
transferred from an
electric stove to the bottom of a pot is an example of conduction.
On a microscopic scale, conduction occurs as rapidly moving or
vibrating atoms and
molecules interact with neighboring particles, transferring some of
their kinetic energy.
Heat is transferred by conduction when adjacent atoms vibrate
against one another, or as
electrons move from one atom to another. Conduction is the most
significant means of
heat transfer within a solid or between solid objects in thermal
contact. Conduction is
greater in solids because the network of relatively close fixed
spatial relationships
between atoms helps to transfer energy between them by vibration.
Fluids and gases are less conductive than solids. This is due to the
large distance
between atoms in a fluid or (especially) a gas: fewer collisions
between atoms means less
conduction.
The (average) kinetic energy of a molecule in the hot body is higher
than in the colder
body. If two molecules collide, an energy transfer from the hot to the
cold molecule occurs
(see the above figure). The cumulative effect from all collisions
results in a net flux of heat
from the hot body to the colder body. The heat flux thus depends on
the temperature
difference T = THot−TCold. Therefore, you will get a more severe
burn from boiling water
than from hot tap water. Conversely, if the temperatures are the
same, the net heat
transfer rate falls to zero, and equilibrium is achieved. Owing to the
fact that the number of
collisions increases with increasing area, heat conduction depends
on the cross-sectional
area. If you touch a cold wall with your palm, your hand cools faster
than if you just touch it
with your fingertip.
Heat transferred by the process of conduction:
Q = kA (THot−TCold) t/d
Where:
• Q Heat transferred
• K Thermal Conductivity
• THOT Hot temperature
• TCOLD Cold Temperature
• t Time
• d The thickness of the material
• AArea of surface
➢ Convection
Convection is the heat transfer by the macroscopic movement of a
fluid, such as a car’s
engine kept cool by the water in the cooling system. Convection
(illustrated in ) is the
concerted, collective movement of ensembles of molecules within
fluids (e.g., liquids,
gases). Convection of mass cannot take place in solids, since
neither bulk current flows
nor significant diffusion can occur in solids. Instead heat diffusion in
solids is called heat
conduction, which we’ve just reviewed.
Convection is driven by large-scale flow of matter. In the case of
Earth, the atmospheric
circulation is caused by the flow of hot air from the tropics to the
poles, and the flow of cold
air from the poles toward the tropics. (Note that Earth’s rotation
causes changes in the
direction of airflow depending on latitude.). An example of
convection is a car engine kept
cool by the flow of water in the cooling system, with the water pump
maintaining a flow of
cool water to the pistons.
While convection is usually more complicated than conduction, we
can describe
convection and perform some straightforward, realistic calculations
of its effects. Natural
convection is driven by buoyant forces: hot air rises because density
decreases as
temperature increases. This principle applies equally with any fluid.
For example, the pot
of water on the stove in is kept warm in this manner; ocean currents
and large-scale
atmospheric circulation transfer energy from one part of the globe
to another.
Heat transferred by the process of convection:
Q = HCA(THot−TCold)
Where,
• Q Heat transferred
• HC Heat Transfer Coefficient
• THot Hot temperature
• TCold Cold Temperature
• AArea of surface
➢ Radiation
Radiation is the transfer of heat through electromagnetic energy.
You can feel heat
transfer from a fire or the Sun. Yet the space between Earth and the
Sun is largely empty,
without any possibility of heat transfer by convection or conduction.
Similarly, you can tell
that an oven is hot without touching it or looking inside—it just
warms you as you walk by.
In these examples, heat is transferred by radiation. The hot body
emits electromagnetic
waves that are absorbed by our skin, and no medium is required for
them to propagate.
We use different names for electromagnetic waves of different
wavelengths: radio waves,
microwaves, infrared radiation, visible light, ultraviolet radiation, Xrays, and gamma rays .
Radiation from a Fire: Most of the heat transfer from this fire to the
observers is through
infrared radiation. The visible light, although dramatic, transfers
relatively little thermal
energy. Convection transfers energy away from the observers as hot
air rises, while
conduction is negligibly slow here. Skin is very sensitive to infrared
radiation so that you
can sense the presence of a fire without looking at it directly.
The energy of electromagnetic radiation depends on its wavelength
(color) and varies
over a wide range; a smaller wavelength (or higher frequency)
corresponds to a higher
energy. We can write this as:
E = hf = hc/λ
where E is the energy, f is the frequency, λ is the wavelength, and h
is a constant.
Because more heat is radiated at higher temperatures, a
temperature change is
accompanied by a color change. For example, an electrical element
on a stove glows
from red to orange, while the higher-temperature steel in a blast
furnace glows from
yellow to white. The radiation you feel is mostly infrared, which is
lower in temperature
still.
The radiated energy depends on its intensity, which is represented
by the height of the
distribution .
The Heat transferred by the process of radiation:
Q = σ(THot−TCold)A
Where,
• Q Heat transferred
• σ Stefan Boltzmann Constant
• THot Hot temperature
• TCold Cold Temperature
• AArea of surface
Effects of Surface Films on Heat Transfer.
Heat transfer processes are particularly affected by fluid properties
which themselves
depend on temperature.
When heat transfer occurs from a surface into the body of a fluid,
natural convection
currents are weakest at the surface, which is covered by what is
effectively a static film.
Consequently, heat transfer across this film can only occur by
conduction, and, as
mentioned above, thermal conductivity in fluids is low. Hence, the
main resistance to heat
transfer into fluid in a pipe is this film adjacent to the pipe wall. An
increase in the velocity
of the fluid moving through the pipe will
reduce the thickness of this static film and give rise to an overall
increase in the heat
transfer into the fluid.
In theory, the heat load transferred across this film is defined as in
eqn (1). However, in
practice, it is difficult to calculate the film thickness, X, and so the
following
relationship is used:
Q = −h A T
where h is the heat-transfer coefficient.
The Ideal Radiator
An ideal thermal radiator of uniform temperature whose radiant
exitance in all parts of the
spectrum is the maximum obtainable from any thermal radiator at
the same temperature is
called a blackbody. Although no material reaches the theoretical
maximum of a blackbody,
it is sometimes convenient to describe the emissive properties of a
material by specifying,
on a wavelength-by-wavelength basis, the fraction of light it
generates with respect to a
blackbody. For example, solar radiation arrives at the Earth's
atmosphere with a spectral
energy distribution similar to a blackbody radiator of 5,800°K.
Luminescent emissions are due to energy arriving from elsewhere,
which is stored in the
material and emitted (after a short period of time) as photons. The
incident energy,
primarily due to factors other than temperature, causes the
excitation of electrons of the
material. These electrons in the outer and incomplete inner shells
move to a higher
energy state within the atom. When an electron returns to the
ground state, a photon is
emitted. The wavelength of the emitted photon will depend on the
atomic structure of the
material and the magnitude of the incoming energy. Typically, an
electron remains in its
excited state for about 10−9s. If there is a much longer delay and
the electron emits a
photon in the visible range, having being originally excited by a
photon of differing (usually
shorter) wavelength, the process is called phosphorescence. The
distinction between
phosphorescence and fluorescence is a matter of scale (time), with
the latter usually
taking less than 10−8s.
A phosphor is defined as a luminescent material that absorbs energy
and reemits it over
some period of time, which is associated with the lifetime of the
excited electron. Most
phosphors are inorganic, i.e., carbon-free, crystals that contain
structural and impurity
defects. Some of these materials are used in TV screens and
computer monitors (cathode
ray tube [CRT]).
As described by Williamson and Cummins, atoms can be excited in
many ways other than
absorbing a photon. The term phosphorescence was originally
applied to light given off by
the reactive element phosphorous and chemically similar
substances when left exposed
to air. They spontaneously combine with oxygen in a slow reaction
and in the process emit
light. This process of light emission as a result of a chemical
reaction is called
chemiluminescence. A related effect is bioluminescence, when light
is produced by
chemical reactions associated with biological activity. When one
hard object is sharply
stricken against another, we may observe a “spark” or light
emission termed
triboluminescence. Excitation is also possible due to the impact of
high-energy particles,
which may cause impressive light emissions such as those found in
aurorae and space
nebulae.
Theory of Exchanges
Prevost’s Theory of Heat Exchanges
Prevost applied the idea of thermal equilibrium to radiation.
According to him, the rate at
which a body radiates or absorbs heat depends on the nature of its
surface, its
temperature and the temperature of the surroundings. The total
amount of heat radiated
by a body increases as its temperature rises. He suggested that all
bodies radiate energy
but hot bodies radiate more heat than the cooler bodies. A body at a
higher temperature
radiates more heat energy to the surroundings than it receives from
the surroundings.
That is why we feel warm when we stand before the furnace. For
example, if you touch
someone, they might feel your skin as either hot or cold.
A body at high temperature radiates more heat to the surroundings
than it receives from it.
Similarly, a body at a lower temperature receives more heat energy
than it loses to the
surroundings. That is why we feel cold when we stand before an ice
block. In this case,
the amount of heat absorbed by the body from the enclosure per
second is greater than
that emitted by the body at the same time so that there is a net gain
of heat by the body. At
one point in time, the rate of exchange of heat from both bodies will
become the same.
Now the bodies are said to be in ‘thermal equilibrium’.
Thus the rise or fall of temperature is due to the exchange of heat
radiation. When the
temperature of the body is the same as that of surroundings, the
exchanges of heat do not
stop. In such a case, the amount of heat energy radiated by the body
is equal to the
amount of heat energy absorbed by it. A body at a higher
temperature than the
surroundings radiates heat at a faster rate than it absorbs. The rate
of emission of heat by
a body depends upon its absolute temperature.
Only at absolute zero temperature, a body will stop emitting. A body
will stop emitting
radiation only when it is at absolute zero. Therefore, Prevost theory
states that all bodies
emit thermal radiation at all temperatures above absolute zero
irrespective of the nature of
the surroundings. Example: 0 K or -273° C. At this temperature the
kinetic energy of the
molecule is zero. The rate of emission of heat by a body does not
depend upon the
temperature of its surroundings. So, It loses heat and its
temperature falls. And it gains
heat, its temperature rises. A body that is at a lower temperature
than the surroundings
absorbs heat at a faster rate than it radiates. Hence, there is no net
loss or gain of heat. Its
temperature is unchanged.
Therefore, Prevost’s theory states that all bodies emit thermal
radiation at all
temperatures above absolute zero, irrespective of the nature of the
surroundings. In the
case of thermal equilibrium, the process of radiation and absorption
continues to take
place. Every material body, at any temperature above absolute zero,
radiates heat to the
surroundings and at the same time absorbs heat from the
surroundings.
Rate of Radiation
The rate of heat transfer by emitted radiation is determined by the
Stefan- Boltzmann law
of radiation:
Q /t = σeAT 4
where σ = 5.67 × 10 −8 J/s · m 2 · K 4 is the Stefan-Boltzmann
constant, A is the
surface area of the object, and T is its absolute temperature in
kelvin. The
symbol e stands for the emissivity of the object, which is a measure
of how well it radiates.
An ideal jet-black (or black body) radiator has e = 1, whereas a
perfect reflector has e = 0.
Real objects fall between these two values. Take, for example,
tungsten light bulb
filaments which have an e of about 0.5, and carbon black (a material
used in printer toner),
which has the (greatest known) emissivity of about 0.99.
The radiation rate is directly proportional to the fourth power of the
absolute
temperature—a remarkably strong temperature dependence.
Furthermore, the radiated
heat is proportional to the surface area of the object. If you knock
apart the coals of a fire,
there is a noticeable increase in radiation due to an increase in
radiating surface area.
Stefan-Boltzmann Law
The Stefan–Boltzmann law describes the power radiated from a
black body in terms of
its temperature. Specifically, the Stefan–Boltzmann law states that
the
total energy radiated per unit surface area of a black body across all
wavelengths per
unit time j* (also known as the black-body radiant emittance) is
directly proportional to the
fourth power of the black body&#39;s thermodynamic temperature T:
j* = σ T 4
Stefan-Boltzmann law, statement that the total radiant heat power
emitted from a surface
is proportional to the fourth power of its absolute temperature.
Formulated in 1879 by
Austrian physicist Josef Stefan as a result of his experimental
studies, the same law was
derived in 1884 by Austrian physicist Ludwig Boltzmann from
thermodynamic
considerations: if E is the radiant heat energy emitted from a unit
area in one second (that
is, the power from a unit area) and T is the absolute temperature (in
kelvins), then E = σT
4 , the Greek letter sigma (σ) representing the
constant of proportionality, called the Stefan-Boltzmann constant.
This constant has the
value 5.670374419 × 10 −8 watt per metre 2 per K 4 . The law
applies only to blackbodies,
theoretical surfaces that absorb all incident heat radiation.
Newton's Law of Cooling
Newton’s law of cooling states that the rate at which an object
cools is proportional to the
difference in temperature between the object and the object’s
surroundings. Simply put, a
glass of hot water will cool down faster in a cold room than in a hot
room.
Newton’s Law of Cooling Formula
Greater the difference in temperature between the system and
surrounding, more rapidly
the heat is transferred i.e. more rapidly the body temperature of
body changes.
Newton’s law of cooling formula is expressed by,
T(t) = Ts + (To – Ts) e-kt
Where,
• t = time,
• T(t) = temperature of the given body at time t,
• Ts = surrounding temperature,
• To = initial temperature of the body,
• k = constant.
Example: Water is heated to 80oC for 10 min. How much would be
the temperature if k
= 0.56 per min and the surrounding temperature is 25oC?
Solution:
Given:
• Ts = 25oC,
• To = 80oC,
• t = 10 min,
• k = 0.56
Now, substituting the above data in Newton’s law of cooling formula,
T(t) = Ts + (To – Ts) × e-kt
= 25 + (80 – 25) × e-0.56 = 25 + [55 × 0.57] = 56.35 oC
Temperature cools down from 80oC to 56.35 oC after 10 min.
Planck's Quantum Theory of Radiation
Planck’s quantum theory states that an atom can only emit or
absorb energy in discrete
amounts. Planck’s constant relates a particle’s energy and
frequency.
Max Planck was able to explain the phenomenon of blackbody
radiation not rising
continuously. Planck stated that energy emitted by an object was
not continuous and
indefinite. In fact, energy can only be emitted in definite amounts or
quanta.
Energy is defined as quantized meaning that it changes by definite
amounts. Think about
climbing a ladder versus a ramp. On a ramp the change is
continuous from start to end. A
ladder must be climbed one rung at a time, you cannot step inbetween rungs.
Planck’s Constant
The energy of radiation can be described with the equation:
E= hν
or
E= hc / λ
where, ν=c / λ
The amount of energy (E) is affected by the frequency of the
radiation (ν). Planck’s
constant is represented by h where h=6.636×10 -34 J ⋅ s. This
constant relates light or a
photon’s energy to the frequency. The constant c is the speed of
light
and c=2.99792458×10 8 m/s.
Wien's Displacement Law
Wien's displacement law states that the black-body radiation curve
for different
temperatures will peak at different wavelengths that are inversely
proportional to the
temperature. The shift of that peak is a direct consequence of the
Planck radiation law,
which describes the spectral brightness of black-body radiation as a
function of
wavelength at any given temperature. However, it had been
discovered by Wilhelm Wien
several years before Max Planck developed that more general
equation, and describes
the entire shift of the spectrum of black-body radiation toward
shorter
wavelengths as temperature increases.
Black-body radiation as a function of wavelength for various
temperatures. Each
temperature curve peaks at a different wavelength and Wien's law
describes the shift
of that peak.
Waves
I. Wave Motion
1. Concept of Wave Motion
Wave motion is the transfer of energy and momentum from one point
of the
medium to another point of the medium without actual transport of
matter
between two points. Wave motion is classified into three different
ways they
are,
• The medium of propagation,
• The dimensions in which a wave propagates energy,
• The energy transfer
Wave motion, propagation of disturbances—that is, deviations from
a state of
rest or equilibrium— from place to place in a regular and organized
way. Most
familiar are surface waves on water, but both sound and light travel
as
wavelike disturbances, and the motion of all subatomic particles
exhibits
wavelike properties. The study of waves therefore forms a topic of
central
importance in all physical science and engineering.
The simplest types of wave motion are vibrations of elastic media,
such as air,
crystalline solids, or stretched strings. If, for example, the surface of
a metal
block is struck a sharp blow, the deformation of the surface material
compresses the metal in the vicinity of the surface, and this
transmits the
disturbance to the layers beneath. The surface relaxes back to its
initial
configuration, and the compression propagates on into the body of
the material
at a speed determined by the stiffness of the material. This is an
example of a
compression wave. The steady transmission of a localized
disturbance
through an elastic medium is common to many forms of wave
motion.
2. Types of Waves and their Classification
Different types of waves have a different set of characteristics.
Based on the
orientation of particle motion and direction of energy, there are
three
categories:
• Mechanical waves
• Electromagnetic waves
• Matter waves
Mechanical Wave
• A mechanical wave is a wave that is an oscillation of matter and is
responsible for the transfer of energy through a medium.
• The distance of the wave’s propagation is limited by the medium of
transmission. In this case, the oscillating material moves about a
fixed point,
and there is very little translational motion. One intriguing property
of
mechanical wave is the way they are measured, which is given by
displacement divided by wavelength. When this dimensionless
factor is 1, it
results in the generation of harmonic effects; for example, waves
break on the
beach when this factor exceeds 1, resulting in turbulence.
There are two types of mechanical waves:
• Longitudinal waves – In this type of wave, the movement of the
particle are
parallel to the motion of the energy i.e. the displacement of the
medium is in
the same direction to which the wave is moving. Example – Sound
Waves,
Pressure Waves.
• Transverse waves – When the movement of the particles is at right
angles or
perpendicular to the motion of the energy, then this type of wave is
known as
Transverse wave. Light is an example of a transverse wave. Some of
the other
examples are – ‘Polarized’ waves & Electromagnetic waves.
Electromagnetic Wave
• Electromagnetic waves are created by a fusion of electric and
magnetic fields.
The light you see, the colours around you are visible because of
electromagnetic waves.
• One interesting property here is that unlike mechanical waves,
electromagnetic waves do not need a medium to travel. All
electromagnetic
waves travel through a vacuum at the same speed, 299,792,458 ms1.
Following are the different types of electromagnetic waves:
• Microwaves
• X-ray
• Radio waves
• Ultraviolet waves
Matter Wave
• This concept is a little complicated to understand. The dual nature
of matter;
its ability to exist both as a particle and a wave was first brought to
light by the
founders of the field of Quantum Physics.
• For example, a beam of electrons can be diffracted just like any
other beam
of electromagnetic radiation or water wave. This property of matter
was
brought forward by Louis de Broglie’s Hypothesis.
3. Water Waves
Water waves are surface waves, a mixture of longitudinal and
transverse
waves. Surface waves in oceanography are deformations of the sea
surface.
The deformations propagate with the wave speed, while the water
molecules
remain at the same positions on average. Energy, however, moves
towards
the shore. Most ocean waves are produced by wind, and the energy
from the
wind offshore is carried by the waves towards the shore.
We distinguish between deep-water waves and shallow-water waves.
The
distinction between deep and shallow water waves has nothing to
do with
absolute water depth. It is determined by the ratio of the water's
depth to the
wavelength of the wave.
The water molecules of a deep-water wave move in a circular orbit.
The
diameter of the orbit decreases with the distance from the surface.
The motion
is felt down to a distance of approximately one wavelength, where
the wave's
energy becomes negligible.
The uniqueness of water waves is found in the observation that they
comprise
both transverse and longitudinal wave motion. As a result, the
particles
composing the wave move in clockwise circular motion, as seen in.
Oscillatory
motion is highest at the surface and diminishes exponentially with
depth.
Waves are generated by wind passing over the surface of the sea.
As long as
the waves propagate slower than the wind speed just above the
waves, there
is an energy transfer from the wind to the waves. Both air pressure
differences
between the upwind and the lee side of a wave crest, as well as
friction on the
water surface by the wind (making the water to go into the shear
stress),
contribute to the growth of the waves.
4. Wave Properties
The basic properties (parts) of a wave include: frequency, amplitude,
wavelength and speed.
Frequency
• Frequency is a measure of how many waves pass a point in a
certain amount
of time.
• The higher the frequency, the closer the waves are together and
the greater
the energy carried by the waves will be.
Amplitude
• Amplitude is a measure of the distance between a line through the
middle of
a wave and a crest or trough.
• The greater the force that produces a wave, the greater the
amplitude of the
wave and the greater the energy carried by the wave.
• The highest point of a transverse wave is the crest and the lowest
point is
called a trough.
• In a transverse wave the higher the wave, the higher the amplitude.
• Sounds with greater amplitude will be louder; light with greater
amplitude
will be brighter.
Wavelength
• Wavelength is a measure of the distance from the crest on one
wave to the
crest on the very next wave.
• Shorter wavelengths are influenced by the frequency.
• A higher frequency causes a shorter wavelength and greater
energy.
Speed
• Speed is a measure of the distance a wave travels in an amount of
time.
• The speed of a wave is determined by the type of wave and the
nature of
the medium.
• As a wave enters a different medium, the wave’s speed changes.
Waves
travel at different speeds in different media.
• All frequencies of electromagnetic waves travel at the same speed
in empty
space.
5. General Wave Motion
Wave motion is a disturbance in a material or medium where the
individual
parts of the material may only show periodic motion, while the
waveform itself
moves through the material. All waves have similar characteristics,
and since
all forms of wave motion follow the same laws and principles,
knowing the
fundamentals of wave motion is important in understanding sound,
light, and
other types of waves.
A wave is a disturbance or oscillation that travels through space
and matter,
accompanied by a transfer of energy.
Wave motion, propagation of disturbances—that is, deviations from
a state of
rest or equilibrium— from place to place in a regular and organized
way. Most
familiar are surface waves on water, but both sound and light travel
as
wavelike disturbances, and the motion of all subatomic particles
exhibits
wavelike properties.
6. Transmission of Energy
Water wave transfers energy through the vibration of the water
particles,
sound waves travel through the vibration of air particles or the
particles of a
liquid or solid, and electromagnetic and magnetic fields vibrate to
transfer
energy through electromagnetic waves.
Wave is a common term for a number of different ways in which
energy is
transferred: In electromagnetic waves, energy is transferred through
vibrations
of electric and magnetic fields. In sound waves, energy is
transferred through
vibration of air particles or particles of a solid through which the
sound travels.
7. Superposition of Waves
The principle of superposition may be applied to waves whenever
two (or
more) waves travelling through the same medium at the same time.
The waves
pass through each other without being disturbed. The net
displacement of the
medium at any point in space or time, is simply the sum of the
individual wave
displacements. This is true of waves which are finite in length (wave
pulses) or
which are continuous sine waves. When two or more waves arrive at
the same
point, they superimpose themselves on one another. More
specifically, the
disturbances of waves are superimposed when they come
together—a
phenomenon called superposition. Each disturbance corresponds to
a force,
and forces add. If the disturbances are along the same line, then the
resulting
wave is a simple addition of the disturbances of the individual
waves—that is,
their amplitudes add.
8. Huygen's Principle
Huygen’s Principle states that every point on a wavefront is a
source of
wavelets, which spread forward at the same speed. In 1678,
Huygens
proposed a model where each point on a wavefront may be regarded
as a
source of waves expanding from that point. The expanding waves
may be
demonstrated in a ripple tank by sending plane waves toward a
barrier with a
small opening. If waves approaching a beach strike a barrier with a
small
opening, the waves may be seen to expand from the opening.
9. Refraction and Dispersion of Waves
In physics, refraction is the change in direction of a wave passing
from one
medium to another or from a gradual change in the medium.
Refraction of light
is the most commonly observed phenomenon, but other waves such
as sound
waves and water waves also experience refraction. How much a
wave is
refracted is determined by the change in wave speed and the initial
direction of
wave propagation relative to the direction of change in speed.
Refraction, in physics, the change in direction of a wave passing
from one
medium to another caused by its change in speed. For example,
waves travel
faster in deep water than in shallow. If an ocean wave approaches a
beach
obliquely, the part of the wave farther from the beach will move
faster than the
part closer in, and so the wave will swing around until it moves in a
direction
perpendicular to the shoreline. The speed of sound waves is greater
in warm
air than in cold. At night, air is cooled at the surface of a lake, and
any sound
that travels upward is refracted down by the higher layers of air that
still remain
warm. Thus, sounds, such as voices and music, can be heard much
farther
across water at night than in the daytime.
Dispersion, in wave motion, any phenomenon associated with the
propagation of individual waves at speeds that depend on their
wavelengths.
Ocean waves, for example, move at speeds proportional to the
square root of
their wavelengths; these speeds vary from a few feet per second for
ripples to
hundreds of miles per hour for tsunamis. A wave of light has a speed
in a
transparent medium that varies inversely with the index of
refraction (a
measure of the angle by which the direction of a wave is changed as
it moves
from one medium into another). Any transparent medium—e.g., a
glass prism—will cause an incident parallel beam of light to fan out
according
to the refractive index of the glass for each of the component
wavelengths, or
colours. Dispersion is sometimes called the separation of light into
colours, an
effect more properly called angular dispersion.
II. Stationary Waves
1. Stationary or Standing Waves
In physics, a standing wave, also known as a stationary wave, is a
wave
which oscillates in time but whose peak amplitude profile does not
move in
space. The peak amplitude of the wave oscillations at any point in
space is
constant with time, and the oscillations at different points
throughout the wave
are in phase. The locations at which the absolute value of the
amplitude is
minimum are called nodes, and the locations where the absolute
value of the
amplitude is maximum are called antinodes.
Standing wave, also called stationary wave, combination of two
waves
moving in opposite directions, each having the same amplitude and
frequency.
The phenomenon is the result of interference; that is, when waves
are
superimposed, their energies are either added together or canceled
out. In the
case of waves moving in the same direction, interference produces
a traveling
wave. For oppositely moving waves, interference produces an
oscillating
wave fixed in space.
2. Reflection of Waves
Reflection is the abrupt change in the direction of propagation of a
wave that
strikes the boundary between two different media. At least some
part of the
incoming wave remains in the same medium. Assume the incoming
light ray
makes an angle θi with the normal of a plane tangent to the
boundary. Then
the reflected ray makes an angle θr with this normal and lies in the
same plane
as the incident ray and the normal.
Law of reflection: θi = θr
3. Modes of Vibration
The modes of vibration associated with resonance in extended
objects like
strings and air columns have characteristic patterns called standing
waves.
These standing wave modes arise from the combination of reflection
and
interference such that the reflected waves interfere constructively
with the
incident waves. An important part of the condition for this
constructive
interference for stretched strings is the fact that the waves change
phase upon
reflection from a fixed end. Under these conditions, the medium
appears to
vibrate in segments or regions and the fact that these vibrations are
made up
of traveling waves is not apparent - hence the term "standing wave".
III. Sound Waves
1. Nature of Sound
In physics, sound is a vibration that propagates as an acoustic wave,
through a
transmission medium such as a gas, liquid or solid.
Sound, a mechanical disturbance from a state of equilibrium that
propagates
through an elastic material medium. A purely subjective definition of
sound is
also possible, as that which is perceived by the ear, but such a
definition is not
particularly illuminating and is unduly restrictive, for it is useful to
speak of
sounds that cannot be heard by the human ear, such as those that
are
produced by dog whistles or by sonar equipment.
2. Vibrating Sources
The vibrations can be carried through air, water or solid materials.
Mechanical,
electrical, or other forms
of energy make objects vibrate. When this happens, the energy
radiates as
sound.
3. Forced Vibration and Resonance
An object when forced to vibrate at a certain frequency by an input
periodic
force, is called forced vibration. Resonance occurs if the object is
forced to
vibrate at its natural frequency.
Forced vibrations as the name implies, happens when an object is
forced by
an input force (periodic in nature) to vibrate at a certain frequency.
Objects that are free to vibrate have their natural frequencies in
which they
vibrate when left for a duration of time.
Resonance occurs when objects are forced to vibrate at their
natural
frequency. The force will create vibrations of very large amplitude
when object
is vibrating at resonance.
4. Transmitting Medium
A transmission medium is something that can mediate the
propagation of
signals for the purposes of telecommunication.
Signals are typically imposed on a wave of some kind suitable for
the chosen
medium. For example, data can modulate sound, and a transmission
medium
for sounds may be air, but solids and liquids may also act as the
transmission
medium. Vacuum or air constitutes a good transmission medium for
electromagnetic waves such as light and radio waves. While
material
substance is not required for electromagnetic waves to propagate,
such waves
are usually affected by the transmission media they pass through,
for instance,
by absorption or reflection or refraction at the interfaces between
media.
Technical devices can therefore be employed to transmit or guide
waves. Thus,
an optical fiber or a copper cable is used as transmission media.
5. Speed of Sound
The speed of sound is the distance travelled per unit of time by a
sound wave
as it propagates through an elastic medium. At 20 °C (68 °F), the
speed of
sound in air is about 343 metres per second (1,235 km/h; 1,125 ft/s;
767 mph;
667 kn), or a kilometre in 2.9 s or a mile in 4.7 s. It depends strongly
on
temperature as well as the medium through which a sound wave is
propagating. At 0°C/32°F, the speed- of-sound is 1192 km/h, 741 mph.
The speed of sound in an ideal gas depends only on its temperature
and
composition. The speed has a weak dependence on frequency and
pressure
in ordinary air, deviating slightly from ideal behavior.
6. Refraction of Sound
Refraction is the bending of waves when they enter a medium where
their
speed is different. Refraction is not so important a phenomenon with
sound as
it is with light where it is responsible for image formation by lenses,
the eye,
cameras, etc. But bending of sound waves does occur and is an
interesting
phenomena in sound. When sound waves move from one medium to
another,
there will be changes to the velocity (or speed), frequency and
wavelength of
the sound wave. This change in velocity can also result in a change
of
direction of the sound wave - also known as refraction.
7. Reflection of Sound Waves
Just like the reflection of light, the reflection of sound is similar as it
follows
the laws of reflections, where the angle of reflection is equal to the
angle of
incidence and the reflected sound, the incident sound, and the
normal sound
belong in the same plane. Sound bounces off the surface of the
medium which
can be a solid or a liquid. In order to make the reflection of sound to
occur, the
surface can be of large size and can be either rough or polished.
Laws of Reflection of Sound
• The angle of reflection is always equal to the angle of incidence .
• The reflected sound, the incident sound, and the normal sound
belong in the
same plane.
8. Interference of Waves; Beats
Beat, in physics, the pulsation caused by the combination of two
waves of
slightly different frequencies. The principle of beats for sound waves
can be
demonstrated on a piano by striking a white key and an adjacent
black key at
the bass end of the keyboard. The resulting sound is alternately soft
and
loud— that is, having characteristic pulsations, or throbs, called
beats. Toward
the treble end of the keyboard, the beat frequency is greater
because the
difference in frequency of adjacent keys is more than at the lower
end. The
Figure depicts two waves n1 and n2 with respective frequencies of
24 and 30
vibrations per second (hertz); the beat frequency N is their
difference, 6 beats
per second.
Beats are the periodic and repeating fluctuations heard in the
intensity of a
sound when two sound waves of very similar frequencies interfere
with one
another. The diagram below illustrates the wave interference
pattern resulting
from two waves (drawn in red and blue) with very similar
frequencies. A beat
pattern is characterized by a wave whose amplitude is changing at a
regular
rate. Observe that the beat pattern (drawn in green) repeatedly
oscillates from
zero amplitude to a large amplitude, back to zero amplitude
throughout the
pattern. Points of constructive interference (C.I.) and destructive
interference
(D.I.) are labeled on the diagram. When constructive interference
occurs
between two crests or two troughs, a loud sound is heard. This
corresponds to
a peak on the beat pattern (drawn in green). When destructive
interference
between a crest and a trough occurs, no sound is heard; this
corresponds to a
point of no displacement on the beat pattern. Since there is a clear
relationship
between the amplitude and the loudness, this beat pattern would be
consistent
with a wave that varies in volume at a regular rate.
9. The Doppler Effect
The Doppler effect or Doppler shift (or simply Doppler, when in
context) is the
change in frequency of a wave in relation to an observer who is
moving relative
to the wave source. It is named after the Austrian physicist
Christian Doppler,
who described the phenomenon in 1842. A common example of
Doppler shift is
the change of pitch heard when a vehicle sounding a horn
approaches and
recedes from an observer. Compared to the emitted frequency, the
received
frequency is higher during the approach, identical at the instant of
passing by,
and lower during the recession.
The reason for the Doppler effect is that when the source of the
waves is
moving towards the observer, each successive wave crest is
emitted from a
position closer to the observer than the crest of the previous wave.
Therefore,
each wave takes slightly less time to reach the observer than the
previous
wave. Hence, the time between the arrivals of successive wave
crests at the
observer is reduced, causing an increase in the frequency. While
they are
traveling, the distance between successive wave fronts is reduced,
so the
waves "bunch together". Conversely, if the source of waves is
moving away
from the observer, each wave is emitted from a position farther from
the
observer than the previous wave, so the arrival time between
successive
waves is increased, reducing the frequency. The distance between
successive
wave fronts is then increased, so the waves "spread out".
10.Sonic Booms
A sonic boom is a sound associated with shock waves created when
an
object travels through the air faster than the speed of sound. Sonic
booms
generate enormous amounts of sound energy, sounding similar to an
explosion or a thunderclap to the human ear. The crack of a
supersonic bullet
passing overhead or the crack of a bullwhip are examples of a sonic
boom in
miniature. Sonic booms due to large supersonic aircraft can be
particularly
loud and startling, tend to awaken people, and may cause minor
damage to
some structures. They led to prohibition of routine supersonic flight
over land.
Although they cannot be completely prevented, research suggests
that with
careful shaping of the vehicle, the nuisance due to the sonic booms
may be
reduced to the point that overland supersonic flight may become a
feasible
option.
A sonic boom does not occur only at the moment an object crosses
the speed
of sound; and neither is it heard in all directions emanating from the
supersonic
object. Rather the boom is a continuous effect that occurs while the
object is
travelling at supersonic speeds. But it affects only observers that
are
positioned at a point that intersects a region in the shape of a
geometrical cone
behind the object. As the object moves, this conical region also
moves behind
it and when the cone passes over the observer, they will briefly
experience
the boom.
11. Ultrasonics VS. Supersonics
Ultrasonics, vibrations of frequencies greater than the upper limit of
the
audible range for humans— that is, greater than about 20 kilohertz.
The term
sonic is applied to ultrasound waves of very high amplitudes.
Hypersound,
sometimes called praetersound or microsound, is sound waves of
frequencies
greater than 1013 hertz. At such high frequencies it is very difficult
for a sound
wave to propagate efficiently; indeed, above a frequency of about
1.25 × 1013
hertz it is impossible for longitudinal waves to propagate at all, even
in a liquid
or a solid, because the molecules of the material in which the waves
are
traveling cannot pass the vibration along rapidly enough.
Ultrasonic is used for ultrasound waves and is defined as waves
with
frequency more than 20 kHz. They cannot be heard by human beings.
Supersonic is used for objects which travel at a speed greater than
the speed
of sound. These objects create sonic boom (sound associated with
supersonic
objects) generating enormous amounts of energy sounding like an
explosion.
Supersonic speed is the speed of an object that exceeds the speed
of sound
(Mach 1). For objects traveling in dry air of a temperature of 20 °C
(68 °F) at
sea level, this speed is approximately 343.2 m/s (1,126 ft/s; 768 mph;
667.1 kn;
1,236 km/h). Speeds greater than five times the speed of sound
(Mach 5) are
often referred to as hypersonic. Flights during which only some
parts of the air
surrounding an object, such as the ends of rotor blades, reach
supersonic
speeds are called transonic. This occurs typically somewhere
between Mach
0.8 and Mach 1.2.
ELECTROSTATICS
A. Electric Charges and Fields
1. Phenomena of Electrostatics
Electrostatics, the study of electromagnetic phenomena that occur
when
there are no moving charges—i.e., after a static equilibrium has
been
established. Charges reach their equilibrium positions rapidly,
because the
electric force is extremely strong. The mathematical methods of
electrostatics
make it possible to calculate the distributions of the electric field
and of
the electric potential from a known configuration of charges,
conductors,
and insulators. Conversely, given a set of conductors with known
potentials, it
is possible to calculate electric fields in regions between the
conductors and to
determine the charge distribution on the surface of the conductors.
The electric
energy of a set of charges at rest can be viewed from the standpoint
of
the work required to assemble the charges; alternatively, the energy
also can
be considered to reside in the electric field produced by this
assembly of
charges. Finally, energy can be stored in a capacitor; the energy
required to
charge such a device is stored in it as electrostatic energy of the
electric field.
Electrostatics is a branch of physics that studies electric charges at
rest.
Since classical physics, it has been known that some materials,
such as amber,
attract lightweight particles after rubbing. The Greek word for
amber, ήλεκτρον, or electron, was thus the source of the word
“electricity”
Electrostatic phenomena arise from the forces that electric charges
exert on
each other. Such forces are described by Coulomb’s law. Even
though
electrostatically induced forces seem to be rather weak, some
electrostatic
forces such as the one between an electron and a proton, that
together make
up a hydrogen atom, is about 36 orders of magnitude stronger than
the gravitational force acting between them.
There are many examples of electrostatic phenomena, from those
as simple
as the attraction of the plastic wrap to one’s hand after it is
removed from a
package to the apparently spontaneous explosion of grain silos, the
damage of
electronic components during manufacturing, and photocopier
&amp; laser
printer operation. Electrostatics involves the buildup of charge on
the surface of objects due to contact with other surfaces. Although
charge
exchange happens whenever any two surfaces contact and separate,
the
effects of charge exchange are usually only noticed when at least
one of the
surfaces has a high resistance to electrical flow. This is because
the charges
that transfer are trapped there for a time long enough for their
effects to be
observed. These charges then remain on the object until they either
bleed off
to ground or are quickly neutralized by a discharge: e.g., the familiar
phenomenon of a static “shock” is caused by the neutralization of
charge built
up in the body from contact with insulated surfaces.
2. Electrification
Electrification refers to the process of replacing technologies that
use fossil
fuels (coal, oil, and natural gas) with technologies that use
electricity as a
source of energy. Depending on the resources used to generate
electricity,
electrification can potentially reduce carbon dioxide (CO2 )
emissions from the
transportation, building, and industrial sectors, which account for 63
percent of all US greenhouse gas emissions. Addressing emissions
from
these sectors is critical to decarbonizing the economy and,
ultimately,
mitigating the impacts of climate change. This explainer reviews
how
electrification can reduce emissions; possibilities and potential
challenges of
electrification in the transportation, building, and industrial sectors;
and policy
options for encouraging electrification.
3. Positive and Negative Charges
Electric charge is the physical property of matter that causes it to
experience
a force when placed in an electromagnetic field. There are two
types of electric
charge: positive and negative (commonly carried by protons and
electrons
respectively). Like charges repel each other and unlike charges
attract each
other. An object with an absence of net charge is referred to as
neutral. Early
knowledge of how charged substances interact is now called
classical
electrodynamics, and is still accurate for problems that do not
require
consideration of quantum effects.
4. Displacement of Charges
Electric displacement, auxiliary electric field or electric vector that
represents that aspect of an
electric field associated solely with the presence of separated free
electric
charges , purposely excluding the contribution of any electric
charges bound
together in neutral atoms or molecules. If electric charge is
transferred
between two originally uncharged parallel metal plates, one
becomes
positively charged and the other negatively charged by the same
amount, and
an electric field exists between the plates. If a slab of insulating
material is
inserted between the charged plates, the bound electric
charges comprising the internal structure of the insulation are
displaced
slightly, or polarized; bound negative charges (atomic electrons)
shift a fraction
of an atomic diameter toward the positive plate, and bound positive
charges
shift very slightly toward the negative. This shift of charge, or
polarization,
reduces the value of the electric field that was present before the
insertion of
the insulation. The actual average value of the electric field E,
therefore, has a
component P that depends on the bound polarization charges and a
component D, electric displacement, that depends on the free
separated
charges on the plates. The relationship among the three vectors D, E,
P in the
metre-kilogram-second (mks) or SI system is: D = ε 0 E + P (ε 0 is a
constant,
the permittivity of a vacuum). In the centimetre- gram-second (cgs)
system
the relationship is: D = E + 4πP. The value of the electric
displacement D may
be thought of as equal to the amount of free charge on one plate
divided by the
area of the plate. From this point of view D is frequently called the
electric
flux density, or free charge surface density, because of the close
relationship
between electric flux and electric charge. The dimensions of
electric
displacement, or electric fluxdensity, in the metre-kilogram- second
system are
charge per unit area, and the units are coulombs per square metre.
In the
centimetre-gram-second system the dimensions of D are the same
as those of
the primary electric field E, the units of which are dynes per
electrostatic unit,
or statvolts per centimetre.
5. Electron Theory and Atomic Structure
Atomic structure refers to the structure of an atom comprising
a nucleus (centre) in which the protons (positively charged)
and neutrons (neutral) are present. The negatively charged particles
called electrons revolve around the centre of the nucleus. The
history of
atomic structure and quantum mechanics dates back to the times of
Democritus, the man who first proposed that matter is composed of
atoms.
The study about the structure of an atom gives a great insight into
the entire
class of chemical reactions, bonds and their physical properties.
The first
scientific theory of atomic structure was proposed by John Dalton in
the
1800s.The atomic structure of an element refers to the constitution
of its
nucleus and the arrangement of the electrons around it. Primarily,
the atomic
structure of matter is made up of protons , electrons and
neutrons.The protons and neutrons make up the nucleus of the atom,
which is
surrounded by the electrons belonging to the atom. The atomic
number of an
element describes the total number of protons in its nucleus.
6. Conductors and Insulators
Conductors
Conductors are materials that permit electrons to flow freely from
particle to particle. An object made of a conducting material will
permit charge
to be transferred across the entire surface of the object. If charge is
transferred
to the object at a given location, that charge is quickly distributed
across the
entire surface of the object.
In a conductor, electric current can flow freely, in an insulator it
cannot.
Metals such as copper typify conductors, while most non-metallic
solids are
said to be good insulators, having extremely high resistance to the
flow of
charge through them implies that the outer electrons of the atoms
are loosely
bound and free to move through the material. Most atoms hold on to
their
electrons tightly and are insulators. In copper, the valence electrons
are
essentially free and strongly repel each other. Any external
influence which
moves one of them will cause a repulsion of other electrons which
propagates,
through the conductor.
Insulators
Most solid materials are classified as insulators because they offer
very
large resistance to the flow of electric current. Metals are classified
as conductors because their outer electrons are not tightly bound,
but in most
materials even the outermost electrons are so tightly bound that
there is
essentially zero electron flow through them with ordinary voltages.
Some
materials are particularly good insulators and can be characterized
by their
high resistivities.
7. The Leaf Electroscope
The leaf electroscope is a common piece of demonstration
equipment found in
many high school and introductory college physics laboratories. Its
simplicity
allows a compelling demonstration of electrostatic forces, and its
versatility
makes it useful in the demonstration of a number of physical
phenomena. The
electroscope has a long history; a device for detecting net static
charge using a
rotating needle, the versorium, was described by Gilbert in De
Magnete in 1600.
The leaf electroscope was invented by Bennet and described in a
letter
published by the Royal Society in 1787.
The interaction between charged objects is a non-contact force that
acts over
some distance of separation. Charge, charge and distance. Every
electrical
interaction involves a force that highlights the importance of these
three
variables. Whether it is a plastic golf tube attracting paper bits, two
like-charged balloons repelling or a charged Styrofoam plate
interacting with
electrons in a piece of aluminum, there is always two charges and a
distance
between them as the three critical variables that influence the
strength of the
interaction. In this section of Lesson 3, we will explore the
importance of these
three variables.
The gold-leaf electroscope was developed in 1787 by British
clergyman and
physicist Abraham Bennet, as a more sensitive instrument than pith
ball
or straw blade electroscopes then in use. It consists of a vertical
metal rod,
usually brass, from the end of which hang two parallel strips of thin
flexible gold leaf. A disk or ball terminal is attached to the top of the
rod, where
the charge to be tested is applied. To protect the gold leaves from
drafts of air
they are enclosed in a glass bottle, usually open at the bottom and
mounted
over a conductive base. Often there are grounded metal plates or
foil strips in
the bottle flanking the gold leaves on either side. These are a safety
measure;
if an excessive charge is applied to the delicate gold leaves, they
will touch the
grounding plates and discharge before tearing. They also capture
charge
leaking through the air that accumulates on the glass walls,
increasing the
sensitivity of the instrument. In the precision instruments the inside
of the bottle
was occasionally evacuated, to prevent the charge on the terminal
from
leaking off through the ionization of the air.
8. Force between Point Charges
Coulomb’s law, or Coulomb’s inverse-square law, is an
experimental law of physics that quantifies the amount of force
between two
stationary, electrically charged particles. The electric force
between charged
bodies at rest is conventionally called electrostatic force or
Coulomb
force. The law was first discovered in 1785 by French physicist
CharlesAugustin de Coulomb, hence the name. Coulomb&#39;s law was
essential to
the development of the theory of electromagnetism, maybe even its
starting
point, as it made it possible to discuss the quantity of electric
charge in a
meaningful way.
The law states that the magnitude of the electrostatic force of
attraction or
repulsion between two point charges is directly proportional to the
product of
the magnitudes of charges and inversely proportional to the square
of the
distance between them.
Here, k e is Coulomb’s constant (k e ≈ 8.988×10 9 N⋅ m 2 ⋅ C −2 ), q
1 and
q 2 are the signed magnitudes of the charges, and the scalar r is the
distance
between the charges.
The force is along the straight line joining the two charges. If the
charges have
the same sign, the electrostatic force between them is repulsive; if
they have
different signs, the force between them is attractive.
Being an inverse-square law, the law is analogous to Isaac Newton’s
inverse-square law of universal gravitation, but gravitational forces
are always
attractive, while electrostatic forces can be attractive or repulsive.
Coulomb’s
law can be used to derive Gauss’s law, and vice versa. In the case of
a single
stationary point charge, the two laws are equivalent, expressing the
same
physical law in different ways. The law has been tested extensively,
and
observations have upheld the law on the scale from 10 −16 m to 10 8
m.
9. System of Units in Electrostatics
The electrostatic system of units (CGS-ESU) is a system of units
used to
measure quantities of electric charge, electric current, and voltage
within
the centimeter–gram–second (or “CGS”) system of metric units. In
electrostatic units, electrical charge is defined by the force that it
exerts on
other charges.
Although the CGS units have mostly been supplanted by the MKSA
(meter–kilogram–second–ampere) or International System of Units
(SI) units,
the electrostatic units are still in occasional use in some
applications, most
notably in certain fields of physics such as in particle physics and
astrophysics.
The main electrostatic units are:
for electric
charge
The CGS-ESU units for magnetic quantities are seldom used, and
don’t have
special names. Sources tend to just use “esu” or the derived unit
expressed in
terms of the CGS base units. For example, the unit for magnetic
induction is g
1/2 /cm 3/2 , corresponding to c cgs gauss, and corresponding to c
cgs × 10
−4 tesla, where c cgs = c / (cm/s) = 29979245800 is the numeric part
of speed of light c expressed in CGS units.
10. Electric Fields and Electric Foeld Intensity
An electric field (sometimes E-field) is the physical field that
surrounds
electrically-charged particles and exerts force on all other charged
particles in
the field, either attracting or repelling them. It also refers to the
physical field for
a system of charged particles.Electric fields originate from electric
charges, or
from time-varying magnetic fields. Electric fields and magnetic
fields are both
manifestations of the electromagnetic force, one of the four
fundamental forces
(or interactions) of nature.
Electric fields are important in many areas of physics, and are
exploited
practically in electrical technology. In atomic physics and chemistry,
for
instance, the electric field is the attractive force holding the atomic
nucleus and
electrons together in atoms. It is also the force responsible for
chemical
bonding between atoms that result in molecules. Other applications
of electric
fields include motion detection via electric field proximity sensing
and an
increasing number of diagnostic and therapeutic medical uses. The
electric
field is defined mathematically as a vector field that associates to
each point in
space the (electrostatic or Coulomb) force per unit of charge
exerted on an
infinitesimal positive test charge at rest at that point.[4][5][6] The
derived SI
units for the electric field are volts per meter (V/m), exactly
equivalent to
newtons per coulomb (N/C).
The magnitude and direction of the electric field are expressed by
the value of
E, called electric field strength or electric field intensity or simply
the electric
field. Knowledge of the value of the electric field at a point, without
any specific
knowledge of what produced the field, is all that is needed to
determine what
will happen to electric charges close to that particular point.
What is Electric Field Intensity?
The space around an electric charge in which its influence can be
felt is known
as the electric field. The electric field intensity at a point is the
force
experienced by a unit positive charge placed at that point.
• Electric Field Intensity is a vector quantity.
• It is denoted by ‘E’.
• Formula: Electric Field = F/q.
• Unit of E is NC-1 or Vm-1
.
The electric field intensity due to a positive charge is always
directed away
from the charge and the intensity due to a negative charge is always
directed
towards the charge.
Due to a point charge q, the intensity of the electric field at a point d
units away
from it is given by the expression:
Electric Field Intensity (E) = q/[4πεd2] NC-1
The intensity of the electric field at any point due to a number of
charges is
equal to the vector sum of the intensities produced by the separate
charges.
Force Experienced by a Charge in Electric Field
The force experienced by a charge in an electric field is given by,
\vec{F}=Q\vec{E}F=QE. where E is the electric field intensity.
Special Cases:
• If Q is a positive charge, the force \vec{F}F acts in the direction of
\vec{E}E .
Acceleration a = F/m = QE/m.
• If Q is a negative charge, the force acts in a direction opposite to
\vec{E}E.
Acceleration a = F/m = – QE/m
A charge in an electric field experiences a force whether it is at rest
or moving.
The electric force is independent of the mass and velocity of the
charged
particle, it depends upon the charge.
11. Electric Field Intensity near an Isolated Point Charge.
Electric Field is a force produce by a charge near its surroundings.
This force is
exerted on other charges when brought in the vicinity of this field.
The SI unit of
Electric field is N/C. Electric field due to a charge at a point is the
force that a
unit positive charge would experience if placed at that point.
12. Lines of Force and Electric Field Intensity
Line of force, in physics, path followed by an electric charge free to
move in
an electric field or a mass free to move in a gravitational field, or
generally any
appropriate test particle in a given force field. More abstractly, lines
of force are
lines in any such force field the tangent of which at any point gives
the field
direction at that point and the density of which gives the magnitude
of the field.
The concept of lines of force was introduced into physics in the
1830s by the
English scientist Michael Faraday, who considered magnetic and
electric
effects in the region around a magnet or electric charge as a
property of the
region rather than an effect taking place at a distance from a cause.
The electric lines of force that represent the field of a positive
electric charge in
space consist of a family of straight lines radiating uniformly in all
directions
from the charge where they originate. A second positive charge
placed in the
field would travel radially away from the first charge.
In the case of a magnetic field, since no isolated unit pole has ever
been
discovered, the field lines are called lines of force only in the sense
that a small
magnet is forced to align itself in the direction of these field lines.
An electric
charge traveling along a magnetic field line undergoes no magnetic
force.
13. Gauss' Law
Gauss’s law states that the net flux of an electric field in a closed
surface is
directly proportional to the enclosed electric charge. It is one of the
four
equations of Maxwell’s laws of electromagnetism. It was initially
formulated by
Carl Friedrich Gauss in the year 1835 and relates the electric fields
at the
points on a closed surface and the net charge enclosed by that
surface.
14. Electrostatics in Nature: Lightning
One of the most fantastic displays of electricity in nature is lightning.
Lightning
occurs when large amounts of electrostatic energy builds up in
clouds from the
energy of storms. When electrically charged regions of clouds
discharge their
energy, a large flash of electricity can be seen in the sky. Lightning
may occur
from cloud to cloud or it can occur from cloud to the ground.
Lightning strikes carry huge amounts of energy. A typical lightning
strike carries
an electric current of over 30,000 amps and delivers 500 megajoules
of energy.
Lightning also creates a loud noise called thunder. This is because
the air
within lightning gets so hot, that it transforms into plasma for a
short period of
time. When the molecules of air turn from gas to plasma, their
expansion
causes a shockwave that we hear as thunder.
Lightning is a naturally occurring electrostatic discharge during
which
two electrically charged regions, both in the atmosphere or with one
on the
ground, temporarily equalize themselves, causing the instantaneous
release of
as much as one gigajoule of energy. [1][2][3] This discharge may
produce a
wide range of electromagnetic radiation, from very hot plasma
created by the
rapid movement of electrons, to brilliant flashes of visible light in
the form
of black-body radiation.
Lightning causes thunder, a sound from the shock wave which
develops as
gases in the vicinity of the discharge experience a sudden increase
in pressure.
Lightning occurs commonly during thunderstorms as well as other
types of
energetic weather systems, but volcanic lightning can also occur
during
volcanic eruptions.
The three main kinds of lightning are distinguished by where they
occur:
either inside a single thundercloud, between two different clouds, or
between
a cloud and the ground. Many other observational variants are
recognized,
including “heat lightning”, which can be seen from a great distance
but not
heard; dry lightning, which can cause forest fires; and ball lightning,
which is
rarely observed scientifically.
Humans have deified lightning for millennia. Idiomatic expressions
derived
from lightning, such as the English expression “bolt from the blue”,
are
common across languages.
B. Electric Potential
1. Concept of Electrical Phenomena
Electrical phenomena are commonplace and unusual events that
can be
observed and that illuminate the principles of the physics of
electricity and are
explained by them. Electrical phenomena are a somewhat arbitrary
division
of electromagnetic phenomena.
Electrical phenomena result from a fundamental property of matter:
electric charge. The atoms that constitute most matter we
encounter contain
charged particles. Protons and electrons each have one-unit charge,
but of
opposite sign. Atoms are generally neutral because the number of
electrons
and protons are the same.
2. Energy in an Electric Field
The energy of an electric field results from the excitation of the
space
permeated by the electric field. It can be thought of as the potential
energy that
would be imparted on a point charge placed in the field.
The electric energy of a set of charges at rest can be viewed from
the
standpoint of the work required to assemble the charges;
alternatively, the
energy also can be considered to reside in the electric field
produced by this
assembly of charges. Finally, energy can be stored in a capacitor;
the energy
required to charge such a device is stored in it as electrostatic
energy of the
electric field.
3. Potential Difference
Potential difference is the difference in the amount of energy that
charge
carriers have between two points in a circuit.
**Measured in Volts: **Potential difference (p.d.) is measured in
volts (V) and
is also called voltage. The energy is transferred to the electrical
components in
a circuit when the charge carriers pass through them. We use a
voltmeter to
measure potential difference (or voltage).
Potential Difference formula:** V = I x R**
The potential difference (which is the same as voltage) is equal to
the amount
of current multiplied by the resistance. A potential difference of one
Volt is
equal to one Joule of energy being used by one Coulomb of charge
when it
flows between two points in a circuit.
Measurements in Circuits
**Ammeters: **An ammeter measures the flow of current that
passes through it.
Ammeters have to be connected in series (in the same loop of the
circuit) with
the electrical component whose current you are measuring. For
example
component X above.
**Voltmeters: **Voltmeters measure the potential difference
(voltage) between
two points in a circuit. For example between two points either side
of
component X above. Voltmeters must always be connected in
parallel (on a
separate branch of the circuit) with the two points being measured.
**Current vs potential difference: **The current is a flow of charge.
Current is
measured through a component. Potential difference is the energy
used
between two points in a circuit, therefore it is measured between
two points
either side of a component. We describe this as the potential
difference
measured across a component.
4. Reference Point for Potential Difference and Potential at a
Point.
Electric potential, the amount of work needed to move a unit charge
from a
reference point to a specific point against an electric field .
Typically, the
reference point is Earth , although any point beyond the influence of
the
electric field charge can be used.
The potential energy U,U of a body at some point x,x is defined to be
the work
done on the object by an extra, imposed force to move it from a
reference
position to its current position. The reference point is called the
“zero point”of
potential energy as the potential energy will be zero there by
definition.
The zero point can be chosen by the physicist solving the problem.
Depending
on the situation this could be infinitely far away (often described as
“at infinity”,
the centre of the earth, sea level, the floor, the table top or any
other point. This
is because only potential energy differences between two locations
are ever
physically measurable.
5. Electric Potential near an Isolated Point Charge.
Point charges, such as electrons, are among the fundamental
building blocks
of matter. Furthermore, spherical charge distributions (like on a
metal sphere)
create external electric fields exactly like a point charge. The
electric potential
due to a point charge is, thus, a case we need to consider. Using
calculus to
find the work needed to move a test charge q from a large distance
away to a
distance of r from a point charge Q, and noting the connection
between work
and potential (W = −qΔV), it can be shown that the electric potential
V of a
point charge is V=kQ/r (Point Charge), where k is a constant equal to
9.0 × 109
N · m2/C2.
The potential at infinity is chosen to be zero. Thus V for a point
charge
decreases with distance, whereas E for a point charge decreases
with
distance squared:
E=F/q=kQ/r2
Recall that the electric potential V is a scalar and has no direction,
whereas the
electric field E is a vector. To find the voltage due to a combination
of point
charges, you add the individual voltages as numbers. To find the
total electric
field, you must add the individual fields as vectors, taking magnitude
and
direction into account. This is consistent with the fact that V is
closely
associated with energy, a scalar, whereas E is closely associated
with force, a
vector.
6. Equipotential Surfaces
The surface which is the locus of all points which are at the same
potential is
known as the equipotential surface. No work is required to move a
charge from
one point to another on the equipotential surface. In other words,
any surface
with the same electric potential at every point is termed as an
equipotential
surface.
If the points in an electric field are all at the same electric potential,
then they
are known as the equipotential points. If these points are connected
by a line
or a curve, it is known as an equipotential line. If such points lie on
a surface, it
is called an equipotential surface. Further, if these points are
distributed
throughout a space or a volume, it is known as an equipotential
volume.
7. Potential Gradient
In physics , chemistry and biology , a potential gradient is the local
rate
of change of the potential with respect to displacement, i.e. spatial
derivative, or gradient. This quantity frequently occurs in equations
of physical
processes because it leads to some form of flux . The vector that
represents
the rate at which a potential changes with position in a specified
direction
specifically : the rate of change with height of the atmospheric
electric potential.
The potential gradient represents the rate of change of potential
along with
displacement. In other words, it represents the slope along which
potential is
changing. ... Potential - The potential between 2 points can be
defined as the
difference between the electric potential energies between the 2
points.
8. Distribution of Charge on Irregular Conductor
Excess charges on a nonuniform conductor become concentrated at
the
sharpest points. Additionally, excess charge may move on or off the
conductor
at the sharpest points. The electrostatic repulsion of like charges is
most
effective in moving them apart on the flattest surface, and so they
become
least concentrated there. This is because the forces between
identical pairs of
charges at either end of the conductor are identical, but the
components of the
forces parallel to the surfaces are different. The component parallel
to the
surface is greatest on the flattest surface and, hence, more
effective in moving
the charge.
9. The Faraday Ice-Pail Experiments
Faraday’s ice pail experiment is a simple electrostatics experiment
performed
in 1843 by British scientist Michael Faraday that demonstrates the
effect
of electrostatic induction on a conducting container. For a container,
Faraday
used a metal pail made to hold ice, which gave the experiment its
name. The
experiment shows that an electric charge enclosed inside a
conducting shell
induces an equal charge on the shell, and that in an electrically
conducting
body, the charge resides entirely on the surface. It also
demonstrates the
principles behind electromagnetic shielding such as employed in the
Faraday
cage. The ice pail experiment was the first precise quantitative
experiment on
electrostatic charge. It is still used today in lecture demonstrations
and physics
laboratory courses to teach the principles of electrostatics.
10. The Van De Graaff Generator
A Van de Graaff generator is an electrostatic generator which uses
a
moving belt to accumulate electric charge on a hollow metal globe
on the top
of an insulated column, creating very high electric potentials . It
produces
very high voltage direct current (DC) electricity at low current levels.
It was
invented by American physicist Robert J. Van de Graaff in
1929.The potential difference achieved by modern Van de Graaff
generators
can be as much as 5 megavolts. A tabletop version can produce on
the order of
100,000 volts and can store enough energy to produce visible
electric sparks .
Small Van de Graaff machines are produced for entertainment, and
for physics
education to teach electrostatics ; larger ones are displayed in
some science
museums .
The Van de Graaff generator was developed as a particle
accelerator for
physics research; its high potential is used to accelerate subatomic
particles to great speeds in an evacuated tube. It was the most
powerful type
of accelerator of the 1930s until the cyclotron was developed. Van
de Graaff
generators are still used as accelerators to generate energetic
particle
and X-ray beams for nuclear research and nuclear medicine .
Particle-beam Van de Graaff accelerators are often used in a
“tandem”
configuration: first, negatively charged ions are injected at one end
toward the
high potential terminal, where they are accelerated by attractive
force toward
the terminal. When the particles reach the terminal, they are
stripped of some
electrons to make them positively charged and are subsequently
accelerated
by repulsive forces away from the terminal. This configuration
results in two
accelerations for the cost of one Van de Graaff generator, and has
the added
advantage of leaving the complicated ion source instrumentation
accessible
near ground potential. The voltage produced by an open-air Van de
Graaff
machine is limited by arcing and corona discharge to about 5
megavolts.
Most modern industrial machines are enclosed in a pressurized tank
of
insulating gas; these can achieve potentials of as much as about 25
megavolts.
MAGNETISM
1. Magnitsm in Matter
All matter exhibits magnetic properties when placed in an external
magnetic field.
Even substances like copper and aluminum that are not normally
thought of as having magnetic
properties are affected by the presence of a magnetic field such as
that produced by either pole of a
bar magnet. Depending on whether there is an attraction or
repulsion by the pole of a magnet,
matter is classified as being either paramagnetic or diamagnetic,
respectively. A few materials,
notably iron, show a very large attraction toward the pole of a
permanent bar magnet; materials of
this kind are called ferromagnetic.
In 1845 Faraday became the first to classify substances as either
diamagnetic or paramagnetic. He
based this classification on his observation of the force exerted on
substances in an inhomogeneous
magnetic field. At moderate field strengths, the magnetization M of
a substance is linearly
proportional to the strength of the applied field H. The magnetization
is specified by the magnetic
susceptibility χ (previously labeled χ m ), defined by the relation M =
χ H. A sample of
volume V placed in a field H directed in the x- direction and
increasing in that direction at a
rate dH/dx will experience a force in the x-direction of F = χμ 0 VH
(dH/dx). If the magnetic
susceptibility χ is positive, the force is in the direction of increasing
field strength, whereas if χ is
negative, it is in the direction of decreasing field strength.
Measurement of the force F in a known
field H with a known gradient dH/dx is the basis of a number of
accurate methods of determining
χ.
Substances for which the magnetic susceptibility is negative (e.g.,
copper and silver) are classified
as diamagnetic. The susceptibility is small, on the orde of −10 −5 for
solids and liquids and −10
−8 for gases. A characteristic feature of diamagnetism is that the
magnetic moment per unit mass
in a given field is virtually constant for a given substance over a very
wide range of temperatures.
It changes little between solid, liquid, and gas; the variation in the
susceptibility between solid or
liquid and gas is almost entirely due to the change in the number of
molecules per unit volume.
This indicates that the magnetic moment induced in each molecule
by a given field is primarily a
property characteristic of the molecule.
2. Magnetic Field Strength (Intensity)
Magnetic field strength, also called magnetic intensity or magnetic
field intensity, the part of
the magnetic field in a material that arises from an external current
and is not intrinsic to the
material itself. It is expressed as the vector H and is measured in
units of amperes per metre. The
definition of H is H = B/μ − M, where B is the magnetic flux density, a
measure of the actual
magnetic field within a material considered as a concentration of
magnetic field lines, or flux, per
unit cross-sectional area; μ is the magnetic permeability; and M is
the magnetization. The
magnetic field H might be thought of as the magnetic field produced
by the flow of current in wires
and the magnetic field B as the total magnetic field including also
the contribution M made by the
magnetic properties of the materials in the field. When a current
flows in a wire wrapped on a
soft-iron cylinder, the magnetizing field H is quite weak, but the
actual average magnetic field (B)
within the iron may be thousands of times stronger because B is
greatly enhanced by the alignment
of the iron’s myriad tiny natural atomic magnets in the direction of
the field.
3. Magnetic Permeability
Magnetic permeability also referred to as permeability in
electromagnetism is a property of a
magnetic material which supports the formation of a magnetic field.
The term was coined by Oliver
Heaviside in the year 1885. Magnetic permeability is a property that
basically allows magnetic lines
of force to pass through a material.
In other words, the magnetic permeability of a material can also be
said to be its magnetization
capability. This helps in determining how much of magnetic flux can
the material support which
will pass through it.
Magnetic permeability is defined as the ratio of the magnetic
induction to the magnetic intensity. It
is a scalar quantity and denoted by the symbol μ . Magnetic
permeability helps us measure a
material’s resistance to the magnetic field or measure of the degree
to which magnetic field can
penetrate through a material.
If the material has greater magnetic permeability, greater will be the
conductivity for
magnetic lines of force.
* Factors Affecting Magnetic Permeability
Permeability also depends on several factors such as the nature of
the material,
humidity, position in the medium, temperature, and frequency of the
applied force.
Magnetic permeability is always positive and can vary with a
magnetic field. Meanwhile,
the opposite of magnetic permeability is magnetic reluctivity.
* Magnetic Permeability Formula
Magnetic permeability formula is given as;
Magnetic permeability (μ) = B/H
Where B = magnetic intensity and H = magnetising field.
The SI unit of magnetic permeability is henries per meter (H/m) or
newtons per ampere
squared (N * A−2).
4. Types of Magnetic Substances
All types of materials and substances posses some kind of magnetic
properties which are listed
further down in this article. But normally the word “ magnetic
materials ” is used only for
ferromagnetic materials ( description below), however, materials
can be classified into following
categories based on the magnetic properties shown by them:
A. Paramagnetic materials
The materials which are not strongly attracted to a magnet are
known as paramagnetic material.
For example: aluminium, tin magnesium etc. Their relative
permeability is small but positive. For
example: the permeability of aluminium is: 1.00000065. Such
materials are magnetized only when
placed on a super strong magnetic field and act in the direction of
the magnetic field.
B. Diamagnetic materials
The materials which are repelled by a magnet such as zinc. mercury,
lead, sulfur, copper, silver,
bismuth, wood etc., are known as diamagnetic materials. Their
permeability is slightly less than
one. For example the relative permeability of bismuth is 0.00083,
copper is 0.000005 and wood is
0.9999995. They are slightly magnetized when placed in a very
string magnetic field and act in
the direction opposite to that of applied magnetic field. In
diamagnetic materials , the two relatively
weak magnetic fields caused due to the orbital revolution and and
axial rotation of
electrons around nucleus are in opposite directions and cancel each
other. Permanent magnetic
dipoles are absent in them, Diamagnetic materials have very little to
no applications in electrical
engineering.
C. Ferromagnetic materials
The materials which are strongly attracted by a magnetic field or
magnet is known as ferromagnetic
material for eg: iron, steel , nickel, cobalt etc. The permeability off
these materials is very very
high ( ranging up to several hundred or thousand). The opposite
magnetic effects of electron orbital
motion and electron spin do not eliminate each other in an atom of
such a material. There is a
relatively large contribution from each atom which aids in the
establishment of an internal magnetic
field, so that when the material is placed in a magnetic field, it’s
value is increased many times thee
value that was present in the free space before the material was
placed there.
D. Ferrites
Ferrites are a special group of ferromagnetic materials that occupy
an intermediate position
between ferromagnetic and non-ferromagnetic materials. They
consist of extremely fine particles
of a ferromagnetic material possessing high permeability , and are
held together with a binding
resin. The magnetization produced in ferrites is large enough to be
of commercial value but their
magnetic saturation are not as high as those of ferromagnetic
materials. As in the case of ferro
magnetics, ferrites may be soft or hard ferrites.
5. Atomic Theory of Magnetism
The atomic theory of magnetism was given by Weber and modified
by Ewing. According to this
theory:
magnet in itself, having a north
pole and a south pole of equal strength.
randomly oriented such that they form
closed chains.
realigned so that north poles of all
molecular magnets point in one direction and south poles of all
molecular magnets point in
the opposite direction.
e molecular magnets are fully aligned, the substance is
said to be saturated with
magnetism
acquire some kinetic energy. Some
of the molecules may get back to the closed chain arrangement.
That is why magnetism of the
specimen would reduce on heating.
6. Diamagnetism, Paramagnetism and Ferromagnetism
In most atoms, electrons occur in pairs. Electrons in a pair spin in
opposite directions. So, when
electrons are paired together, their opposite spins cause their
magnetic fields to cancel each other.
Therefore, no net magnetic field exists. Alternately, materials with
some unpaired electrons will
have a net magnetic field and will react more to an external field.
Most materials can be classified
as diamagnetic, paramagnetic, or ferromagnetic.
➢ Diamagnetic Materials
Diamagnetic materials have a weak, negative susceptibility to
magnetic fields. Diamagnetic
materials are slightly repelled by a magnetic field and do not retain
the magnetic properties when
the external field is removed. In diamagnetic materials all the
electrons are paired so there is no
permanent net magnetic moment per atom. Diamagnetic properties
arise from the realignment of
the electron paths under the influence of an external magnetic field.
Most elements in the periodic
table, including copper, silver, and gold, are diamagnetic.
➢ Paramagnetic Materials
Paramagnetic materials have a small, positive susceptibility to
magnetic fields. These materials are
slightly attracted by a magnetic field and do not retain the magnetic
properties when the external
field is removed. Paramagnetic properties are due to the presence of
some unpaired electrons, and
from the realignment of the electron paths caused by the external
magnetic field. Paramagnetic
materials include magnesium, molybdenum, lithium, and tantalum.
➢ Ferromagnetic Materials
Ferromagnetic Materials have a large, positive susceptibility to an
external magnetic field. They
exhibit a strong attraction to magnetic fields and are able to retain
their magnetic properties after
the external field has been removed. Ferromagnetic materials have
some unpaired electrons so their
atoms have a net magnetic moment. They get their strong magnetic
properties due to the presence
of magnetic domains. In these domains, large numbers of atom’s
moments (1012 to 1015) are
aligned parallel so that the magnetic force within the domain is
strong. When a ferromagnetic
material is in the unmagnetized state, the domains are nearly
randomly organized and the net
magnetic field for the part as a whole is zero. When a magnetizing
force is applied, the domains
become aligned to produce a strong magnetic field within the part.
Iron, nickel, and cobalt are
examples of ferromagnetic materials. Components with these
materials are commonly inspected
using the magnetic particle method.
7. Magnetic Poles and Dipoles
Magnetic pole, region at each end of a magnet where the external
magnetic field is strongest. A bar
magnet suspended in Earth ’ s magnetic field orients itself in a north
– south direction. The
north-seeking pole of such a magnet, or any similar pole, is called a
north magnetic pole. The
south-seeking pole, or any pole similar to it, is called a south
magnetic pole. Unlike poles of
different magnets attract each other; like poles repel each other.
The magnetic force between a pole of one long bar magnet and that
of another was described by an
inverse square law as early as 1750. If, for example, the separation
between the two poles is doubled,
the magnetic force diminishes to one-fourth its former value.
Breaking a magnet in two does not isolate its north pole from its
south pole. Each half is found to
have its own north and south poles. Magnetic forces, in fact, cannot
be traced to unit magnetic
poles of submicroscopic size in direct contrast to electric forces
that are caused by actual discrete
electric charges, such as electrons and protons. Indeed, magnetic
forces themselves also
fundamentally arise between electric charges when they are in
motion.
Magnetic dipole, generally a tiny magnet of microscopic to
subatomic dimensions, equivalent to a
flow of electric charge around a loop. Electrons circulating around
atomic nuclei, electrons
spinning on their axes, and rotating positively charged atomic nuclei
all are magnetic dipoles. The
sum of these effects may cancel so that a given type of atom may
not be a magnetic dipole. If they
do not fully cancel, the atom is a permanent magnetic dipole, as are
iron atoms. Many millions of
iron atoms spontaneously locked into the same alignment to form a
ferromagnetic domain also
constitute a magnetic dipole. Magnetic compass needles and bar
magnets are examples of
macroscopic magnetic dipoles.
The strength of a magnetic dipole, called the magnetic dipole
moment, may be thought of as a
measure of a dipole’s ability to turn itself into alignment with a given
external magnetic field. In a
uniform magnetic field, the magnitude of the dipole moment is
proportional to the maximum
amount of torque on the dipole, which occurs when the dipole is at
right angles to the magnetic
field. The magnetic dipole moment, often simply called the magnetic
moment, may be defined then
as the maximum amount of torque caused by magnetic force on a
dipole that arises per unit value of
surrounding magnetic field in vacuum.
When a magnetic dipole is considered as a current loop, the
magnitude of the dipole moment is
proportional to the current multiplied by the size of the enclosed
area. The direction of the dipole
moment, which may be represented mathematically as a vector, is
perpendicularly away from the
side of the surface enclosed by the counterclockwise path of
positive charge flow. Considering the
current loop as a tiny magnet, this vector corresponds to the
direction from the south to the north
pole. When free to rotate, dipoles align themselves so that their
moments point predominantly in
the direction of the external magnetic field. Nuclear and electron
magnetic moments are quantized,
which means that they may be oriented in space at only certain
discrete angles with respect to the
direction of the external field.
8. Magnetic Resonance
Magnetic resonance, absorption or emission of electromagnetic
radiation by electrons or atomic
nuclei in response to the application of certain magnetic fields. The
principles of magnetic
resonance are applied in the laboratory to analyze the atomic and
nuclear properties of matter.
Electron-spin resonance (ESR) was first observed in 1944 by a
Soviet physicist, Y.K. Zavoysky, in
experiments on salts of the iron group of elements. ESR has made
possible the study of such
phenomena as the structural defects that give certain crystals their
colour, the formation and
destruction of free radicals in liquid and solid samples, the
behaviour of free or conduction
electrons in metals, and the properties of metastable states
(excited states that are long-lived
because energy transfer from them by radiation does not occur) in
molecular crystals. Nuclear
magnetic resonance (NMR) of protons was first observed in the
United States in 1946 by Felix
Bloch, William W. Hansen, and Martin E. Packard and independently
by Edward M. Purcell,
Robert V. Pound, and Henry C. Torrey. Scientists soon observed NMR
in practically all the stable
nuclei with nuclear moments greater than zero (about 100 species).
Later discoveries with NMR
included electric quadrupole effects; an important shift of NMR
frequencies in metals; and the
splitting of energy levels in liquids resulting from variations in
chemical structure and the influence
of one nuclear spin on another.
A particle of matter that is spinning about its own axis or moving in
an orbit around some external
point acts like a gyroscope: it resists forces that tend to change its
state of motion. The measure of
this resistance is the mechanical angular momentum, which
depends on the mass of the particle, its
size or that of its orbit, and the angular velocity (the number of
revolutions per unit time). The
angular momentum is represented by a vector directed along the
axis of rotation. An electric charge
in such motion creates a magnetic field with strength and direction
represented by a magnetic
vector denoted μ. This vector, which is proportional to the
magnitude of the charge (instead of the
mass of a particle), measures the tendency of the charge ’s axis of
rotation to align itself in the
direction of an external magnetic field. The motion of a particle that
has both mass and charge is
characterized by both of these vectors, which will be collinear but
may be oppositely directed,
depending on the sign of the charge. If a bar magnet that is not
spinning is placed in a magnetic
field, its north pole seeks the south pole of the field, and it comes to
rest with its own field aligned
with the external field. Work would be required to change its
orientation; this means that the system
can store potential energy. The energy associated with the magnet
depends, therefore, on its
magnetic moment, the strength of the external magnetic field, and
the angle between the direction
of the moment of the magnet and the direction of the external field.
In magnetic-resonance devices, a weak oscillating field (H′) is
superimposed on a strong constant
field (H), and its vector rotates with an angular velocity (ω) in a
plane perpendicular to the direction
of the strong field. If the rate of rotation (ω) of the weak
superimposed field is different from the
Larmor frequency (ωL) of the precessing particle, the two rotating
fields will be out of phase; the
axis of the particle will successively be attracted and repelled by
the superimposed rotating field
during complete revolutions and will wobble only slightly. When they
are synchronized, however, a
steady force will act on the axis. In this situation, called resonance,
the orientation angle (and with
it the magnetic energy state) of the particle will suddenly change.
When a system is raised to a
higher state, energy is extracted from the superimposed field, and
vice versa. The use of an
oscillating field to produce resonance is sometimes called “driving a
resonance.”
9. The Magnetic Circuit
Magnetic circuit, closed path to which a magnetic field, represented
as lines of magnetic flux, is
confined. In contrast to an electric circuit through which electric
charge flows, nothing actually
flows in a magnetic circuit. In a ring-shaped electromagnet with a
small air gap, the magnetic field
or flux is almost entirely confined to the metal core and the air gap,
which together form the
magnetic circuit. In an electric motor, the magnetic field is largely
confined to the magnetic
pole pieces, the rotor, the air gaps between the rotor and the pole
pieces, and the metal frame. Each
magnetic field line makes a complete unbroken loop. All the lines
together constitute the total flux.
If the flux is divided, so that part of it is confined to a portion of the
device and part to another, the
magnetic circuit is called parallel. If all the flux is confined to a
single closed loop, as in a
ring-shaped electromagnet, the circuit is called a series magnetic
circuit.
10. Reluctances in Series and Parallel
The reluctance of a magnetic circuit is analogous to the resistance
of an electric circuit. Reluctance
depends on the geometrical and material properties of the circuit
that offer opposition to the
presence of magnetic flux. Reluctance of a given part of a magnetic
circuit is proportional to its
length and inversely proportional to its cross-sectional area and a
magnetic property of the given
material called its permeability. Iron, for example, has an extremely
high permeability as compared
to air so that it has a comparatively small reluctance, or it offers
relatively little opposition to the
presence of magnetic flux. In a series magnetic circuit, the total
reluctance equals the sum of the
individual reluctances encountered around the closed flux path. In a
magnetic circuit, in summary,
the magnetic flux is quantitatively equal to the magnetomotive force
divided by the reluctance.
11. Terrestrial Magnetism
The science of Terrestrial Magnetism is based on the fact that a
magnet, free to move about its
center of gravity, tends to assume a position of relative rest in an
approximately definite direction
with respect to the geographical meridian and the vertical at the
place of observation. That it should
do so must be due to the existence of a field of force which is
known as the terrestrial magnetic
field [1], and the systematized knowledge of that field, of its
variations in time and space, and of its
relations with the subject-matter of other branches of terrestrial and
cosmical physics, constitutes
the science of Terrestrial Magnetism.
Terrestrial Magnetism is the magnetism of the earth. It is also a
branch of geophysics that deals
with the phenomena of the earth’s magnetic condition.
Terrestrial magnetism is caused by the action of permanent sources,
which are located within the
earth and experience only slow secular variations, and by external
(variable) sources, which are
located in the earth ’ s magnetosphere and ionosphere.
Correspondingly, a distinction is made
betweenthe main geomagnetic field (~ 99 percent) and the variable
geomagnetic field
(~ I percent).
OPTICS
(Eye and Optical Instruments)
1. The Human Eye
The human eye is a sense organ that reacts to light and
allows vision . Rod and cone cells in the retina are photoreceptive
cells which are able
to detect visible light and convey this information to the brain . Eyes
signal information which
is used by the brain to elicit the perception of color, shape, depth,
movement, and other features.
The eye is part of the sensory nervous system . Similar to the eyes
of other mammals , the human
eye’s non-image- forming photosensitive ganglion cells in the retina
receive light signals
which affect adjustment of the size of the pupil, regulation and
suppression of the
hormone melatonin , and entrainment of the circadian rhythm .The
eye has six muscles which
control the eye movement, all providing different tension and torque.
The eye works a lot like a
camera, the pupil provides the f-stop, the iris the aperture stop, the
cornea resembles a lens. The
way that the image is formed is much like the way a convex lens
forms an image.
2. Refractive Errors
Refractive error means that the shape of your eye does not bend
light correctly, resulting in a
blurred image. The main types of refractive errors
are myopia (nearsightedness), hyperopia (farsightedness),
presbyopia (loss of near vision with age),
and astigmatism.
3. Sensitivity of the Eye and Persistence of Vision
The sensitivity of the human eye to light of a certain intensity varies
strongly over
wavelengths between 380 nm and 800 nm. Under daylight conditions,
the average normal sighted
human eye is most sensitive at a wavelength of 555 nm, resulting in
the fact that green light at
this wavelength produces the impression of highest “ brightness ”
when compared to light at
other wavelengths. The spectral sensitivity function of the average
human eye under daylight
conditions (photopic vision) is defined by the CIE spectral luminous
efficiency function V(λ). Only
in very rare cases is the spectral sensitivity of the human eye under
dark adapted
conditions (scotopic vision), defined by the spectral luminous
efficiency function V&#39;( λ ),
technically relevant. By convention, these sensitivity functions are
normalized to a value of 1 in
their maximum. Persistence of vision traditionally refers to the
optical illusion that occurs when
visual perception of an object does not cease for some time after
the rays of light proceeding from it
have ceased to enter the eye. The illusion has also been described
as “retinal persistence”,
“ persistence of impressions “, simply “ persistence” and other
variations.
Persistence of vision is the optical phenomenon where the illusion
of motion is created because the
brain interprets multiple still images as one. When multiple images
appear in fast enough
succession, the brain blends them into a single, persistent, moving
image.
4. Stereoscopic Vision and Limitations of Vision
Stereoscopic vision describes the ability of the visual brain to
register a sense of three-dimensional
shape and form from visual inputs. In current usage, stereoscopic
vision often refers uniquely to the
sense of depth derived from the two eyes. This usage excludes a
number of things that might be
considered stereoscopic vision, such as the sense of depth arising
from the motion parallax
generated when subjects translate themselves through the visual
environment. This article is
primarily concerned with binocular stereoscopic vision. The human
eye is a remarkable biological
invention, a shining triumph of the process of evolution. Although
the human eye was the detector
that started us on mankind’s exploration of the Cosmos, it has some
shortcomings that ultimately
limit that exploration:
a. The eye has limited size and therefore limited light-gathering
power.
b. The eye has limited frequency response, since it can only see
electromagnetic
radiation in the visible wavelengths.
c. The eye distinguishes a new image multiple times a second, so it
cannot be used
to accumulate light over a long period in order to intensify a faint
image.
d. The eye cannot store an image for future reference like a
photographic plate can.
5. Magnifier and the Microscope
It is instinctive, when one wishes to examine the details of an
object, to bring it as near as possible
to the eye. The closer the object is to the eye, the larger the angle
that it subtends at the eye, and
thus the larger the object appears. If an object is brought too close,
however, the eye can no longer
form a clear image. The use of the magnifying lens between the
observer and the object enables the
formation of a “virtual image” that can be viewed in comfort. To
obtain the best possible image,
the magnifier should be placed directly in front of the eye. The
object of interest is then brought
toward the eye until a clear image of the object is seen.
Without lenses, the highest possible magnification is when the
object is brought to the closest
position at which a clear virtual image is observed. For many people,
this image distance is about
25 cm (10 inches). As people age, the nearest point of distinct vision
recedes to greater distances,
thus making a magnifier a useful adjunct to vision for older people.
The magnifying power, or extent to which the object being viewed
appears enlarged, and the field
of view, or size of the object that can be viewed, are related by the
geometry of the optical system.
A working value for the magnifying power of a lens can be found by
dividing the least distance of
distinct vision by the lens’ focal length, which is the distance from
the lens to the plane at which
the incoming light is focused. Thus, for example, a lens with a least
distance of distinct vision of 25
cm and a focal length of 5 cm (2 inches) will have a magnifying
power of about 5×. If the diameter
of the magnifying lens is sufficient to fill or exceed the diameter of
the pupil of the eye, the virtual
image that is viewed will appear to be of substantially the same
brightness as the original object.
The field of view of the magnifier will be determined by the extent to
which the magnifying lens
exceeds this working diameter and also by the distance separating
the lens from the eye. The clarity
of the magnified virtual image will depend upon the aberrations
present in the lens, its contour, and
the manner in which it is used.
6. Refracting and Reflecting Telescopes
A refracting telescope (also called a refractor) is a type of optical
telescope that uses a lens as its
objective to form an image (also referred to a dioptric telescope).
The refracting telescope design
was originally used in spy glasses and astronomical telescopes but
is also used for long-focus
camera lenses.
A reflecting telescope (also called a reflector) is a telescope that
uses a single or a combination
of curved mirrors that reflect light and form an image . The
reflecting telescope was invented
in the 17th century by Isaac Newton as an alternative to the
refracting telescope which, at that
time, was a design that suffered from severe chromatic aberration .
Each telescope has its own
advantage, for instance the refractor is better for observing the
planets and the moon and
the reflector for deep-sky objects (e.g. galaxies). However, the
refractor suffers from what is known
as chromatic aberration.
The difference between Reflecting and Refracting Telescopes is
that a reflecting telescope has a
single or a combination of curved mirrors in order to reflect the light
rays and form an image. On
the other hand, an refracting telescope has a lens as its objective
lens to form an image.
Each telescope has its own advantage, for instance the refractor is
better for observing the planets
and the moon and the reflector for deep-sky objects (e.g. galaxies).
However, the refractor suffers
from what is known as chromatic aberration.
7. The Prism Binocular
In 1851 an Italian, Ignatius Porro, devised a very ingenious and yet
simple arrangement of prisms
by which the simple astronomical telescope might yield an erect
image. An instrument was
constructed with these prisms by Boulanger, in 1859, and again in
1875 by Nachet, the firm so well
known in connection with the binocular microscope. Neither of
those makers succeeded in making
it popular, however, probably partly because of the quality of the
glass of which the prisms were
made, and partly because the prisms were not well enough worked
to give good images— the light
is four times reflected, and it is obvious that if the reflecting faces
are not all perfectly flat the
definition will be seriously impaired. In 1893 Ernst Abbe designed an
instrument, making use of the
new glass obtained by Schott; the resulting “prism-binoculars “made
under the modern conditions
were an immediate success. The faces of the prisms are tested by
Newton's bands of colour. These
bands must be perfectly straight right up to the edge. The refracting
surfaces are tested, as well as
the reflecting, though perfection of the latter is the more important.
8. The Photographic Camera
A camera is an optical instrument that captures a visual image. At
their most basic, cameras are
sealed boxes (the camera body) with a small hole (the aperture) that
allows light in to capture an
image on a light-sensitive surface (usually photographic film or a
digital sensor). Cameras have
various mechanisms to control how the light falls onto the lightsensitive surface. Lenses focus the
light entering the camera, the size of the aperture can be widened or
narrowed to let more or less
light into the camera, and a shutter mechanism determines the
amount of time the photo-sensitive
surface is exposed to the light. The still image camera is the main
instrument in the art
of photography and captured images may be reproduced later as a
part of the process of
photography, digital imaging, photographic printing. The similar
artistic fields in the moving image
camera domain are film, videography, and cinematography. The
word camera comes from camera
obscura, which means “dark chamber”
and is the Latin name of the original device for projecting an image
of external reality onto a flat
surface. The modern photographic camera evolved from the camera
obscura. The functioning of the
camera is very similar to the functioning of the human eye. The first
permanent photograph was
made in 1825 by Joseph
Nicéphore Niépce.