Why is the definition of work = force times displacement?

[robphy]

In my opinion,
I think it is important to distinguish
the "work done by a force"
vs
the "net work done by all the forces on an object" [= the work done by the net force]
...over a distance along a path.

Starting from forces and Newton's Laws.. but not yet at energies...
It could be argued that
"work done by a force" is only a definition
but doesn't really mean anything physical
until
one forms the net work done by all of the forces on an object,
then applies Newton's Second Law,
then observes the appearance of an interesting quantity, the difference of "what is defined as the kinetic energy." (This is the first appearance of "energy".)

So, then,
the work done by a force is
the contribution of that force to changing the kinetic energy of the object.

(Then, potential energy can be defined from those forces whose work is independent of path.)Similarly, the impulse done by a force is
the contribution of that force to changing the momentum of the object.Other similar quantities that might be dreamt up (like $\int_{path} \vec F \cdot d\vec a$)
would likely not have any meaningful interpretation... and so would not be useful for physics.

 

[sophiecentaur]

"work done by a force" is only a definition
but doesn't really mean anything physical

I could agree with that but 'not meaning anything physical' should not be a problem. Imo, the stipulation of physicality need only be brought in at either end of analysis. Work by and work on are concepts that keep giving PF members problems but is it really relevant, once we have launched on the mathematical model?

In the analogous systems that we use in Electrical calculations, we ignore the problem when we can. Admittedly, this can cause confusion when multiple 'generators' are connected into a circuit but not a problem as long as you stick rigidly with the signs of the currents and PDs until the end. At that point you may find that one of your 'generators' must have actually been acting as a 'motor'. The mechanical world is the same.

 

[robphy]

I could agree with that but 'not meaning anything physical' should not be a problem. Imo, the stipulation of physicality need only be brought in at either end of analysis.

Yes, it’s not a problem.
But my point is that
the physics of “work done by a [single] force” comes in with its use in Newton’s Second Law (in the forces-first energy-later storyline from the typical introductory textbook.)

I did say “ doesn't really mean anything physical until …. “

Work by and work on are concepts that keep giving PF members problems but is it really relevant, once we have launched on the mathematical model?

My key point is not about “work done by” vs “work done on”….
But about
“work by one force”
vs
“work by the net force”.

Similarly,
A force contributes to the acceleration of an object.
It’s not that each force (as part of a system of forces) contributes its own acceleration.
It's $\vec F_{net}=m\vec a$, not "$\vec F_{1}=m\vec a_1$, $\vec F_{2}=m\vec a_2$, etc...".

 

[sophiecentaur]

My key point is not about “work done by” vs “work done on”….

I feel that's just what you are implying. Work is done when a force is displaced. If a complicated system involves forces (or components) acting against each other then the result may be that the 'weaker' force moves less - or even backwards. There will still be an identifiable amount of work done, associated with the source of each force.
But, as I have commented, this really doesn't have to matter unless the work (or power) is relevant to the details of implementing the system. The actual Power rating of some of the force providers will relate to the cost of the motor, for example. The total work done can be calculated, as you say, by net force times net distance (often the easiest option).

Having re-read your post I think you seem to be talking as if there's a heirarchy involved but there are many cases where the 'doer' and 'done to' are not distinguishable. Your idea of 'net' is not necessary - it woulds be just a matter of re-arranging an equation and then changing which is the dependent variable.

 

[robphy]

@sophiecentaur I don't know how this became a "doer" vs "done to" discussion.

My point is:
Starting from forces and displacements,
what is the physical meaning of "work" associated with a "force"
without first defining or invoking energy or using Newton's Second Law?

Certainly one can (and some textbooks do) invoke energy before it is officially defined.
If one is merely focused on getting answers and solving problems, then that is fine.

But in trying to give a storyline (here, a force-first energy-later storyline)
about how we know certain things and what things are deeper/more-fundamental than other things,
I am making this point about
"work and one force" (which has no a priori relation to energy via Newton's Second Law in a force-first energy-later storyline.. so work is just a definition of something)
and
"work and the net force" (which does have a relation to energy).(In my experience, I am trying to avoid student misconceptions
about (1) physics being a bunch of definitions [somehow related] that need to be memorized,
(2) each force having its own acceleration [with the accelerations somehow adding when the forces do],
(3) kinetic energy somehow having components like the vectors in the problem do,
etc...
)

 

[Mister T]

Starting from forces and Newton's Laws.. but not yet at energies...
It could be argued that
"work done by a force" is only a definition
but doesn't really mean anything physical
until
one forms the net work done by all of the forces on an object,
then applies Newton's Second Law,
then observes the appearance of an interesting quantity, the difference of "what is defined as the kinetic energy." (This is the first appearance of "energy".)

The concept of the work done by a force has physical utility. It is used, for example, to introduce beginners to some of the basic principles of simply machines such as levers.

 

[robphy]

The concept of the work done by a force has physical utility. It is used, for example, to introduce beginners to some of the basic principles of simply machines such as levers.

Without using the notion of energy, can you give an introductory definition?

 

[vanhees71]

I think it is also important to distinguish between work done by a force, which I would exclusively define as the line integral of the force along the trajectory (i.e., the solutions of Newton's equation of motion) of the particle.

If the force is conservative, i.e., if it can be derived from a potential,
$$\vec{F}(\vec{x})=-\vec{\nabla} V(\vec{x}),$$
it's potential energy, and the work is given by the difference of the potential between the two times the work is calculated for. The simplifying thing of this case is that then the potential can be calculated independent from knowing the solutions of the equation of motion, because it's a line integral along any curve $C$ connecting a fixed point $\vec{x}_0$ with $\vec{x}$,
$$V(\vec{x})=-\int_{C} \mathrm{d} \vec{x}' \cdot \vec{F}(\vec{x}').$$
If $V$ (and $\vec{F}$) are not explicitly time-dependent then the energy-conservation law applies, i.e., then
$$E=\frac{m}{2} \vec{v}^2+V(\vec{x})$$
is constant along the trajectory of the particle.

It may sound a bit pedantic, but making these concepts clear from the very beginning, avoids a lot of misunderstandings and confusion about work, energy, and the energy-conservation law.

 

[votingmachine]

I agree with the utility side.

There is some naming and units argument. If you ask, "how far is it to the store". 3 miles and 5 kilometers are the same answer. If you know your walking speed is 5 km per hour, you know how long it takes to walk there.

We define "speed" as distance over time. We could have defined another thing that was the square root of distance over time. But no one would use it.

Work, as we define it, is a valuable metric.

 

[Terrakron]

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.

I believe it was Joule who came with this definition. I am looking for a reference to this.

 

[jtbell]

The term work was introduced in 1826 by the French mathematician Gaspard-Gustave Coriolis

The word "work" was actually coined by Carnot in "Reflections on the motive power of fire" (1824)

Try the early 1600s, as I described in this post:
https://www.physicsforums.com/threads/what-is-up-with-work.584812/post-3804085

 

[hutchphd]

A primary motivator for the invention of mechanical engines was to pump water from mines. The concept of moving a certain volume of water by a certain vertical distance is therefore the natural unit by which to compare such devices, and to call it "work" is similarly obvious. The fact that this measure can be related to other forms of what we call energy involves some spectacular physics, and usually Carnot gets the broadest acclaim.

 

[vanhees71]

I think this goes as far back to a debate between the arch enemies Newton and Leibnitz whether (in modern terms) $m v$ or $m v^2$ is the right measure for "motion" ;-)). Thanks to the work of later mathematicians and physicists who worked out what we today call "calculus" and "analysis" and formulated mechanics in the form we know it today (Euler et la), we know that both quantities have their merits in Newtonian mechanics as momentum ($\vec{p}=m \vec{v}$) and kinetic energy ($E_{\text{kin}}=m \vec{v}^2/2$). Why this is so, we have discussed at length above!

 

[hutchphd]

Although these theoretical arguments are essentially contemporaneous with the development of the first mining pump engines. Did Newton talk about "work" per se as an object of interest?