# Why work is force times distance?

Chaitanya Gaur
So, here we are trying our best to describe physical universe. We start from displacement, time, velocity and then force, but I don't get why would we define a physical quantity called 'work' as force times distance?

Is it just some quantity we defined in physics because it turns out to be useful and it doesn't have much of reference with idea of 'work' in usual meaning itself?

Aufbauwerk 2045

Gold Member
2021 Award
Is it just some quantity we defined in physics because it turns out to be useful and it doesn't have much of reference with idea of 'work' in usual meaning itself?
Yes, It is often the case that science uses common words in different or more restrictive or differently defined ways than regular English does. Get used to it.

Dale
Mentor
why would we define a physical quantity called 'work' as force times distance?
Why not? If your car runs out of gas and you have to push it 50 m. to a gas station you are doing something -- let's call that work. If you have to push it twice as far, you are doing twice the work. Or if you have to push a car of only half the mass for 50 m., you are doing only half the work.

Strictly speaking, work isn't force times distance. This is true only if the force applied is constant and in the same direction as the object is moving. If the force varies or is applied at a varying angle, then the formula is more complication.

Monsterboy and russ_watters
Gold Member
Is it just some quantity we defined in physics because it turns out to be useful and it doesn't have much of reference with idea of 'work' in usual meaning itself?
If you consider 4 cases, I think you will agree that the definition agrees well with our concept of physical work:
1) Push an object a short distance and each inch is easy: little work
2) Push an object a great distance and each inch is hard: a lot of work
3) Push an object a short distance and each inch is hard: a medium amount of work
4) Push an object a great distance and each inch is easy: a medium amount of work

PS. I see from the following post by @stevendaryl that the question may be much more profound than I thought. Quantaties that are conserved can have be very profound significance.

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Aufbauwerk 2045 and SchroedingersLion
Staff Emeritus
So, here we are trying our best to describe physical universe. We start from displacement, time, velocity and then force, but I don't get why would we define a physical quantity called 'work' as force times distance?

Is it just some quantity we defined in physics because it turns out to be useful and it doesn't have much of reference with idea of 'work' in usual meaning itself?

The reason that work is important in physics is because of the conservation of energy. If you perform work on a system, the energy you expend must go into either increasing the kinetic energy, or increasing the potential energy, or increasing the thermal energy (heating up the system), or maybe some other forms of energy.

If there is no potential energy or thermal energy that is changing, then this becomes:

##W = \Delta KE##

Aufbauwerk 2045, bhobba, PeterDonis and 2 others
Gold Member
it doesn't have much of reference with idea of 'work' in usual meaning itself?

You will get into difficulty if you want to argue that one vs another English word should have been chosen to classify a defined scientific thing. Your question about what makes force times displacement worth considering as a 'thing' is to me a great question.

Why call it "work" is to me a red herring. In any case, welcome to PF. :-)

Puneet Tanwar
Of course force, energy, and work can be confusing concepts. Here's a somewhat related question which I've even heard discussed on radio. I'm curious how people would explain this one to a science student in primary or secondary school.

We know that gravity acts on a refrigerator magnet. There is a downward force on the magnet, yet it does not fall. Will the magnet ever "run out" of magnetism, and if so, is this because it is expending energy to keep itself attached to the refrigerator? If it is in fact expending energy, why do we say it's not doing any "work" to stay attached?

After all, as my version of the question goes, if I am a first-grader standing near the top of a hill, pulling on my red wagon, where my little sister happens to be sitting, just enough to keep it from moving so it won't run downhill, because I'm too weak to pull it any farther uphill, then I am doing "work" as we normally understand it. My muscles certainly ache after holding on to that wagon until my dad comes along and takes over. I am applying a force to the wagon, I am expending energy to keep it from moving, my energy is running low, yet physics says I am only doing "work" when I pull the wagon up the hill. Why?

Cathr
Gold Member
2021 Award
After all, as my version of the question goes, if I am a first-grader standing near the top of a hill, pulling on my red wagon, where my little sister happens to be sitting, just enough to keep it from moving so it won't run downhill, because I'm too weak to pull it any farther uphill, then I am doing "work" as we normally understand it. My muscles certainly ache after holding on to that wagon until my dad comes along and takes over. I am applying a force to the wagon, I am expending energy to keep it from moving, my energy is running low, yet physics says I am only doing "work" when I pull the wagon up the hill. Why?
Because as has already been explained several times in this thread English and science do not use words the same way.

russ_watters
Homework Helper
Gold Member
My guess is that the first example of people understanding the concept that work = force * distance came from ploughing a field.

Cathr
Because as has already been explained several times in this thread English and science do not use words the same way.

It depends what we mean by "explain." Some students will not be satisfied with that answer. Or they may think physics uses words in a sloppy or confusing fashion. Is there an alternative word that can be used instead of "work" in such cases? Clearly something is going on, so what do we call it?

How would a teacher explain this, other than just saying "that's how we define 'work', no reason." I can hear the students after class saying, "that teacher really didn't explain it." Why do we use the word the way we do? Maybe it comes out of the development of mechanics theory as applied to machines? Maybe someone said a machine is not doing work if nothing is moving? I don't have a good answer, so I am curious about this from a history of science or teaching standpoint if nothing else.

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Gold Member
2021 Award
It depends what we mean by "explain." Some students will not be satisfied with that answer. Or they may think physics uses words in a sloppy or confusing fashion. Is there an alternative word that can be used instead of "work" in such cases? Clearly something is going on, so what do we call it?

How would a teacher explain this, other than just saying "that's just how we define 'work', no reason." I can just hear the students after class saying, "that teacher really didn't explain it." Why do we use the word the way we do? Maybe it comes out of the development of mechanics theory as applied to machines? Maybe someone said a machine is not doing work if nothing is moving? I don't have a good answer, so I am curious about this from a history of science or teaching standpoint if nothing else.

Homework Helper
Is there an alternative word that can be used instead of "work" in such cases? Clearly something is going on, so what do we call it?
We could call it "grobnitz". But that lacks the hook to experience that "work" possesses.

We could call it "grobnitz". But that lacks the hook to experience that "work" possesses.

You may well be on to something.

Maybe there is a really long compound German word that means something like

"effectthatkeepsmagnetfromfallingofftherefrigeratorbutisnotdoingwork."

Speaking of work, I must go now. :)

Staff Emeritus
2021 Award
Some students will not be satisfied with that answer.

Well, the physicists of the world are not going to change their language because some freshman doesn't like it.

Mentor
Will the magnet ever "run out" of magnetism, and if so, is this because it is expending energy to keep itself attached to the refrigerator?

This is an easy one: no.

My muscles certainly ache after holding on to that wagon until my dad comes along and takes over. I am applying a force to the wagon, I am expending energy to keep it from moving, my energy is running low, yet physics says I am only doing "work" when I pull the wagon up the hill. Why?

Because the wagon isn't moving. The only reason you're expending energy is that the human body is not designed to just assume a static position and stay there. Your muscles and bones can't just lock into place; you have to continually shift them around just to keep pushing on the wagon. But that's a particular property of the human body, not a general physical property of anything that could possibly hold the wagon motionless. So your confusion here is because you're focusing on an irrelevant feature of the situation as you've described it.

To remove the irrelevant feature, chock the wagon's wheels so they can't roll, and stop trying to push on it and walk way. Then the wagon can sit there motionless indefinitely without anything expending any energy at all (just as the magnet can stay stuck to your refrigerator indefinitely without anything expending any energy at all). Which is why physicists say no work is being done.

Monsterboy, Aufbauwerk 2045 and bhobba
Mentor
See the work-energy theorem - the change in a particles kinetic energy between points x1 and x2 equals the work done on the particle between x1 and x2. So for the theorem to apply you must define work in the usual way.

You can find a discussion on it in - Classical Mechanics - David Morin - Chapter 5. Noether's Theorem determines the energy of a free particle as its Kinetic Energy by the definition of energy as the conserved quantity related to time translation invariance and the work energy theorem determines why its useful to then define this thing called work. Noether etc is discussed in Chapter 6 of the same book.

Strangely I couldn't find it in Landau which was interesting - it may be there but tucked away as a problem or something like that. However Morin was written to be more sophisticated than your usual first year classical mechanics treatment for use at Harvard, and the author writes it could even be used at High School where he thinks it will be - what were his words - a hoot. I agree - but you need to have done calculus which of course you would expect virtually everyone to have done going to a school like Harvard.

Thanks
Bill

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Aufbauwerk 2045
Mentor
Of course force, energy, and work can be confusing concepts.

Only if you do not know Noether and the Lagrangian formulation (which of course you need to understand Noether) - then its a snap. That's why I think it should be taught as early as possible - but you do need calculus after which a book like Morin's is accessible for motivated students even at High School.

Thanks
Bill

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Only if you do not know Noether and the Lagrangian formulation (which of course you need to understand Noether) - then its a snap. That's why I think it should be taught as early as possible - but you do need calculus after which a book like Morin's is accessible for motivated students even at High School.

Thanks
Bill

Actually I was thinking of a simpler explanation! But anyway, thanks for the suggestions. I had in mind what I believe Feynman said about explaining something to a 10 year old.

I guess you people are right. I will just say it's how work is defined, then encourage people to learn more about physics if they want a deeper understanding. Same for force and energy. However, if I come across an explanation a ten year old can understand, I will pass it along.

If you consider 4 cases, I think you will agree that the definition agrees well with our concept of physical work:
1) Push an object a short distance and each inch is easy: little work
2) Push an object a great distance and each inch is hard: a lot of work
3) Push an object a short distance and each inch is hard: a medium amount of work
4) Push an object a great distance and each inch is easy: a medium amount of work

PS. I see from the following post by @stevendaryl that the question may be much more profound than I thought. Quantaties that are conserved can have be very profound significance.

Mentor
Of course force, energy, and work can be confusing concepts.
I don't see why. Work is force times distance, which is the mechanical expending of energy. And force is just what a scale reads.

So why is that difficult? ....unless one doesn't want to accept the simple definitions, in which case they are arguing against a definition, which is pointless.
Here's a somewhat related question which I've even heard discussed on radio. I'm curious how people would explain this one to a science student in primary or secondary school.

We know that gravity acts on a refrigerator magnet. There is a downward force on the magnet, yet it does not fall. Will the magnet ever "run out" of magnetism, and if so, is this because it is expending energy to keep itself attached to the refrigerator? If it is in fact expending energy, why do we say it's not doing any "work" to stay attached?
The magnet is not expending energy. See the definitions above. Why is this hard?
After all, as my version of the question goes, if I am a first-grader....
I don't mean to be rude, but you aren't a first grader, are you? I don't think it is wise to pretend you are. This thread has gone weird and it doesn't seem to me that there is a good reason why. The definitions of these terms are short sentences. They are simple. We're not trying to define "photons" or "life" or "existence" here. What's the problem?
My muscles certainly ache after holding on to that wagon until my dad comes along and takes over. I am applying a force to the wagon, I am expending energy to keep it from moving, my energy is running low, yet physics says I am only doing "work" when I pull the wagon up the hill. Why?
Because that's the definition of "work". I don't get it: you used the definition of "force" correctly and the definition of "energy" correctly; so why don't you use the definition of "work" correctly? Is it because the energy and the work aren't equal? Well that's just efficiency.

Dale and bhobba
Mentor
Actually I was thinking of a simpler explanation!

You can explain it all to a 10 year old - but it's trickier - that's all eg what do you do if they ask why does the change in Kinetic energy equal the work done or why is kinetic energy 1/2mv^2 - even what is energy anyway - smart ass 10 year olds . Tell them we will return to it when you have studied calculus, hopefully by which time you are out of the picture and someone else will do it.

Thanks
Bill

I don't see why. Work is force times distance, which is the mechanical expending of energy. And force is just what a scale reads.

So why is that difficult? ....unless one doesn't want to accept the simple definitions, in which case they are arguing against a definition, which is pointless.

The magnet is not expending energy. See the definitions above. Why is this hard?

I don't mean to be rude, but you aren't a first grader, are you? I don't think it is wise to pretend you are. This thread has gone weird and it doesn't seem to me that there is a good reason why. The definitions of these terms are short sentences. They are simple. We're not trying to define "photons" or "life" or "existence" here. What's the problem?

Because that's the definition of "work". I don't get it: you used the definition of "force" correctly and the definition of "energy" correctly; so why don't you use the definition of "work" correctly? Is it because the energy and the work aren't equal? Well that's just efficiency.

I think it's obvious that I'm asking from a pedagogical standpoint. I heard a brilliant physicist try to explain the fridge magnet issue to a radio talk show caller, and he had a terrible time putting it into plain English. I am curious if other people can explain it, even to the proverbial ten year old. Are some people not familiar with that concept? Actually I've heard other versions, such as it was Einstein and it was a six year old. But I think most people really do understand the question. Sorry, if it's not clear to people by now, I can't clarify it any more. Thanks.

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Mentor
I think it's obvious that I'm asking from a pedagogical standpoint..
It wasn't obvious to me and I suspect was not obvious to others who responded to you. The devils' advocate act is not productive if we already have a devil. It obfuscates the lesson we're trying to teach.
I heard a brilliant physicist try to explain the fridge on the magnet issue to a radio talk show caller, and he had a terrible time putting it into plain English.
That's shocking. The issue is breathtakingly simple.
I am curious if other people can explain it, even to the proverbial ten year old. Are some people not familiar with that concept? Actually I've heard other versions, such as it was Einstein and it was a six year old. But I think most people really do understand the question. Sorry, if it's not clear to people by now, I can't clarify it any more. Thanks.
Is this more devil's advocate or are you actually not clear on the issue? I really can't tell. If I explain it in a way a 10 year old can understand, will you come back and say that's not good enough because you're only six?

mfig
We start from displacement, time, velocity and then force, but I don't get why would we define a physical quantity called 'work' as force times distance?

Work connects the concepts of net force and energy, as was eluded to earlier. Recall:

##W_{a,b} = \int_a^b \textbf{F}\cdot d \textbf{x} = \int_a^b m \frac{d \textbf{v}}{dt}\cdot d \textbf{x} = \int_a^b m d \textbf{v} \cdot \frac{d \textbf{x}}{dt} = \int_a^b m d \textbf{v} \cdot \textbf{v} = \int_a^b \frac{m}{2} d (\textbf{v} \cdot \textbf{v}) = \int_a^b \frac{m}{2} d (v^2) = \int_a^b d( \frac{m v^2}{2}) = \Delta KE_{a,b}##

This connection is vital to a fuller view of the physics. That is good enough reason to define work the way it is, but I think there are other reasons too. It does comport with our experience that when we push something, i.e., we exert a force on an object through a distance, we get tired and feel like we have done work. Think about throwing a baseball from 0 to 90 mph over a 2.5 meter distance, as a pitcher does. It takes work to do that.

Mentor
It does comport with our experience that when we push something, i.e., we exert a force on an object through a distance, we get tired and feel like we have done work.

But, as @Aufbauwerk 2045 pointed out, we also get tired and feel like we have done work if we haven't actually moved anything through any distance--i.e., in a case where, to a physicist, we haven't done any work. So it's important to keep the physicist's concept of work separate from the ordinary common language concept of work. They're not the same.

bhobba, FactChecker and russ_watters
mfig
But, as @Aufbauwerk 2045 pointed out, we also get tired and feel like we have done work if we haven't actually moved anything through any distance--i.e., in a case where, to a physicist, we haven't done any work. So it's important to keep the physicist's concept of work separate from the ordinary common language concept of work. They're not the same.

I agree that they are not the same, that is why I said the definition comports with our experience. I didn't mean to imply that it exactly matches our experience. Also, I am not sure I agree that to a physicist there is no work done when you strain against an immovable object. Even though your muscle fibers are not moving the wall, say, they are moving as they contract against your bones while you exert yourself. Thus I think a case can be made that is work being done by your muscles, it just doesn't translate through the usual channels to impact an external object.

Dale
Gold Member
2021 Award

http://www.feynmanlectures.caltech.edu/I_14.html
Where he says exactly what has been said repeatedly in this thread:
The word “work” in physics has a meaning so different from that of the word as it is used in ordinary circumstances that it must be observed carefully that there are some peculiar circumstances in which it appears not to be the same. For example, according to the physical definition of work, if one holds a hundred-pound weight off the ground for a while, he is doing no work. Nevertheless, everyone knows that he begins to sweat, shake, and breathe harder, as if he were running up a flight of stairs. Yet running upstairs is considered as doing work (in running downstairs, one gets work out of the world, according to physics), but in simply holding an object in a fixed position, no work is done. Clearly, the physical definition of work differs from the physiological definition, for reasons we shall briefly explore.

Homework Helper
Gold Member
Of course force, energy, and work can be confusing concepts. Here's a somewhat related question which I've even heard discussed on radio. I'm curious how people would explain this one to a science student in primary or secondary school.

We know that gravity acts on a refrigerator magnet. There is a downward force on the magnet, yet it does not fall. Will the magnet ever "run out" of magnetism, and if so, is this because it is expending energy to keep itself attached to the refrigerator? If it is in fact expending energy, why do we say it's not doing any "work" to stay attached?

The magnet is held up by friction. The magnet supplies the normal force needed for there to be friction force to counter gravity. It is not necessary to expend energy to create a force. For example a book exerts a force on a book shelf without expending any energy. Likewise the fridge magnet expends no energy creating a force between it and the fridge.

After all, as my version of the question goes, if I am a first-grader standing near the top of a hill, pulling on my red wagon, where my little sister happens to be sitting, just enough to keep it from moving so it won't run downhill, because I'm too weak to pull it any farther uphill, then I am doing "work" as we normally understand it. My muscles certainly ache after holding on to that wagon until my dad comes along and takes over. I am applying a force to the wagon, I am expending energy to keep it from moving, my energy is running low, yet physics says I am only doing "work" when I pull the wagon up the hill. Why?

If you put a rock under the wheels or tie the waggon to a rock it wouldn't roll down the hill either right? The rock wouldn't consume any energy to do this so there must be something different about you and a rock. The answer is that a human being is a very inefficient machine. It burns energy when creating forces even when those forces do no useful work.

Rx7man, Dale, Monsterboy and 2 others
Mentor
I don't get it: you used the definition of "force" correctly and the definition of "energy" correctly; so why don't you use the definition of "work" correctly? Is it because the energy and the work aren't equal? Well that's just efficiency.

All you wrote is true. But the following are legit issues depending on how on the ball the person you are explaining it to is. They are:
1. What is energy?
2. Why is 1/2 mv^2 a form of energy?
3. Why is the work energy theorem true? That one just requires a bit of algebra (plus using a few very early calculus concepts such as Δx so small you can neglect (Δx)^2 to prove the first equation):
https://courses.lumenlearning.com/boundless-physics/chapter/work-energy-theorem/

The other two are trickier - requiring calculus and Noether which would not be understandable to a 10 yo (unless you are Terry Tao).

For 1 and 2 you just have to assure them there is a general definition of energy but it requires more advanced math (tell them it's calculus - it may motivate them to study it later). Show them the interesting story of Noether and explain the theory in words. Tell them for a free particle applying Noether leads to energy of 1/2mv^2 but to prove it will need to be deferred until later.

Its interesting there is a book accessible to grade 12's that does that - it was designed for that exact purpose - first year Harvard students with Calculus BC could be exposed to what is considered advanced classical mechanics from the start.

Thanks
Bill

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Mentor
I am not sure I agree that to a physicist there is no work done when you strain against an immovable object.

There is no work done on the object, which was the point under discussion. It is true that your muscles are doing work if you include the internal motions of parts of your body in the model.

CWatters, russ_watters and bhobba
Monsterboy
After all, as my version of the question goes, if I am a first-grader standing near the top of a hill, pulling on my red wagon, where my little sister happens to be sitting, just enough to keep it from moving so it won't run downhill, because I'm too weak to pull it any farther uphill, then I am doing "work" as we normally understand it. My muscles certainly ache after holding on to that wagon until my dad comes along and takes over. I am applying a force to the wagon, I am expending energy to keep it from moving, my energy is running low, yet physics says I am only doing "work" when I pull the wagon up the hill. Why?

This has sparked of an interesting conversation although some might find it silly. It is difficult to explain this to a six year old without proper context.
Let me give an alternative solution. You are doing work by pulling the wagon but you are not doing work on the wagon, work is being done inside your own body by expending the internal energy present in your body to breathe air using your lungs and pump blood through your blood vessels to transport oxygen and glucose to your muscles. Now, that is considered as work because you are pumping the fluid throughout your body i.e flow work.

It is just like holding down the accelerator and applying the breaks fully at the same time, you are not any work but you are wasting the fuel i.e energy. But inside the car the crank shaft keeps rotating and that is rotational work being done inside the car body without the car as a whole doing any work.

You might argue that you are doing flow work in your body even while sitting still in your room, that is also correct, we are constantly consuming our internal energy in our body to run our internal organs and the body as a whole, that requires our blood to do flow work, but you will be doing less work in this case because your muscles aren't consuming as much energy.

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Aufbauwerk 2045