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PE s = 1 2 kx 2 , size 12{"PE" rSub { size 8{s} } = { {1} over {2} } ital "kx" rSup { size 8{2} } } {}

where k size 12{k} {} is the spring’s force constant and x size 12{x} {} is the displacement from its undeformed position. The potential energy represents the work done on the spring and the energy stored in it as a result of stretching or compressing it a distance x size 12{x} {} . The potential energy of the spring PE s size 12{"PE" rSub { size 8{s} } } {} does not depend on the path taken; it depends only on the stretch or squeeze x size 12{x} {} in the final configuration.

An undeformed spring fixed at one end with no potential energy. (b) A spring fixed at one end and stretched by a distance x by a force F equal to k x. Work done W is equal to one half k x squared. P E s is equal to one half k x squared. (c) A graph of force F versus elongation x in the spring. A straight line inclined to x axis starts from origin. The area under this line forms a right triangle with base of x and height of k x. Area of this triangle is equal to one half k x squared.
(a) An undeformed spring has no PE s size 12{"PE" rSub { size 8{s} } } {} stored in it. (b) The force needed to stretch (or compress) the spring a distance x size 12{x} {} has a magnitude F = kx size 12{F= ital "kx"} {} , and the work done to stretch (or compress) it is 1 2 kx 2 size 12{ { {1} over {2} } ital "kx" rSup { size 8{2} } } {} . Because the force is conservative, this work is stored as potential energy ( PE s ) size 12{ \( "PE" rSub { size 8{s} } \) } {} in the spring, and it can be fully recovered. (c) A graph of F size 12{F} {} vs. x size 12{x} {} has a slope of k size 12{k} {} , and the area under the graph is 1 2 kx 2 size 12{ { {1} over {2} } ital "kx" rSup { size 8{2} } } {} . Thus the work done or potential energy stored is 1 2 kx 2 .

The equation PE s = 1 2 kx 2 size 12{"PE" rSub { size 8{s} } = { {1} over {2} } ital "kx" rSup { size 8{2} } } {} has general validity beyond the special case for which it was derived. Potential energy can be stored in any elastic medium by deforming it. Indeed, the general definition of potential energy    is energy due to position, shape, or configuration. For shape or position deformations, stored energy is PE s = 1 2 kx 2 size 12{"PE" rSub { size 8{s} } = { {1} over {2} } ital "kx" rSup { size 8{2} } } {} , where k size 12{k} {} is the force constant of the particular system and x size 12{x} {} is its deformation. Another example is seen in [link] for a guitar string.

A six-string guitar is placed vertically. The left-most string is plucked in the left direction with a force F shown by an arrow pointing left. The displacement of the string from the mean position is d. The plucked string is labeled P E sub string, to represent the potential energy of the string.
Work is done to deform the guitar string, giving it potential energy. When released, the potential energy is converted to kinetic energy and back to potential as the string oscillates back and forth. A very small fraction is dissipated as sound energy, slowly removing energy from the string.

Conservation of mechanical energy

Let us now consider what form the work-energy theorem takes when only conservative forces are involved. This will lead us to the conservation of energy principle. The work-energy theorem states that the net work done by all forces acting on a system equals its change in kinetic energy. In equation form, this is

W net = 1 2 mv 2 1 2 mv 0 2 = Δ KE. size 12{W rSub { size 8{"net"} } = { {1} over {2} } ital "mv" rSup { size 8{2} } - { {1} over {2} } ital "mv" rSub { size 8{0} rSup { size 8{2} } } =Δ"KE" "." } {}

If only conservative forces act, then

W net = W c , size 12{W rSub { size 8{"net"} } =W rSub { size 8{c} } } {}

where W c is the total work done by all conservative forces. Thus,

W c = Δ KE. size 12{W rSub { size 8{c} } =Δ"KE"} {}

Now, if the conservative force, such as the gravitational force or a spring force, does work, the system loses potential energy. That is, W c = Δ PE size 12{W rSub { size 8{c} } = +- D"PE"} {} . Therefore,

Δ PE = Δ KE size 12{ - Δ"PE"=Δ"KE"} {}

or

Δ KE + Δ PE = 0 . size 12{Δ"KE"+Δ"PE"=0} {}

This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces. That is,

KE + PE = constant     or KE i + PE i = KE f + PE f } (conservative forces only),

where i and f denote initial and final values. This equation is a form of the work-energy theorem for conservative forces; it is known as the conservation of mechanical energy    principle. Remember that this applies to the extent that all the forces are conservative, so that friction is negligible. The total kinetic plus potential energy of a system is defined to be its mechanical energy    , ( KE + PE ) size 12{ \( "KE"+"PE" \) } {} . In a system that experiences only conservative forces, there is a potential energy associated with each force, and the energy only changes form between KE size 12{"KE"} {} and the various types of PE size 12{"PE"} {} , with the total energy remaining constant.

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
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Tamia
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Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Une: physics for the health professions. OpenStax CNX. Aug 20, 2014 Download for free at http://legacy.cnx.org/content/col11697/1.1
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