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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

Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
Giovani Reply
What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
Brhanu Reply
omega= omega not +alpha t derivation
Provakar Reply
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
Brhanu Reply
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
Saeed Reply
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
Saeed
define variable velocity
Ali Reply
displacement in easy way.
Mubashir Reply
binding energy per nucleon
Poonam Reply
why God created humanity
Manuel Reply
Because HE needs someone to dominate the earth (Gen. 1:26)
Olorunfemi
why god made humenity
Ali
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
Jun Reply
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
Abrar
when we face electric shock these elements work as a conductor that's why we got this shock
Abrar
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
ximena Reply
why electromagnetic induction is not used in room heater ?
Gopi Reply
room?
Abrar
What is position?
Amoah Reply
What is law of gravition
sushil Reply
Practice Key Terms 2

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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