1) The document discusses kinetic and potential energy, introducing concepts like conservative and non-conservative forces. Conservative forces have potential energies and their work is path independent.
2) Examples are provided to illustrate these concepts, such as calculating the work done by different forces on a particle moving between two points along different paths.
3) The principle of conservation of energy is described for systems with only conservative forces, where the total mechanical energy (kinetic plus potential) remains constant. For systems with non-conservative forces, the change in mechanical energy equals the work done by those forces.
7-1 KINETIC ENERGY
After reading this module, you should be able to . . .
7.01 Apply the relationship between a particle’s kinetic
energy, mass, and speed.
7.02 Identify that kinetic energy is a scalar quantity.
7-2 WORK AND KINETIC ENERGY
After reading this module, you should be able to . . .
7.03 Apply the relationship between a force (magnitude and
direction) and the work done on a particle by the force
when the particle undergoes a displacement.
7.04 Calculate work by taking a dot product of the force vector and the displacement vector, in either magnitude-angle
or unit-vector notation.
7.05 If multiple forces act on a particle, calculate the net work
done by them.
7.06 Apply the work–kinetic energy theorem to relate the
work done by a force (or the net work done by multiple
forces) and the resulting change in kinetic energy. etc...
7-1 KINETIC ENERGY
After reading this module, you should be able to . . .
7.01 Apply the relationship between a particle’s kinetic
energy, mass, and speed.
7.02 Identify that kinetic energy is a scalar quantity.
7-2 WORK AND KINETIC ENERGY
After reading this module, you should be able to . . .
7.03 Apply the relationship between a force (magnitude and
direction) and the work done on a particle by the force
when the particle undergoes a displacement.
7.04 Calculate work by taking a dot product of the force vector and the displacement vector, in either magnitude-angle
or unit-vector notation.
7.05 If multiple forces act on a particle, calculate the net work
done by them.
7.06 Apply the work–kinetic energy theorem to relate the
work done by a force (or the net work done by multiple
forces) and the resulting change in kinetic energy. etc...
Charge Quantization and Magnetic MonopolesArpan Saha
This talk, given as a part of the Annual Seminar Weekend 2011, IIT Bombay, dealt with a homotopy-based
variant of the argument Dirac provided to show that the existence of a single magnetic monopole in the Universe
is a sufficient condition for the quantization of electric charge.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
11. kinetics of particles work energy methodEkeeda
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
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Fundamental of Physics "Potential Energy and Conservation of Energy"Muhammad Faizan Musa
8-1 POTENTIAL ENERGY
After reading this module, you should be able to . . .
8.01 Distinguish a conservative force from a nonconservative
force.
8.02 For a particle moving between two points, identify that
the work done by a conservative force does not depend on
which path the particle takes.
8.03 Calculate the gravitational potential energy of a particle
(or, more properly, a particle–Earth system).
8.04 Calculate the elastic potential energy of a block–spring
system.
8-2 CONSERVATION OF MECHANICAL ENERGY
After reading this module, you should be able to . . .
8.05 After first clearly defining which objects form a system,
identify that the mechanical energy of the system is the
sum of the kinetic energies and potential energies of those
objects.
8.06 For an isolated system in which only conservative forces
act, apply the conservation of mechanical energy to relate
the initial potential and kinetic energies to the potential and
kinetic energies at a later instant. etc...
Charge Quantization and Magnetic MonopolesArpan Saha
This talk, given as a part of the Annual Seminar Weekend 2011, IIT Bombay, dealt with a homotopy-based
variant of the argument Dirac provided to show that the existence of a single magnetic monopole in the Universe
is a sufficient condition for the quantization of electric charge.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
11. kinetics of particles work energy methodEkeeda
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes. Visit us: https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
Ekeeda Provides Online Video Lectures for Civil Engineering Degree Subject Courses for All Engineering Universities. Visit us: https://ekeeda.com/streamdetails/stream/civil-engineering
Ekeeda Provides Online Civil Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
Fundamental of Physics "Potential Energy and Conservation of Energy"Muhammad Faizan Musa
8-1 POTENTIAL ENERGY
After reading this module, you should be able to . . .
8.01 Distinguish a conservative force from a nonconservative
force.
8.02 For a particle moving between two points, identify that
the work done by a conservative force does not depend on
which path the particle takes.
8.03 Calculate the gravitational potential energy of a particle
(or, more properly, a particle–Earth system).
8.04 Calculate the elastic potential energy of a block–spring
system.
8-2 CONSERVATION OF MECHANICAL ENERGY
After reading this module, you should be able to . . .
8.05 After first clearly defining which objects form a system,
identify that the mechanical energy of the system is the
sum of the kinetic energies and potential energies of those
objects.
8.06 For an isolated system in which only conservative forces
act, apply the conservation of mechanical energy to relate
the initial potential and kinetic energies to the potential and
kinetic energies at a later instant. etc...
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The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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http://sandymillin.wordpress.com/iateflwebinar2024
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Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The Roman Empire A Historical Colossus.pdfkaushalkr1407
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
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physics430_lecture07.ppt
1. Physics 430: Lecture 7
Kinetic and Potential Energy
Dale E. Gary
NJIT Physics Department
2. September 22, 2009
We are now going to take up the conservation of energy, and its
implications. You have all seen this before, but now we will use a powerful,
more mathematical description.
You will see that the discussion is more complicated that the other
conservation laws for linear and angular momentum. The main reason is
that each type of momentum comes in only one flavor, whereas there are
many forms of energy (kinetic, several kinds of potential, thermal, etc.).
Processes transform one type of energy into another, and it is only the total
energy that is conserved, hence the additional complication.
We will be introducing new mathematical tools of vector calculus, such as
the gradient and the curl, which you may be familiar with, or not. I will
give you the needed background as they come up.
Chapter 4—Energy
3. September 22, 2009
An obvious form of energy is energy of motion, or kinetic energy. We will
use the symbol T, which is perhaps strange to you but is very much
standard in Classical Mechanics. The kinetic energy of a particle of mass m
traveling at speed v is defined to be:
Consider such a particle moving on some trajectory through space while its
kinetic energy changes, on moving from position r1 to r1 + dr. We can take
the time derivative of the kinetic energy, after writing , so that
But the first term on the right is the force . Thus, we can write
the derivative of kinetic energy as
Finally, multiplying both sides by dt and noting that v dt = dr, we have
4.1 Kinetic Energy and Work
2
2
1
mv
T
v
v
2
v
v
v
v
v
v
v
v
v
m
m
dt
d
m
dt
dT
)
(
)
( 2
1
2
1
v
p
F
m
v
F
dt
dT
r
F d
dT
r1
r1+dr
dr
Work-KE Theorem
4. September 22, 2009
The equation just derived is only valid for an infinitesimal displacement, but
we can extend this to macroscopic displacements by integrating, to get:
which says that the change in kinetic energy of a particle is equal to the
sum of force (in the direction of the displacement) times the incremental
displacement.
However, note that this is the displacement along the path of the particle.
Such an integral is called a line integral. In evaluating the integral, it is
usually possible to convert it into an ordinary integral over a single variable,
as in the following example (which we will look at in a moment).
With the notation of the line integral
where the last is a definition, defining the work done by F in moving from
point 1 to point 2. Note that F is the net force on the particle, but we can
also add up the work done by each force separately and write:
Line Integrals and Work
2
1
r
r
r
F d
T
)
2
1
(
2
1
1
2
W
d
T
T
T r
F
i
i
W
T
T )
2
1
(
1
2
5. September 22, 2009
Evaluate the line integral for the work done by the 2-d force F = (y, 2x)
going from the origin O to the point P = (1, 1) along each of the three paths:
a) OQ then QP
b) OP along x = y
c) OP along a circle
Path a):
Path b):
Example 4.1: Three Line Integrals
2
2
0
2
1
0
1
0
1
0
dy
xdy
ydx
d
d
d
W
P
Q
Q
O
a
a r
F
r
F
r
F
P
O Q
5
.
1
2
3
3
2
1
0
2
1
0
0
x
xdx
xdy
ydx
d
d
W
P
P
O
b
b r
F
r
F
6. September 22, 2009
Path c): This is a tricky one. Path c can be expressed as
so
This is a parametric equation, using q as a parameter along the path. With
this parameter, F = (sin q, 2(1cos q)). With this substitution:
The point here is that the line integral depends on the path, in general (but
not for special kinds of forces, which we will introduce in a moment).
Example 4.1: Three Line Integrals
21
.
1
4
/
2
cos
)
cos
1
(
2
sin
2
/
0
2
q
q
q
q
d
d
W
c
c r
F
)
sin
,
cos
1
(
)
,
( q
q
y
x
r
)
cos
,
(sin
)
,
( q
q
dy
dx
dr
7. September 22, 2009
We must now introduce the concept of potential energy corresponding to
the forces on an object. As you know, not every force lends itself to a
corresponding potential energy. Those that do are called conservative
forces.
There are two conditions that a force must satisfy to be considered a
conservative force.
The first condition for a force F to be conservative is that F depends only
on the position r of the object on which it acts. It cannot depend on
velocity, time, or any other parameter.
Although this is restrictive, there are plenty of forces that satisfy this
condition, such as gravity, the spring force, the electric force. You can often
see this directly, such as for the gravitational force:
4.2 Potential Energy and
Conservative Forces
r
r
F ˆ
)
( 2
r
GmM
Depends only on r.
8. September 22, 2009
The second condition for a force to be conservative concerns the work done
by the force as the object on which it acts moves between two points r1
and r2 (or just points 1 and 2, for short)
Reusing our earlier figure, we saw in Example 4.1 that the force described
there was NOT conservative, because it did different amounts of work for
the three paths a, b, and c.
Forces involving friction, obviously are not conservative, because if you
were sliding a box, say, on a surface with friction along the three paths
shown, the friction would do work , where L is different
for the three paths. Such forces are non-conservative.
Non-Conservative Forces
2
1
)
2
1
( r
F d
W
2
1
L
f
W fric
fric )
2
1
(
9. September 22, 2009
The force of gravity, on the other hand, has the property that the work
done is independent of the path. You know that if the height of point 1 and
point 2 differ by an amount h, then you will drop in height by h no matter
what path you take. In fact
independent of path.
The conditions for a force to be conservative, then, are:
Conservative Forces
mgh
W
)
2
1
(
grav
Conditions for a Force to be Conservative
A force F acting on a particle is conservative if and only if it satisfies
two conditions:
1. F depends only on the particle’s position r (and not on the velocity
v, or the time t, or any other variable); that is, F = F(r).
2. For any two points 1 and 2, the work W(1 2) done by F is the
same for all paths between 1 and 2.
10. September 22, 2009
The reason that forces meeting these conditions are called conservative is
that, if all of the forces on an object are conservative we can define a
quantity called potential energy, denoted U(r), a function only of position,
with the property that the total mechanical energy
is constant, i.e. is conserved.
To define the potential energy, we must first choose a reference point ro, at
which U is defined to be zero. (For gravity, we typically choose the
reference point to be ground level.) Then U(r), the potential energy, at any
arbitrary point r, is defined to be
In words, U(r) is minus the work done by F when the particle moves from
the reference point ro to the point r.
Potential Energy
)
(r
U
T
E
Potential Energy
r
r
r
r
F
r
r
r
o
)
(
)
(
)
( o d
W
U
11. September 22, 2009
Statement of the problem:
A charge q is placed in a uniform electric field pointing in the x direction with
strength Eo, so that the force on q is . Show that this force is
conservative and find the corresponding potential energy.
Solution:
The work done by F in going between any two points 1 and 2 along any path
(which is negative potential energy) is:
This work done is independent of the path, because the electric force depends
only on position, i.e. the force is conservative. To find the corresponding
potential energy, we must first choose a reference point at which U is zero. A
natural choice is to choose our origin (the point 1), in which case the potential
energy is
You may recall that Eo x is the electric potential V, so that qV is the potential
energy.
Example 4.2: Potential Energy of a
Charge in a Uniform Electric Field
)
(
ˆ
)
2
1
( 1
2
o
2
1
o
2
1
o
2
1
x
x
qE
dx
qE
d
qE
d
W
r
x
r
F
x
E
F ˆ
o
qE
q
x
qE
W
r
U o
)
0
(
)
(
r
12. September 22, 2009
The potential energy can be defined even when more than one force is
acting, so long as all of the forces are conservative. An important example
is when both gravity Fgrav and a spring force Fspr are acting (so long as the
spring obeys Hooke’s Law, F(r) = kr).
The work-kinetic energy theorem says that if we move an object subject to
these two forces along some path, the forces will do work independent of
the path (depending only on the two end-points of the path) given by
Rearrangement shows that
hence total mechanical energy is conserved. Extended to n such forces:
Several Forces
)
( spr
grav
spr
grav U
U
W
W
T
Principle of Conservation of Energy for One Particle
If all of the n forces Fi (i=1…n) acting on a particle are conservative, each with its
corresponding potential energy Ui(r), the total mechanical energy defined as
is constant in time.
)
(
)
(
1 r
r n
U
U
T
U
T
E
0
)
( spr
grav
U
U
T
13. September 22, 2009
As we have seen, not all forces are conservative, meaning we cannot define
a corresponding potential energy. As you might guess, in that case we
cannot define a conserved mechanical energy.
Nevertheless, if there are some conservative forces acting, for which a
potential energy can be defined, then we can divide the forces into a
conservative part Fcons, and a nonconservative part Fnc, such that
which allows us to write
What this says is that mechanical energy (T + U) is no longer conserved,
but any changes in mechanical energy are precisely equal to the work done
by the nonconservative forces.
In many problems, the only nonconservative force is friction, which acts in
the direction opposite the motion so that the work is negative.
Nonconservative Forces
nc
nc
cons
W
U
W
W
T
nc
W
U
T
)
(
r
f d
14. September 22, 2009
Example 4.3: Block Sliding Down
an Incline
We did this problem using forces in lecture 2. Let’s now apply these ideas
of energy to arrive at the same result.
As before, we have to identify the forces, and set a
coordinate system, but this time we write down the
potential and kinetic energies in the problem.
The kinetic energy, as always, is T = ½ mv2.
The gravitational potential energy is U = mgy, where we can set y = 0 (and
hence U = 0) at the ground level.
The friction force does negative work Wfric = fd, but recall that f = mN where
. Putting all of this together, becomes
where d is the distance along the incline, and y is the change in height.
If the block starts out with zero initial velocity at the top of the incline, and
we ask what is the speed v at the bottom, then y = h = d sin q, so
or
q
cos
mg
N fric
)
( W
U
T
q
mg
N
f
h
q
m cos
2
2
1
2
2
1
mgd
y
mg
mv
mv i
f
q
m
q cos
sin
2
2
1
mgd
mgd
mv
)
cos
(sin
2 q
m
q
gd
v
15. September 22, 2009
Comparison with Example 1.1
When we did this problem using forces, we obtained equation of motion
from which, after integration, we got the expression
Comparing with our just derived expression
they may seem quite different. What is happening is that using forces we
can get the velocity versus time, whereas with energy we are only getting
the speed at the end points. Energy considerations are very powerful if
you just want to know the result at a particular point, in which case you
can ignore the details of the motion in getting there. If you instead need
to know the path taken, or the details along the path, you have to use the
tools of Newton’s Laws.
However, we will find in a few weeks that these energy considerations do
contain all of the information of Newton’s Laws, and we will build the
tools necessary in Lagrangian mechanics to get the equation of motion
starting from energy. This allows us to attack much more complicated
problems. For this reason, it is important to get good at energy
problems. Here is an example you probably have seen before.
)
cos
(sin q
m
q
g
x
t
g
x )
cos
(sin q
m
q
)
cos
(sin
2 q
m
q
gd
v
)
cos
(sin
2
)
cos
(sin 2
2
1
q
m
q
q
m
q
g
d
t
t
g
d
16. September 22, 2009
Statement of the problem:
(a) The force exerted by a one-dimensional spring, fixed at one end, is F = kx,
where x is the displacement of the other end from its equilibrium position.
Assuming that this force is conservative (which it is) show that the corresponding
potential energy is U = ½ kx2, if we choose U = 0 at its equilibrium position.
Solution to (a):
We start with the definition of potential energy:
But we choose U = 0 at x = x1, which amounts to choosing x1 = 0, so that
Problem 4.9
)
(
)
2
1
(
)
( 2
1
2
2
2
1
2
1
2
1
x
x
k
xdx
k
Fdx
W
x
U
2
2
1
)
( kx
x
U
17. September 22, 2009
Statement of the problem:
(b) Suppose this spring is hung vertically from the ceiling with a mass m suspended
from the other end, and constrained to move in the vertical direction only. Find the
extension xo of the new equilibrium position with the suspended mass. Show that
the total potential energy (spring plus gravity) has the same form ½ ky2 if we use
the coordinate y equal to the displacement measured from the new equilibrium
position at x = xo (and redefine our reference point so that U = 0 at y = 0).
Solution to (b):
The new equilibrium position is reached when the force of the stretched
spring kxo equals the force of gravity on the mass mg. Thus
To define the potential energy at the new equilibrium position, we
have to examine the work done in displacing the mass a distance y:
Problem 4.9, cont’d
g
k
m
xo
2
2
1
2
2
1
o
0
o
0
spr
grav
)
(
(
)
(
)
0
(
)
(
ky
ky
y
kx
mgy
y
d
y
x
k
mg
y
d
F
F
y
W
y
U
y
y
xo
y=0