# Force times distance (if the force is constant)

1. Jun 11, 2004

### speeding electron

The idea of energy is, it could be argued, intuitive. If a object is moving, or if it is hot, then it is natural to think of some sort vibrancy, some sort of rousing inner property of the object. As I understand it is defined by: force times distance (if the force is constant). Who discovered that is was this that was conserved, i.e. that it was this product that gave the fundemental property of energy?

2. Jun 11, 2004

### Gokul43201

Staff Emeritus
F.d is simply the work done on a body. It may lead to a change in energy, but is itself not a measure of energy. So there is no meaning to the statement "F.d is conserved."

3. Jun 12, 2004

### speeding electron

Yes, but why does this particular product equal the energy change?

4. Jun 12, 2004

### JohnDubYa

This is a good opportunity to nail down precisely the definition of energy. I'll start:

We deemed the product of the force and distance an important physical quantity (it in essence tells you how much you accomplished in a day's work). We then defined energy as the ability to produce this amount of Fd. Later it was established that this energy was a conserved quantity. Noether's theorem later proved it.

Are there problems with the above description? I think so, but at this time I am not sure where. Anyone?

5. Jun 12, 2004

### arildno

Let us review Newton's 2.law for an object with constant mass.
This states:
$$\vec{F}=m\vec{a}$$
Multiply this equation with the velocity:
$$\vec{F}\cdot\vec{v}=m\vec{a}\cdot\vec{v}$$

Now, we may rewrite the right-hand side:
$$\vec{F}\cdot\vec{v}=\frac{d}{dt}(\frac{1}{2}m\vec{v}^{2})$$

This equation is rather instructive; it says that the rate of change of an object's kinetic energy equals the net power from the external forces acting upon the object.
We integrate in time, and get:
$$\int_{0}^{t}\vec{F}\cdot\frac{d\vec{x}}{d\tau}d\tau=\bigtriangleup(\mathcal{K}_{E})$$

Recognizing $$\frac{d\vec{x}}{d\tau}d\tau$$ as the infinitesemal distance $$d\vec{x}$$ covered in time interval $$d\tau$$ we finally get the expression:
$$\bigtriangleup(\mathcal{K}_{E})=\oint\vec{F}\cdot{d\vec{x}}$$

which is the formula relating the change in kinetic energy to the work done along the object's path.

6. Jun 12, 2004

### JohnDubYa

Which is the work-energy theorem for kinetic energy. But the question was asked about energy, not just kinetic energy.

The work-energy theorem for potential energy states that the work done by conservative forces is given by the negative change in potential energy. Also, the work done by nonconservative forces is given by the change in total (mechanical) energy of the body. So that has to be included in the definition of energy.

7. Jun 12, 2004

### Gokul43201

Staff Emeritus
It has dimensions of energy.

8. Jun 12, 2004

### jdavel

Gokul said, "It has dimensions of energy."

So does torque.

9. Jun 12, 2004

### TALewis

Good catch.

10. Jun 12, 2004

### speeding electron

So it was shown that energy is conserved after we defined it as equal to Fd? Does that mean that it was just a lucky guess? What was the use of assigning significance to this value before we knew it was conserved?

11. Jun 12, 2004

### JohnDubYa

To lift a box you must apply a force through a distance. Lifting a box to a higher height is an accomplishment, so F X d is an important property.

But looking back, my explanation sucks.

12. Jun 12, 2004

### arildno

"Energy can neither be created nor destoyed.."
I thought that was an axiom modern physics adheres to(?)

13. Jun 13, 2004

### speeding electron

JohnDubYa: I take your point, and that's fair enough. My original point was that in a way they struck it lucky, because for all they knew before the discovery it was conserved, it could have been that in nature what is conserved is Forces times the square of the distance, or something else. Did they suspect it was conserved (Fd that is) when they assigned it a significance?

14. Jun 13, 2004

### JohnDubYa

To be honest, I'm not sure how the concept of work was originally fabricated. Good question.

15. Jun 13, 2004

### JohnDubYa

16. Jun 13, 2004

### TALewis

You might be interested in looking up Joule's experiment. Joule used an apparatus that included a mass that was strung around a paddle, so that the paddle stirred a bucket of water as the mass fell. In this experiment, Joule related the work gravity does on a falling mass (weight*distance, or mgh) to the increase in the water's temperature due to the spinning paddle. His result is known historically as "the mechanical equivalent of heat," showing the relationship between mechanical energy and internal energy.

17. Jun 13, 2004

### Gokul43201

Staff Emeritus
Okay, let me rephrase. It is a scalar quantity with dimensions of energy.

In other words, find me any scalar with dimensions of energy, and there is some way to see that quantity as being the change in energy of a system. F.d is just one such scalar.

How about (dF/dl).A ? That too has dimensions of energy, doesn't it ?

Yes, and it is the surface energy of a liquid, with surface tension= F/l and surface area = A.

Last edited: Jun 13, 2004
18. Jun 13, 2004

### Gokul43201

Staff Emeritus
Please ! F.d is NOT conserved. In fact, while gravity (for example) is a conservative force (integral over a closed path vanishes) this is not true for a general F.dx.

There is nothing special about F.d. It is just the right way to calculate the change in energy of certain systems. Energy is conserved, not because of F.d. Energy would be conserved, no matter how it was calculated.

19. Jun 13, 2004

### jdavel

speeding electron asked, "Did they suspect it was conserved (Fd that is) when they assigned it a significance?"

You've nailed it!

Energy is significant in physics for exactly one reason: it's conserved.

20. Jun 13, 2004

### Gokul43201

Staff Emeritus
Take a horizontal tube of varying cross section, with pistons at the ends, filled with an incompressible fluid. Pascal found that, anywhere along this tube, F/A is conserved. This is the principle on which hydraulic brakes work.

Last edited: Jun 13, 2004