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  • Define conservative force, potential energy, and mechanical energy.
  • Use the work-energy theorem to show how having only conservative forces implies conservation of mechanical energy.

Potential energy and conservative forces

Work is done by a force, and some forces, such as weight, have special characteristics. A conservative force    is one, like the gravitational force, for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken. We can define a potential energy     ( PE ) size 12{ \( "PE" \) } {} for any conservative force, just as we did for the gravitational force. For example, when you wind up a toy, an egg timer, or an old-fashioned watch, you do work against its spring and store energy in it. (We treat these springs as ideal, in that we assume there is no friction and no production of thermal energy.) This stored energy is recoverable as work, and it is useful to think of it as potential energy contained in the spring. Indeed, the reason that the spring has this characteristic is that its force is conservative . That is, a conservative force results in stored or potential energy. Gravitational potential energy is one example, as is the energy stored in a spring. We will also see how conservative forces are related to the conservation of energy.

Potential energy and conservative forces

Potential energy is the energy a system has due to position, shape, or configuration. It is stored energy that is completely recoverable.

A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken.

We can define a potential energy ( PE ) size 12{ \( "PE" \) } {} for any conservative force. The work done against a conservative force to reach a final configuration depends on the configuration, not the path followed, and is the potential energy added.

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

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Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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Cied
types of nano material
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Source:  OpenStax, Concepts of physics. OpenStax CNX. Aug 25, 2015 Download for free at https://legacy.cnx.org/content/col11738/1.5
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