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How the work-energy theorem applies

Now let us consider what form the work-energy theorem takes when both conservative and nonconservative forces act. We will see that the work done by nonconservative forces equals the change in the mechanical energy of a system. As noted in Kinetic Energy and the Work-Energy Theorem , the work-energy theorem states that the net work on a system equals the change in its kinetic energy, or W net = ΔKE size 12{W rSub { size 8{"net"} } =D"KE"} {} . The net work is the sum of the work by nonconservative forces plus the work by conservative forces. That is,

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

so that

W nc + W c = Δ KE , size 12{W rSub { size 8{"nc"} } +W rSub { size 8{c} } =Δ"KE"} {}

where W nc size 12{W rSub { size 8{"nc"} } } {} is the total work done by all nonconservative forces and W c size 12{W rSub { size 8{c} } } {} is the total work done by all conservative forces.

A person pushing a heavy box up an incline. A force F p applied by the person is shown by a vector pointing up the incline. And frictional force f is shown by a vector pointing down the incline, acting on the box.
A person pushes a crate up a ramp, doing work on the crate. Friction and gravitational force (not shown) also do work on the crate; both forces oppose the person’s push. As the crate is pushed up the ramp, it gains mechanical energy, implying that the work done by the person is greater than the work done by friction.

Consider [link] , in which a person pushes a crate up a ramp and is opposed by friction. As in the previous section, we note that work done by a conservative force comes from a loss of gravitational potential energy, so that W c = Δ PE size 12{W rSub { size 8{c} } = - Δ"PE"} {} . Substituting this equation into the previous one and solving for W nc size 12{W rSub { size 8{"nc"} } } {} gives

W nc = Δ KE + Δ PE. size 12{W rSub { size 8{"nc"} } =Δ"KE"+Δ"PE"} {}

This equation means that the total mechanical energy ( KE + PE ) size 12{ \( "KE + PE" \) } {} changes by exactly the amount of work done by nonconservative forces. In [link] , this is the work done by the person minus the work done by friction. So even if energy is not conserved for the system of interest (such as the crate), we know that an equal amount of work was done to cause the change in total mechanical energy.

We rearrange W nc = Δ KE + Δ PE size 12{W rSub { size 8{"nc"} } =D"KE"+D"PE"} {} to obtain

KE i + PE i + W nc = KE f + PE f . size 12{"KE""" lSub { size 8{i} } +"PE" rSub { size 8{i} } +W rSub { size 8{"nc"} } ="KE""" lSub { size 8{f} } +"PE" rSub { size 8{f} } } {}

This means that the amount of work done by nonconservative forces adds to the mechanical energy of a system. If W nc size 12{W rSub { size 8{"nc"} } } {} is positive, then mechanical energy is increased, such as when the person pushes the crate up the ramp in [link] . If W nc size 12{W rSub { size 8{"nc"} } } {} is negative, then mechanical energy is decreased, such as when the rock hits the ground in [link] (b). If W nc size 12{W rSub { size 8{"nc"} } } {} is zero, then mechanical energy is conserved, and nonconservative forces are balanced. For example, when you push a lawn mower at constant speed on level ground, your work done is removed by the work of friction, and the mower has a constant energy.

Applying energy conservation with nonconservative forces

When no change in potential energy occurs, applying KE i + PE i + W nc = KE f + PE f size 12{"KE""" lSub { size 8{i} } +"PE" rSub { size 8{i} } +W rSub { size 8{"nc"} } ="KE""" lSub { size 8{f} } +"PE" rSub { size 8{f} } } {} amounts to applying the work-energy theorem by setting the change in kinetic energy to be equal to the net work done on the system, which in the most general case includes both conservative and nonconservative forces. But when seeking instead to find a change in total mechanical energy in situations that involve changes in both potential and kinetic energy, the previous equation KE i + PE i + W nc = KE f + PE f size 12{"KE""" lSub { size 8{i} } +"PE" rSub { size 8{i} } +W rSub { size 8{"nc"} } ="KE""" lSub { size 8{f} } +"PE" rSub { size 8{f} } } {} says that you can start by finding the change in mechanical energy that would have resulted from just the conservative forces, including the potential energy changes, and add to it the work done, with the proper sign, by any nonconservative forces involved.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Stoney Reply
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Adin Reply
?
Kyle
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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characteristics of micro business
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for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
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CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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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.
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s.
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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
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Source:  OpenStax, Work and energy. OpenStax CNX. Nov 09, 2015 Download for free at http://legacy.cnx.org/content/col11902/1.1
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