Download Elastic potential energy

INTRODUCTION
Energy exist in many forms. From time to time,we hear news about an “energy crisis”.
Usually newscaster are talking about shortage of energy available from burning fossil
fuels,such as oil and natural gas. You might find a barrel of oil and sit down in front it.
Where is the energy in it? It doesn’t seem to be doing anything. Its just a big container
of dark,thick liquid. However,if you light it on fire (don’t) the energy it contains become
vividly apparent. Energy is measured in Joules,just as is work. In fact,definition of
energy is “the capacity to do work”. [Stan Gibilisco,(2002)]
A skier slides down a hill and comes to rest at bottom. What became of potential energy
he or she had at top? The engine of a car is shut off while car is allowed to coast along a
level road. Eventually the car slows down and come to stop. What became of its original
kinetic energy?We can give any example of the apparent disappearance of kinetic or
potential energy. What these examples have is that heat is always produced in an amount
just equivalent to the lost energy.
[Energy can be created or destroyed,although it can be changed from one to form
another]
This generalization is the law of conservaton of energy. It is the principle with widest
application in science,applying equally to distant star and to biological process.[Athur
Beiser(2006)]
[In any energy transformation ,the total amount energy before and after the
transformation is constant]
This tell us something very important about energy. It cannot be created or destroyed.
The total amount of energy is constant. If we measure or calculate the total energy of a
system before a transformation and again afterwards,we will always get the some result.
If we find any difference,we must work for places where energy may be entering the
system or escaping unnoticed.[David Sang,(2001)]
TABLE OF CONTENTS
1.1
THE PRINCIPLE OF CONSERVATION OF ENERGY
1
2.1
KINETIC ENERGY
2
3.1
2.1.1
Kinetic Energy and Work-Kinetic Theorem
2.1.2
Kinetic Energy-Motion
2.1.3
Kinetic Energy
POTENTIAL ENERGY
3.1.1
Gravitational Potential Energy
3.1.2
Elastic Energy
3
4.0
KE-GPE TRANSFORMATIONS
8
5.0
CONSERVATION OF MECHANICAL ENERGY
9
6.0
CONSERVATIVE AND NON-CONSERVATION
11
CONCLUSION
12
BIBLIOGRAPHY
13
1.1
THE PRINCIPLE OF CONSERVATION OF ENERGY
The principle of conservation of energy defined that energy can be converted from to
another. Even the energy is convert but the total energy in an isolated system never
change. As we know,we have two type of energy that related each other . There are
kinetic and potential energy. For example,a coconut that fall from its tree have potential
energy to strike on the ground. From the energy,it will convert to kinetic energy. Here,we
will know the principle of conservation of energy happened. Our first explanation is
about kinetic energy. What is kinetic energy? We will know about it on next our slide.
After we explain about kinetic energy,we will explain about potential energy. Potential
energy have two subtopic. First is elastic potential energy,then second is gravitational
potential energy. Kinetic energy and potential energy are related each other. So,that
means we have to explain what are about of the both energy.
[David Sang,Physics(2001)]He say that kinetic energy is energy that make things to
move. You transfer energy to a ball when you throw it or hit it. A car uses energy from its
fuels to start moving. Elastic energy stored in a stretched piece a rubber is needed to fire a
pellet from catapult. So many object has energy,this energy known as KE(kinetic
energy).
Besides,Zitzewitz said in book of Glenco Phycics,kinetic energy is propotional to the
object’s mass. Thus,a 7.26kg shot thrown through the air has much more kinetic energy
than a 148g baseball with same velocity. The kinetic energy
of an object is also
proportional to the squre of velocity of the object.Zitzewitz,Glencoe (2002).
While potential energy is the energy associated with forces that depend on the position
or configuration of an object relative to the surrounding. Various type of potential
energy(PE) can be defined and each type associated with a particular force.
[Giancoli,Physics International Edition(1980),Sixth edition,United Stated America]
1
2.1
KINETIC ENERGY
2.1.1
Kinetic Energy and Work-Kinetic Theorem
Solving problems with Newton’s Second Law can be difficult if the forces
involved are complex. An alternative approach to such problems is to
relate the speed of an object to its displacement under the influence of
some net force. If the work done by the net force on the object can be
calculated for a given displacement,the change in the object’s speed is
easy to evaluate. [Serway Faughn ,(1998)]
2.1.2
Kinetic Energy-Motion
As we know,equation of kinetic energy is KE=1/2mv².
A moving object can do work on another object it strikes. A moving
object exerts a force on a second object which undergoes a displacement.
An object in motion has the ability to do work and thus can be said to have
energy. The energy of motion is called kinetic energy,from a Greek word
“Konetikos” meaning “motion”.[ Giancoli, (2005)]
Paul Peter said,kinetic energy is a highly visible form of energy associated
with the motion of a particle,single body,or system of object moving
together.[Paul Peter (2001) ]
2.1.3
Kinetic Energy
The kinetic energy defined which a body posseses solely because it is
moving. Its also defined as the amount of work it can do in coming to rest
or what amounts to the same thing,the amount of work that must have
been done on it to increase its velocity from zero to velocity it has.
The kinetic energy(KE) of an object depends on two factors:
(a)
mass of object
(b)
speed of object
2
3.1
POTENTIAL ENERGY
According to Douglas C. Gianacoli(2009), potential energy is the energy associated
with forces that depend on the position or configuration of objects relative to the
surroundings. Various types of potential energy can be defined, and each type is
associated with a particular conservative force.
The wound-up spring of a toy is an example of potential energy. The spring acquired its
potential energy because work was done on it by the person winding the toy. As the
spring unwinds, it exerts a force and does work to make the toy move.
Potential energy is the energy storred in a system. Examples of potential energy are the
gravitational potential energy and elastic potential energy.
3.2
Gravitational Potential Energy
3.2.1
Introduction
An example of a potential energy which is most common is gravitational potential
energy. A heavy brick held above the ground has potential energy because of its position
relative to the Earth. For if its released, the brick has the ability to do work as it will fall
to into the ground due to the gravitational force, and can do work on a stake, driving it
into the ground.
3.2.2
Definition
Based on the introduction above, gravitational potential energy is defined as the amount
of work that was done on it to give it that energy. On this basis, if a body of a mass m, is
at height h, then
3
GRAVITATIONAL POTENTIAL ENERGY = m g h
Where m = mass(kg)
g = (-9.81
)
h = height of the body above some arbitrary reference level(e.g. the
ground or a bench top) where the potential energy is taken to be
zero. (Raymond A. Serway)(1996)
FIGURE 1
An upward force
is being exerted to lift a brick from
Gravitational Potential Energy is the stored energy associated with the height of the
object according to Raymond A. Serway(1996).
4
To lift up an object of mass, m, to a certain height, h , the minimum amount of force is
the weight of the object, mg, as shown in Figure 1.
The work done = Fs cos O = (mg)(h) = mgh. The energy spent in doing this amount of
work is stored as potential energy of the object. Thus, potential energy, P.E. = mgh
When a book is placed on a table, the book has potential energy because it is higher than
the ground. If there is nothing to hold the book, it will fall and gain kinetic energy. In
other words, the energy of the book on the table is stored when it is resting on the table
If an object is at initial height,
, and its final height is at a higher location,
change of potential energy.
Diagram for The Principle of Conservation of Energy
5
there is a
The figure above shows a brick falling from a height,
a)
until it reaches the ground at
The principle of conservation of energy states that the total energy in a
system is constant.
b)
As a result, energy cannot be created or destroyed.
c)
When a brick of mass, m kg falls from a height of h metres to the ground,
it loses its gravitational potential energy which is changed into kinetic
energy of motion.
3.3
d)
Energy can be converted from one form to another
e)
If air resistance is ignored, the kinetic energy of the brick just before it hits
the ground is equal to its potential energy at the beginning.
Elastic potential energy
When a railroad car runs into a spring bumper at the end of the track, the spring is
compressed as the car is brought to a stop. If there is no friction, the bumper springs back
and the car moves away with its original speed in the opposite direction. During the
interaction with the spring, the car’s kinetic energy has been “stored” in the elastic
deformation of the spring, something very similar happens in a rubber-band slingshot.
Work is done on the rubber band by the force that stretches it, and that work is stored in
the rubber band until you let it go. Then the rubber band gives the kinetic energy to the
projectile. ( Hugh D. Young, Roger Freedman, T. R Sandin, A. Lewis Ford )
We continued just as we did for the gravitational potential energy. We should begin with
the work done by the elastic force of the spring and then combine this with the workenergy theorem. The difference is that gravitational potential energy is a shared property
of a body and the earth, but elastic potential energy is stored just in the spring or other
deformable body.
We found that the work we must do on the spring to move one end from an elongation x1
to a different elongation x2 is : 6
W = ½ kx2 ² - ½ kx1 ²
( work done on a spring )
Where k is the force constant of the spring. If we stretch the spring further, we do
positive work on the spring ; if we let the spring relax while holding one end, we do
negative work on it. We also see that this expression for work is still correct if the spring
compressed, not stretched, so that x1 or x2 or both are negative. Then, now we must find
the work done by the spring. As we know that from the Newton's third law ; the two
quantities of work are just negative of each other. Changing the signs in this equation, we
find that in a displacement from x1 to x2 the spring does an amount of work Wel given by :
Wel = ½ kx1 ² - ½ kx2 ²
( work done by a spring )
The subscript “el” means elastic.Just as for gravitational work, we can express the work
done by a spring in terms of a given quantity at the beginning and end of the
displacement. This quantity is ½ kx ².
and we defined it to be the elastic potential energy.
U = ½ kx ² ( elastic potential energy )
The unit of U is the joule (J), the unit used for all energy and work quantities. Then the
units of k are N/m and that 1 N . m = 1 J
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4.1
KINETIC ENERGY-GRAVITATIONAL POTENTIAL ENERGY
A ski-jumper has travelled uphill on a chair-lift. This increase his gravitational potential
energy (GPE). Now he can ski down-hill. His kinetic energy (KE) increases as his GPE
decrease. GPE is being transformed into KE. As he jumps from the ramp, he starts to rise
in the air. Now KE is being transformed into GPE.
There are many other situations where such transforms are going on. The following are
some example:a. An apple falls from the tree it accelerates downwards. GPE is being
transformed into KE.
b. On a roller-coaster ride, a car runs downhill, then back up the next slope.
GPE is being transformed into KE and then back to GPE. Because its
friction, some of energy is wasted as heat, so he car cannot reach its
original height.
c. A pendulum is swinging from side to side. At its highest point, it is
momentarily stationary. Now its GPE is a maximum but it has no KE as
it swings downwards, it speed up. GPE is being transformed into KE. At
lowest point of its swing, it is moving at its fast. It slow again it swings
back up. KE is being transformed back into GPE.
We can use these ideas to perform some useful calculation. We will state the principle of
conservation of energy for falling object in following useful form:
decrease in GPE = increase in KE
We can write this as an equation assuming that speed v is gained as the object
falls from rest through a height,h
mgh=mv²
This equation can be solved to give v when we know h, or to give h when we
know v.
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5.1
COSERVATION OF MECHANICAL ENERGY
Although there are many forms of energy,you will be concerned only with kinetic and
gravitational potential energy while investigating motion in this book. The sum of these
forms of energy is referred to mechanical energy. In any given system,if no other formss
of energy are present,
Mechanical Energy E= K + U g
Imagine
a system consisting of a ball and Earth. Suppose the ball has a weight of 10.0N
and will be released 2.00 m above the ground,which you define to be the reference level.
Because the ball isn't yet moving, it has no kinetic energy. Its potential energy is
represent by the following.
U g=m g h
U=(10N)(2.0)=20 J
The ball.s total mechanical energy is threfore 20.0J. As the ball falls, it loses potential
energy and gains kinetic energy. When it is 1.00 m above Earth's surface,its potential
energy is,
U g= m g h
U=(10N)(1.0m)=10.J
What is the ball's kinetic energy when it is at a height of 1.00 m? The system, which
consist of the ball and Earth,is closed and,with no external forces acting upon it, is
isolated. Thus its total energy remain constant at 20.0 J
E= K + Ug, so K = E-U g
K= 20.0 J– 10.0 J=10.J
When the ball reaches the ground,its potential energy is zero,and its kinetic energy is now
the full 20.0 J. The equation above describes conservation of mechanical energy is
E before = E after, which may be rewritten as follow.
K before + U before = K after + U after
(Paul w. Zitzewitz,2002)
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6.1
CONSERVATIVE AND NON-CONSERVATIVE
In our discussion of potential energy we have talked about “storing” kinetic energy by
converting it to potential energy. We always have in mind that later we may retrieve it
again as kinetic energy. When you throw a ball up in the air, it slows down as kinetic
energy is converted into potential energy. But on the way down, the conversion is
reversed,and the ball speeds up as potential energy is converted back to kinetic energy. If
there is no air resistance, the ball is moving just as fast when you catch it as when you
threw it.
If a glider moving on a frictionless horizontal air track runs into a spring bumper at the
end of the track,the spring compresses and the glider stops. But then it bounces back,and
if there is no friction,the glider has the same speed and kinetic energy it had before the
collision. Again,there is a two-way conversion from kinetic to potential energy and back.
In both cases we find that we can define a potential-energy function so that the total
mechanical,kinetic plus potential, is constant or conserved during the motion.
A force that offers this opportunity of two-way conversion between kinetic and potential
energies is called a conservative force. An essential feature of conservative forces is that
their work is always reversible. Anything that we deposit in the energy “bank” can later
be withdrawn without loss. Another important aspect of conservative forces is that a
body may move from point 1 to point 2 by various paths, but the work done by a
conservative force is the same for all of these paths. Thus,if a body stays close to surface
of the earth, the gravitational force mg is independent of height,and the work done by this
force depends only on the change in height. If the body moves around a closed path,
ending at the same point where it started,the total work done by the gravitational force is
always zero.
The work done by a conservative force always has these properties:
1
2
3
4
It can always be expressed as the difference between the initial and final
values of a potential energy function
It is reversible
It is independent of the path of the body and depends only on starting and
ending points.
When the starting and ending points are the same,the total work is zero.
10
6.2
Reducing Energy Waste
Efforts are made to use waste heat energy. Power stations produce a lot of waste heat, so
onmentthey are an abvious target for attention.The modern power station at Newscastleupon-Tyne in UK has two features designed to make it less damaging to the enviroment:
(a)
(b)
its burns lot of rubbish,collected by the local council
as well as as generating electricity,it uses most of its waste heat to provide
hot water to local homes and offices
Other modern technologies can help to reduce energy wastage. Since cars a major source
of polluting chemicals and waste heat,cleaner cars are most desirable. Some electrical
energy is passed to a battery, which can take over when the Sun is covered by cloud.
Another important technology that will help to clean up cars in the future is provided fuel
cell. These are form of battery. Fuels such as petrol or hydrogen is put into one
compartment,and oxygen into another . The two react and produce electricity. The fuel is
oxidised,just as if it had been burned ,but without producing the intermediate heat.
(David Sang,2001)
11
CONCLUSION
The gravitational potential energy of a particle of mass,m that is elevated a distance y
near the Earth 's surface is
U g=mgy
The elastic potential energy stored in a spring of force constant k is
2
U s=1/2 kx
A force conservative if the work it does on the particle is independent of the path the
particles between two points. Alternatively,a force is conservative if the work it does zero
when the particle moves through an arbitrary closed path and returns to its initial
position. A force that not meet these criteria is said to be non-conservative.
Kinetic energy is the energy of motion and defined to be one half the mass times the
speed squared . If the kinetic energy before a collision is the same as that after collision
,the collision conserves kinetic energy and is said to be elastic. Kinetic energy is
transformed into other forms of energy in inelastic collision.
Work is equal to its product of the force in the direction of motion and the distance
traveled. If the force is perpendicular to the velocity of the object, or if the object does
not move,no work is done by force. Then change in kinetic energy of an object is equal to
the work done on the object. The gravitational energy of an object depends on the weight
and its distance from Earth's surface.
The total energy of a closed, isolated system, energy can change form, but the total
amount of energy doesn't change. Momentum is conserved in collisions if the external
force is zero. The mechanical energy may be unchanged or whether the collision is
elastic or inelastic.
12
BIBLIOGRAPHY
1
Paul Peter,College Physics(2001),Second Edition,Urone.
2
Roger Muncaster,A-Level Physics(1993),Fourth Edition.
3
Stan Gibilisco,Physics Demystified(2002),United States America.
4
Douglas c. Giancoli,Physics for Scientist and Engineers(2009),Fourth
Edition,United States America.
5
Raymond a. Serway,Physics for Science and Engineers with Modern
Physics(1996),Orlando,Florida.
6
Young Freedman,University Physics(1998),Nineth Edition,India.
7
David Sang,Physics(2001),Cambridge University, United Kingdom.
8
Zitzewitz,Glencoe Physics in Principle Problems(2002),Columbus,Ohio.
9
Hugh D. Young,Exended Version with Modern Physics(1996),Addison-wesley.
10
Keith Burrow,Heinemain Physics(1996),Fourth Edition,Malaysia.
13