# Why is the definition of work = force times displacement?

Summary:
How do we define anything in physics ?
Hello everyone,
I was curious about how do we define a physical quantity and mathematical relation between them.
For example:
Work done is defined as product of force along the displacement and displacement i.e W=Fcos(theta)d.
Now this definition is given in books straightforward but my questions is how this definitions comes to be and why we define work as product of two quantities and not division of two quantities.
In general how do we choose a certain definition to be true and holds a particular relation between physical quantities.

Delta2

sophiecentaur
Gold Member
2020 Award
Summary:: How do we define anything in physics ?

how this definitions comes to be
All of Physics is based on measurement. Also, formulae are chosen to be as simple as possible. Experience (millions of experiments) has given us confidence about how Maths produces very good models of the way the quantities we observe are related to each other.

In the case of the Force times displacement, someone will have had an idea that Work is an 'identifiable' quantity that can be transferred from one system to another (i.e. a falling weight can raise a different weight by a different amount - pulley or lever) and the relevant variables could be identified (weight and distance raised or lowered) and the relationship confirmed. It would not have been hard to invent an experiment with a number of weights and heights to confirm the relationship. Friction upsets the accuracy but that can be taken care of with suitable analysis of results.

It would have been the same thing with E=mc2; That relationship was 'suggested' by the initial idea that c is always observed to be the same. Experiment will show that the formula predicts things correctly (always within the limits of experimental error).

When we find a situation in which the simple model doesn't account for what we see, we change (extend) the model. Cosmology needed something to account for the way the galaxies behave the way they do and it was concluded that there must be 'something' extra at work. Present theory puts it down to the presence of two unexpected things we call Dark Matter and Dark Energy. No experimental evidence is available but many many observations from our new telescopes confirm that this is probably a good model.

Abhishek11235, DaveE, pinball1970 and 1 other person
jbriggs444
Homework Helper
Summary:: How do we define anything in physics ?
Feynman answers in this way. It matches what @sophiecentaur is saying.

pinball1970 and Delta2
Dale
Mentor
2020 Award
Summary:: How do we define anything in physics ?

I was curious about how do we define a physical quantity and mathematical relation between them
Basically we just agree on it. That is how any word is defined, in science or in life. Buildings have openings designed to open and close so as to allow people to easily enter or exit. We call such openings “doors”. There is no reason that we have to use that word, but we have agreed to do so.

I use the word because it is convenient and, since we have agreed, when I do so I am understood.

Just as I often need to refer to the openings in a building, scientists often need to refer to the dot product of force and displacement. Therefore, for convenience they have agreed to use the word “work” to refer to the dot product of force and displacement. The ratio of force and displacement is not a quantity that scientists often need to refer to, so they haven’t agreed on any specific word. On the rare occasions that they do need to they will simply describe it, either mathematically or verbally.

So to answer your question: we define terms in science by agreement. Typically some scientist initially proposes the term in a peer reviewed publication, and if other scientists agree they use the term also.

The dot product of force and displacement is something that arises often in physics, so it was convenient to assign it a word. The agreed-upon word is “work”. Hypothetically, another word could have been chosen, and hypothetically the word “work” could have been used for another concept. However, since the dot product of force and displacement is an important concept that arises naturally in many places, if we had not used the word “work” we undoubtedly would have used some other word.

jbriggs444, Vanadium 50, sophiecentaur and 2 others
Orodruin
Staff Emeritus
Homework Helper
Gold Member
Basically we just agree on it. That is how any word is defined, in science or in life. Buildings have openings designed to open and close so as to allow people to easily enter or exit. We call such openings “doors”. There is no reason that we have to use that word, but we have agreed to do so.

I use the word because it is convenient and, since we have agreed, when I do so I am understood.

Just as I often need to refer to the openings in a building, scientists often need to refer to the dot product of force and displacement. Therefore, for convenience they have agreed to use the word “work” to refer to the dot product of force and displacement. The ratio of force and displacement is not a quantity that scientists often need to refer to, so they haven’t agreed on any specific word. On the rare occasions that they do need to they will simply describe it, either mathematically or verbally.

So to answer your question: we define terms in science by agreement. Typically some scientist initially proposes the term in a peer reviewed publication, and if other scientists agree they use the term also.

The dot product of force and displacement is something that arises often in physics, so it was convenient to assign it a word. The agreed-upon word is “work”. Hypothetically, another word could have been chosen, and hypothetically the word “work” could have been used for another concept. However, since the dot product of force and displacement is an important concept that arises naturally in many places, if we had not used the word “work” we undoubtedly would have used some other word.
I would like to add that we tend to define words for useful things and concepts. Experience has taught us that doors in buildings are useful and that work as defined is a useful concept in physics.

Abhishek11235, Dale, sophiecentaur and 2 others
It's not a matter of choice. First we define that a meter is that long, a second is that long and kilogram is that heavy. Then we pick names, we call:
mass * length / time^2 is a force
mass * length^2 / time^2 is a work
you can clearly see that force * length is equal to work. That's not a choice.
Let's look at the force / length, that would be mass / time^2, we don't have a name for it. We could give it a name if you want, let's call it panda.
Panda * length is equal to force. Hey, it's our first formula :)

Now let's take special relativity for example, all it says is that time = distance (length)
t=d, you can square t^2 = d^2, you can add extra dimensions, t^2 = x^2 + y^2 + z^2, you can derive length contraction from it and time dilation. Essentially, it's just t=d. Physics formulas are a mathematical game. One day someone smart will come a long and show that simple definitions match observational data and the game ends, or maybe we will give more names for new things, who knows.

Last edited:
nasu and sophiecentaur
vanhees71
Gold Member
Indeed, we define energy, because it's a very common concept. The deeper reason is that the physical laws (as far as Newtonian mechanics and special relativity) is concerned do not change with time. With any symmetry of this kind there is associated a conservation law, and that's why we give names to the corresponding conserved quantities. Energy is conserved for any closed system within Newtonian and special-relativistic physics because of time-translation invariance, momentem is conserved because of spatial translation invariance, and angular momentum is conserved due to the isotropy of space (no preferred direction). The invariance under "boosts", i.e., changes from one inertial frame to another, moving with constant speed wrt. the former, is a bit trickier: In Newtonian physics it leads to the uniform motion of the center of mass of the closed system, in special relativistic physics it's the center of energy.
Concerning energy conservation for the motion of one particle you can derive the work-energy theorem. The equation of motion reads
$$m\ddot{\vec{x}}=\vec{F}(\vec{x}).$$
Then you multiply with ##\dot{\vec{x}}## and note that you can write the left-hand side of the equation as a time derivative,
$$m \dot{\vec{x}} \cdot \ddot{\vec{x}}=\frac{\mathrm{d}}{\mathrm{d} t} \left (\frac{m}{2} \dot{\vec{x}}^2 \right) =\vec{F}(\vec{x)} \cdot \dot{\vec{x}}.$$
Then integrating over ##t \in (t_1,t_2)## you find
$$\frac{m}{2} (\dot{\vec{x}}^2(t_2)-\dot{\vec{x}}^2(t_1))=\int_{t_1}^{t_2} \mathrm{d} t \dot{\vec{x}}(t) \cdot \vec{F}(\vec{x}).$$
The integral on the right-hand side is called "work". In general you need to know the solution of the equation of motion to calculate the work.
If, however, in addition the force can be derived from a scalar potential via
$$\vec{F}(\vec{x})=-\vec{\nabla} V(\vec{x}),$$
then the work integral becomes independent of the path, because
$$\int_{t_1}^{t_2} \mathrm{d} t \dot{\vec{x}}(t) \vec{F}[\vec{x}(t)]=-[V(\vec{x}_2)-V(\vec{x}_1)],$$
where ##\vec{x}_1=\vec{x}(t_1)## and ##\vec{x}_2=\vec{x}(t_2)##.
Then the work-energy theorem can be written as
$$\frac{m}{2} \dot{\vec{x}}^2(t_2)+ V(\vec{x}_2)=\frac{m}{2} \dot{\vec{x}}^2(t_1)+V(\vec{x}_1)=E=\text{const}.$$
This is the energy-conservation law, i.e., if the force can be derived from a potential, the total energy is conserved, i.e.,
$$E=E_{\text{kin}}+V=\frac{m}{2} \dot{\vec{x}}^2+V(\vec{x})=\text{const}.$$

sophiecentaur
Mister T
Gold Member
Now this definition is given in books straightforward but my questions is how this definitions comes to be and why we define work as product of two quantities and not division of two quantities.
The history of science is filled with definitions of terms that never survived to the modern era and hence don't appear in textbooks. Why? Because they have no utility. People don't use them. It's the definitions that we find useful that survive and thus appear in textbooks.

Abhishek11235, vela, vanhees71 and 3 others
jbriggs444
Homework Helper
you can clearly see that force * length is equal to work. That's not a choice.
We could instead have chosen to call it "torque".

sophiecentaur
Orodruin
Staff Emeritus
Homework Helper
Gold Member
Then we pick names, we call:
mass * length / time^2 is a force
mass * length^2 / time^2 is a work
you can clearly see that force * length is equal to work.
This is not very precise and it is also not true. ML^2/T^2 are the dimensions of work precisely because we define work as force * distance.

Furthermore, the physical dimensions do not define the type of quantity. There are several examples of physical quantities that share physical dimensionality but are inherently different quantities. While work has physical dimension ML^2/T^2, it does not mean that all quantities with those dimensions describe work.

gmax137
Try a summer job as a mason helper. Carrying loads of brick up a ladder to where the mason is working. You will soon develop an intuition about force * displacement. It is a lot of work.

vela, vanhees71, nasu and 3 others
haushofer
In the case you ask, the explanation is that if you define the work for a particle undergoing constant acceleration and hence force, you'll get the change in kinetic energy. So work is the bridge between force, energy and displacement, a connection which is intuitive.

Delta2 and sophiecentaur
Staff Emeritus
It's the definitions that we find useful that survive and thus appear in textbooks.
Like the "arg", the unit of work performed incorrectly.

hutchphd and russ_watters
DaveE
Gold Member
I suspect that you may be thinking about this kind of backwards. Physicists noticed that force * distance had some properties that were useful. They arbitrarily gave it a name for convenience. This concept was connected to many other situations and became ubiquitous. I think it was more discovered than defined.

"The teaching is merely a vehicle to describe the truth. Don’t mistake it for the truth itself. A finger pointing at the moon is not the moon. The finger is needed to know where to look for the moon, but if you mistake the finger for the moon itself, you will never know the real moon." - Thich Nhat Hanh

edit: OK, I wasn't actually there at the time. But my story sounds good to me, so I'm sticking with it.

hutchphd
This is not very precise and it is also not true. ML^2/T^2 are the dimensions of work precisely because we define work as force * distance.
The OP said why work is not force / distance. A simple dimensional analysis shows why this is not possible. Yes, you can have multiple names for things with same dimensions. That doesn't change the fact that dimensions must be respected in formulas. '/' or '*' is not a choice. How about we call a work an energy?
If you think its good to have multiple descriptions for same dimensionalities so be it. IMHO, that's confusing and unnecessary. We have a lot of useless definitions already. This doesn't help anyone, specially not to physicist, who spends time to figure stuff up that doesn't matter at all, instead of spending time on something more fundamental. If anything, things must be simplified and not to be made more complex. I don't like and don't want one. So we have a name for a chance in kinetic energy, let's give another name for a change in potential energy, that would help everyone (sarcasm off)

Last edited:
Orodruin
Staff Emeritus
Homework Helper
Gold Member
The OP said why work is not force / distance. A simple dimensional analysis shows why this is not possible.
No, this is backwards. If work was force/distance, then it would simply have different physical dimension. The physical dimension is not what determines what we call things.

Yes, you can have multiple names for things with same dimensions.
You are missing the point. It is not having multiple names for things with the same physical dimension. It is about physically different types of quantities having the same physical dimension.

That doesn't change the fact that dimensions must be respected in formulas. '/' or '*' is not a choice.
Again, not the point. Work is defined as force*distance, which is why it has the physical dimension it does. If it was defined differently it would have different physical dimension as it is the definition that determines the physical dimension and not the other way around.

If you think its good to have multiple descriptions for same dimensionalities so be it. IMHO, that's confusing and unnecessary.
You are simply wrong. The dimensionality of a quantity is not what determines what we call the type of quantity.

So we have a name for a chance in kinetic energy, let's give another name for a change in potential energy, that would help everyone (sarcasm off)
You may think you are being funny, but it just underlines that you did not understand the issue. It is not about calling the same type of quantity something else, an energy is an energy. It is about things with the same dimensions possibly being completely different types of quantities and you therefore simply cannot identify the type of quantity by its physical dimension.

As an example: I give you the physical quantity 330 GPa. What type of quantity am I describing?

Staff Emeritus
I give you the physical quantity 330 GPa. What type of quantity am I describing?
Excellent! A Young's modulus?

Orodruin
Staff Emeritus
Homework Helper
Gold Member
Excellent! A Young's modulus?
Alas, it is the pressure at the center of the Earth. Too bad, thank you for playing.

Of course, had you said pressure I would have said it is the Young’s modulus of molybdenum so there was really no way of winning here.

vela, hutchphd, russ_watters and 1 other person
sophiecentaur
Gold Member
2020 Award
Like the "arg", the unit of work performed incorrectly.
I remember, from school, the erg is a dyne cm but "arg"? Sounds like the name of a Vogon chief(?).
The cgs system was laughable, even to fourteen year olds because every unit seemed to be more related to insect life than ours.

sophiecentaur
Gold Member
2020 Award
They arbitrarily gave it a name for convenience.
I really don't think there was arbitrariness involved. at the time, they were selling their ideas to the practical and industrial sectors, who all appreciated the idea of work. The "Horsepower" was a lovely bit of steam engine marketing which was based on a pretty feeble equine sample. Most healthy teenage boys can manage almost one Horsepower when running upstairs.

If you are after finding arbitrary names then just look at the lists of fundamental particles that were named by clever researchers who were a bit too smart to be allowed to make up names for their products.

hutchphd
We're basically arguing what comes first, the name or the formula, almost chicken and egg situation.
The term work was introduced in 1826 by the French mathematician Gaspard-Gustave Coriolis
James Prescott Joule (24 December 1818 – 11 October 1889) discovered its relationship to mechanical work (see energy). This led to the law of conservation of energy, which in turn led to the development of the first law of thermodynamics.
Names come first, then we work the math out . Dark energy, dark matter, we already have names for quantities with dimensions, but there's no fundamental understanding about it.
About Young's modulus, it involves dimensionless quantity, that's not fundament, i don't really carry what ratios are. Yes, it is useful approximation for practical purposes, but in order to get better understanding about physics you have to know where those ratios are coming from. Describe the stress in terms of bonding energies.

Last edited:
sophiecentaur
sophiecentaur
Gold Member
2020 Award
Names come first, then we work the math out .
Yes; that's the order of things - until you get to advanced theoretical Physics where they often predict the existence of an entity and then go looking for it. e.g. Higgs Boson

A.T.
Names come first, then we work the math out .
Names are just shorthands for quantities that are useful, and thus often used. The words used and their historical development doesn't really matter.

russ_watters
Yes; that's the order of things - until you get to advanced theoretical Physics where they often predict the existence of an entity and then go looking for it. e.g. Higgs Boson
Well that's one way of looking at it, i guess, before higgs work there was already speculations about similar particle, and other authors contributed towards it too, but we picked higgs name for it. I have no idea how they called the phenomena before higgs? Maybe it was "a particle that gives mass" There had to be the name so that physicists can talk about the problem between each other. Guessing higgs mass was quite interesting to watch, there was around 100 different calculations. 90% of guesses didn't fit in approximation bounds. Physics are complicated. Let's not overcomplicate it.

Last edited:
ergospherical
Gold Member
Story time! The word "work" was actually coined by Carnot in "Reflections on the motive power of fire" (1824) and derives from an analysis of the performance of steam engines used in mining. These were used to pump water out of mines, and the amount of "work" a steam engine can do could be quantified in terms of the product of the weight of the water lifted and the distance raised (out of a deep hole in the ground).

Fig: Newcomen's engine

hutchphd, DaveE and Delta2