# Work done by conservative force

## Main Question or Discussion Point

I know work done by conservative forces= - ##ΔU##=##ΔK.E##
But I have a question.Does it mean work done by all the conservative forces present (in a particular physics problem)= - ##ΔU##=##ΔK.E##
or just work done by a conservative force= - ##ΔU##=##ΔK.E##
I mean let's say a problem in physics involves both electrostatic and gravitational force We know both are conservative forces.Should I equate work done by gravitational force +work done done by electrostatic force=-##ΔU##=##ΔK.E##
or should I treat each force separately.If yes,then I don't now how am I going to treat each force separately.

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Matterwave
Gold Member
It should be the total.

• gracy
vela
Staff Emeritus
Homework Helper
I know work done by conservative forces= - ##ΔU##=##ΔK.E##
You should consult your textbook and see how this equation is derived. That should provide answers to your questions.

It should be the total.
You mean this one is correct
work done by gravitational force +work done done by electrostatic force=-##ΔU##=##ΔK.E##

Last edited:
You should consult your textbook and see how this equation is derived.
My text book does not involve the derivation, it just includes application .

haruspex
Homework Helper
Gold Member
My text book does not involve the derivation, it just includes application .
I confirm it is the total. Think of it this way: the object experiences a net force ##F_{net}##. We know that determines its motion: ##F_{net}=ma##. The acceleration, over a period, determines its change in KE.
By the way, we are here discussing the conservative forces acting on the body which gains this KE. We are not interested in forces acting on other bodies in the same system.

• gracy
I confirm it is the total.
You mean this one is correct
work done by gravitational force +work done done by electrostatic force=-##ΔU##=##ΔK.E##
And similarly in case of non conservative force
Work done by non conservative force=##ΔK.E##+##ΔP.E##
Does it mean Work done by all the non conservative forces present in a (particular physics problem)=##ΔK.E##+##ΔP.E##

vela
Staff Emeritus
Homework Helper
My text book does not involve the derivation, it just includes application .
I doubt that, or you have an incredibly bad textbook.

The work-energy theorem comes from integrating Newton's second law:
$$\int_{\vec{r}_i}^{\vec{r}_f} \left(\sum_i \vec{F}_i\right)\cdot d\vec{r} = \int_{\vec{r}_i}^{\vec{r}_f} m\vec{a}\cdot d\vec{r}.$$ The righthand side is equal to the change in kinetic energy. On the lefthand side, we can pull the summation out of the integral to get
$$\sum_i \int_{\vec{r}_i}^{\vec{r}_f} \vec{F}_i \cdot d\vec{r} = \frac 12 mv_f^2 - \frac 12 mv_i^2 .$$ Each integral in the summation is just the work done by force ##i##, so we have
$$W_1 + W_2 + W_3 + \cdots + W_n = \frac 12 mv_f^2 - \frac 12 mv_i^2$$ where ##W_i = \int \vec{F}_i\cdot d\vec{r}##. If a force is conservative, we bring its contribution to the work over to the other side of the equation and rename it as the change in potential energy. That is, ##\Delta U_i = -W_i##. The stuff left on the lefthand side is the work done by non-conservative forces. So now we have
$$W_{nc} = \Delta KE + \Delta U_1 + \Delta U_2 + \cdots + \Delta U_k.$$ If there are no non-conservative forces, the lefthand side is 0, and you end up with
$$0 = \Delta KE + \Delta U_1 + \Delta U_2 + \cdots + \Delta U_k.$$

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haruspex
Homework Helper
Gold Member
work done by gravitational force +work done done by electrostatic force=-ΔU=ΔK.E
I think you misunderstand the significance of "conservative" here. The work done on an object by any force is ΔKE. What's special about conservative forces is that the work done is independent of the path taken. If the object returns to its original position, the total work done by the force is zero.
And similarly in case of non conservative force
Work done by non conservative force=ΔK.E+ΔP.E
No. PE is not a property of the individual body. It is a property of a system. For gravitational PE 'of' an object in Earth's field, the PE is possessed by the object-Earth system. A force may do work which raises that PE, but is not considered work done on the body.
A force, conservative or not, may do work on the body (ΔKE) and work on the system (ΔPE).

• gracy
work done by gravitational force +work done done by electrostatic force=-ΔU=ΔK.E
Here also I categorized conservative forces ,If it is correct
why not this
work done by Sum of all the non conservative forces=ΔK.E+ΔP.E

haruspex
Homework Helper
Gold Member
Here also I categorized conservative forces ,If it is correct
why not this
work done by Sum of all the non conservative forces=ΔK.E+ΔP.E
Why do you keep referring to conservative and non conservative forces in this thread? I have mentioned several times that these work equivalence equations have nothing to do with whether the forces are conservative. I have provided you with the condition that determines whether a force is conservative or not.

Chestermiller
Mentor
I agree with Haruspex. The source of your confusion lies in the practice of referring to forces as conservative and non-conservative. Whoever taught it to you this way has done you a disservice. How about revealing the name of the book you are using so others can be warned against using it.

Chet

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Chestermiller
Mentor
So how should I solve such questions?
I considered conservative and non conservative ,it was not a problem then.I am lost.
For whatever it's worth, I think you are obsessing over this terminology too much. Your time is much too valuable for that. If you want to specify a specific problem, we can talk about how to solve it.

Chet

• gracy
But It s much convenient for me to use those terminologies (conservative /non conservative)I loved the answer Mister T gave me in https://www.physicsforums.com/threads/definition-of-potential-energy.834808/page-4 post #61/I was having only one doubt which I posted here in this thread but now all are telling me to stop using those terminologies .I desperately want to stick to those terms and formulas (conservative/non conservative)Can not I?Is there no way?

Chestermiller
Mentor
But It s much convenient for me to use those terminologies (conservative /non conservative)I loved the answer Mister T gave me in https://www.physicsforums.com/threads/definition-of-potential-energy.834808/page-4 post #61/I was having only one doubt which I posted here in this thread but now all are telling me to stop using those terminologies .I desperately want to stick to those terms and formulas (conservative/non conservative)Can not I?Is there no way?
You should do whatever works best for you. It's a personal thing.

Chet

• gracy
It's a personal thing.
You mean it is not wrong to use conservative/non conservative forces for work calculations?

Chestermiller
Mentor
Like I said, whatever works best for you.

• gracy
Then I will choose conservative/non conservative force approach for work done calculation,but as I said to apply those formulas I need to clear a doubt
please help me to get the answer of my op and post#7. Chestermiller
Mentor
Then I will choose conservative/non conservative force approach for work done calculation,but as I said to apply those formulas I need to clear a doubt
please help me to get the answer of my op and post#7. I can't help you because the terminology drives be crazy. All you need to remember is that the change in KE is equal to the work done by the net force.

Chestermiller
Mentor
This is all I'm going to say:

The work done by both conservative and non-conservative forces (as you call them) is always equal to the integral of the force specific dotted with the displacement.

For a conservative force, the work can also be expressed in terms of a change in potential, without carrying out the integration.

In post #7, the change in kinetic energy plus potential energy is equal to the work done by the non-conservative force. But you typically can't use this equation to get the force done by the non-conservative force by summing the change in potential energy plus kinetic energy because you typically don't know what these are in advance. To get the force done by the non-conservative force, you still typically need to need to integrate the force with respect to displacement. Only in homework problems where you are told the change in kinetic energy plus potential energy can you back out the work of the non-conservative force. But to do this, you have to be told in advance what the change in kinetic energy is.

• gracy
you have to be told in advance what the change in kinetic energy is.
Why did not you include change in potential energy?I should know (should be given in the question)change in potential energy as well as change in kinetic energy in order to calculate work of the non-conservative forces using formula
##W_nc##=##ΔK.E##+##ΔP.E##
Right?

haruspex
Homework Helper
Gold Member
But It s much convenient for me to use those terminologies (conservative /non conservative)I loved the answer Mister T gave me in https://www.physicsforums.com/threads/definition-of-potential-energy.834808/page-4 post #61/I was having only one doubt which I posted here in this thread but now all are telling me to stop using those terminologies .I desperately want to stick to those terms and formulas (conservative/non conservative)Can not I?Is there no way?
Having read that post, I think I finally understand where the confusion lies. i misunderstood the relationship between the PE term there and the forces under consideration.
In your equation ##W_{nc}=\Delta PE+\Delta KE##, the ##\Delta PE## refers to the sum of all potentials present, so covers all the conservative forces. The work done by those is ##-\Delta PE##. If there are no other forces present then this equals ##\Delta KE##, the work done on the body. The sum of all non-conservative forces present accounts for any discrepancy between ##\Delta PE +\Delta KE## and 0.
That said, the separation of forces into those roles can be hazy. A real spring is mostly conservative, but not entirely. You would need to split the spring force into a conservative component and a non-conservative one.

Chestermiller
Mentor
Why did not you include change in potential energy?
Yes.
I should know (should be given in the question)change in potential energy as well as change in kinetic energy in order to calculate work of the non-conservative forces using formula
##W_nc##=##ΔK.E##+##ΔP.E##
Right?
Yes, but I hope you understand that, when the equation is written and applied in this way, it is typically only being used "diagnostically," to deduce what the non-conservative work must have been to produce the observed change in kinetic energy and potential energy. Ordinarily, you would calculate the non-conservative work directly from the corresponding force and displacement, and then use that to calculate what the change in KE + PE is. So, for this kind of application, you would interchange the right- and left hand sides of the equation.

Chet

• gracy