Concept Definitions In Physics eg. $W=\int \vec{F}\cdot d\vec{x}$

Question

What does it actually mean "this concept is defined as ..."? for example when someone tells me "Work is defined as $W=\int \vec{F}\cdot d\vec{x}$".

What I understand:

• I understand the need for the integral sign, the need for summing up all the small work's that make up the total work done.

What I don't understand:

• why does it have to be this way? Why does $W= \int\vec{F}\cdot d\vec{x}$ have to be the definition for work? why does work done have to be defined as "The product of the force and distance moved in the direction of the force"? Does it have a mathematical derivation or is it just defined empirically (Through experiments)?

On the contrary the concept of electric flux $\phi=\vec{E}\cdot\vec{A}$
I do understand why it has to be this way. I understand that electric flux is proportional to the number of field lines that pass through a surface and that $\mid\vec{E}\mid$ is proportional to the electric flux density-$\dfrac{Number OfFieldLines}{Area}$ thus it makes sense the definition $\phi=\vec{E}\cdot\vec{A}$

Answer 1

The definition of electric flux that you quote "has to be" that way because it is a special case of the notion of flux in general. On the other hand, the definition of work that you quote does not have reference to a more general notion: that is just what work is. The motivation for giving the quantity $\int\vec{F}\cdot d\vec{x}$ a name is that it turns out to be a very useful quantity: the work-energy principle provides the motivation for defining the quantity $W$, but does not itself constitute a definition.

Answer 2

You choose two interesting examples.

The difficulty with the first might be that the mathematical object that we use to represent force, the vector, is really not up to the task. Force and electric field can be represented by vectors, but occasionally a situation arises when we have to recognize that there is something different between the two. There are names for the two types: contravarient (electric field), and covarient (force). As luck would have it, I used this very example, work, to explain why this distinction is necessary in an earlier StackExchange post

But there is another mathematical object that does a better job of representing a force than does a vector. It is the 1-form. In three-dimensional Euclidean space there isn't much of an advantage mathematically for using 1-forms vs. vectors, but in other situations there is. On the other hand, the 1-form does present a metaphor, or intuitive picture, that is similar to the picture of "number of field lines piercing the surface". A 1-form is pictured as a series of parallel sheets, whose normal vector would be perpendicular to the sheets. That takes care of the direction of the force. The magnitude is represented by the spacing between the sheets. The closer the spacing, the larger the magnitude.

With this picture of force as 1-form, we can associate work with "number of sheets pierced by the displacement of the object".

I've often wondered if it would be better to introduce 1-forms right away in "Physics 101" rather than leaving the topic to be discovered much too late to be of any real use. But even if that would be a good idea, and I'm not sure it is, it's not going to happen.

Answer 3

why does work done have to be defined as "The product of the force and distance moved in the direction of the force"? Does it have a mathematical derivation or is it just defined empirically (Through experiments)?

Neither. It's a definition. Definitions are not derived. Definitions are anything you want them to be. I can define the "kinetic momentum" of an object to be the product of the object's momentum and kinetic energy: $$p_K=pK=\frac12m^2v^3$$

And it's done. I defined this value. Nothing else needs to be said about it. I didn't do any derivations, proofs, etc. That's how definitions work. You define something, and that's what it is.

The real question is, "Is this definition useful in describing the world around us?" My new definition probably is not. But work is definitely useful. You can show that is value defined by $\text dW=\mathbf F\cdot\text d\mathbf x$ relates to another defined quantity, the kinetic energy. You can show that the work done by conservative forces relates to another defined quantity called potential energy. And I could go on and on.

So, work is defined the way it is because it's useful.

Answer 4

There is no analytical derivation for any definition. But we can understand it through a simple example.

Assume a mass $m$ kept on a horizontal surface. A force $F$ starts to act at time $t=0$. At time $t=t$ the mass has moved a distance $x$ and acquired a velocity $v$. If one asks the change in energy of the mass due to the force we can say it is $\frac{1}{2}mv^2$. Now one can also ask what does it take for the force $F$ to create the change of energy of the mass. Or we can also say what is the work done by the force $F$ in order to change the energy of the mass.

We know $v^2-0^2=2ax$ which imples $\frac{1}{2}mv^2=\frac{1}{2}m(2ax)=max=Fx$ and we define work by force $F$ as $Fx$.

Answer 5

To expand on Will's and Jitendra's answer:

We define work arbitrarily, to be the sum of the force along the path, $W=\int_L \vec{F}\cdot d\vec{l}$. This is a useful definition because of the work-energy theorem, that shows us that the work is the change in kinetic energy. So we ultimately came up with this concept, "work", and its definition because carving out reality using this concept turned out to be useful.

In the case of flux, we are guided by mathematical and geometric intuition to generically define the flux of any field through a surface, $\Phi=\iint_S \vec{E} \cdot d\vec{a}$. We then later find that this is a useful definition through laws such as Gauss' law for the electric field, $\Phi=q_{enc}/\epsilon_0$. So even if initially our motivation for the definition is mathematical, ultimately we still use this definition because we find it useful to carve-up reality in this way, because the laws of nature are simpler to consider and apply when we use this concept.