# Who Decided that Work = Force * Distance?

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1. Jan 27, 2015

### jon4444

In other words, why is change in kinetic energy defined this way? (In contrast to momentum being defined as Force times time?) I'm just trying to anchor my understanding of kinetic and potential energies into something intuitive?
And, since Force is a vector, why isn't Work a vector quantity?
Thx, Jon

2. Jan 27, 2015

### Orodruin

Staff Emeritus
Force multiplied by distance is the work done on a system. This does not imply that this turns into kinetic energy, but it could also become othher energy forms.

It is simply how we define work. There is another quantity which is force times time, namely impulse. They both have distinct relations with other kinematical quantities, such as energy and momentum. You could of course choose what you call what, but this is the standard definition and if hou define it differently you will get different relations to the other kinematical quantities.

Force is a vector and so is the displacement. If you take the force x distance in several spatial dimensions, the product is an inner product between force and displacement, which gives you a scalar.

3. Jan 27, 2015

### Staff: Mentor

For constant force and therefore constant acceleration, it comes from the following standard relationship for one-dimensional kinematics with constant acceleration: $$v_2^2 - v_1^2 = 2ad$$ Start by writing Fd = mad, then use the equation above to substitute for a.

4. Jan 27, 2015

### jon4444

I'm not sure I understand your point. I get how the kinematics equation leads you to 1/2mv^2, but my question was directed to why you begin with Fd (or mad) as definition of work/energy in the first place

5. Jan 27, 2015

### Staff: Mentor

I think the concept of force times distance being "special" goes back to studies of simple machines, in which you have a small "input" force acting over a large distance producing a large "output" force acting over a small distance. A long time ago (can't find the thread now), I posted a reference to a book published around 1500 (maybe 1600?) which was apparently the first to use "force times distance" in connection with simple machines.

6. Jan 27, 2015

### atyy

A force is a push or a pull.

And energy is the ability to do work.

By defining work as how much a force moves and object over some distance, we can get an equation about how how pushing or pulling produces work :) This is called the work-energy theorem.

Work is not a vector because it is the integral over a path of the dot product between force and displacement - it is the dot product that produces a scalar from two vector quantities.

7. Jan 27, 2015

### Staff: Mentor

Ah, here it is: What is up with work?

8. Jan 27, 2015

### jackwhirl

A table which supports a book is applying a force to that book. Because the book isn't going anywhere (in this book/table/earth system) there is no change to its kinetic or potential energy. No work is being done to the book.

9. Jan 27, 2015

### jon4444

But in that case, I'm assuming work is being converted to heat within your muscles.

10. Jan 27, 2015

### Staff: Mentor

Right, the muscle fibers sort of ratchet back and forth, doing work which increases the thermal energy of the muscles.

11. Jan 27, 2015

### jackwhirl

Work is still not being done to the book, however. (At least in an ideal system where the book is stationary in your arms.

12. Jan 27, 2015

### Staff: Mentor

That isn't correct. The situation is analogous to a car revving its engine with the parking brake on. No work is being done on the car as a whole and all of the mechanical energy generated by the engine is dissipated as heat, mostly in the transmission.

Whether your muscles are continuously vibrating or not (I don't think they are), the amount they are would not necessarily bear any relation to the energy they consume.

13. Jan 27, 2015

### jackwhirl

I have failed to see where we are in disagreement.

14. Jan 28, 2015

### CWatters

Perhaps one example being ploughing a field? How much work it is depends on the difficulty with which the plough can be pulled through the ground and the size of the field.

15. Jan 28, 2015

### Staff: Mentor

Oops. I was tripped up by your reverse/negative wording. Sorry. (That's the second time that's happened to me this week!)

16. Jan 29, 2015

### brainpushups

To respond to your original question:
The definition of work dates back to Leibniz' insistence that vis viva (mv^2) was the true conserved quantity in nature and that Descartes 'quantity of motion' (mv) was not. Leibniz' argument stemmed from Galileo's work that vis viva is proportional to the height from which a falling body is released (very early idea of what we now call gravitational potential energy and kinetic energy).

Fast forward to the beginning of the industrial revolution: Coulomb pioneered work in the machine sciences and used mechanics to model worker productivity during a day of labor. One of the quantities that Coulomb used as a measure of machine efficiency was the product of force and distance, but he did not use the term 'work' to describe this quantity

Coriolis was the first person to coin the term work (travail) and he defined the quantity as the product of force and distance where the force must be the component along the direction of the displacement (as a previous post mentioned - this is elegantly written using the scalar product). I believe that Coriolis was also responsible for introducing the factor of 1/2 into the definition of vis viva, but I'd have to verify that (I'm not sure offhand if he called it by a different name, but I know that the term 'kinetic energy' first used by Lord Kelvin).

The road to the general principles of work and energy as they are stated today is long and winding.

17. Feb 2, 2015

It's because work is the integral of force.
Once you integrate force, it simply happens that W=FD.
I think this is probably correct.

18. Feb 3, 2015

### lutz kayser

brainpushups is correct. But the formula cannot be proven theoretically. It is an empirically proven law. This is superior to any theorists postulate.

19. Feb 3, 2015

### phinds

Who Decided that Work = Force * Distance?

I think it was probably the first guy who had to carry a big rock from one place to another and noticed that the farther he had to carry it, the more tired he got and the heavier it was the more tired he got and he said "gee, heavier and farther is more work!"

20. Feb 3, 2015

### ulianjay

Work is the integral of force over distance. If the force is constant over the distance then of course W = F • D.

If you integrate force with respect to some other quantity, such as time, you do not get work.

21. Feb 3, 2015

### brainpushups

Another 2 cents since this is still going:

Technically work is the line integral of the dot product of force and an infinitesimal displacement. For nonconservative forces work is path dependent (just to be clear).

22. Feb 6, 2015

### andresB

The change in the kinetic energy is not "defined" that way.

If the force is conservative and you define the work as the line integral of the force dot the distance then you can prove using vector calculus that the work is indeed equal to the change of the kinetic energy.

23. Feb 6, 2015

### A.T.

Why does the force have to be conservative for this?

24. Feb 6, 2015

### brainpushups

Indeed. The work-energy theorem does not require the net work to be a result of conservative forces. For example, when the work done by kinetic friction is included in the total work is equal to the change in kinetic energy.

25. Feb 7, 2015

### andresB

I stand corrected.