Work Done: Understanding the Definition and Maths

[Devil Moo]

I know you guys all understand what work done is defined.
I am wonder how work done is defined. Why does it define in this mathematical form?
Also, how do we know 1 energy calculated by work formula = 1 energy of other form

 

[mfb]

Why does it define in this mathematical form?

It is a useful quantity if we define it that way. You can define many things, but only a few will be useful.

Also, how do we know 1 energy calculated by work formula = 1 energy of other form

Energy conservation is an experimental result. There are good reasons to expect it, but we could live in a universe where energy is not conserved.

 

[Devil Moo]

It is a useful quantity if we define it that way. You can define many things, but only a few will be useful.
Energy conservation is an experimental result. There are good reasons to expect it, but we could live in a universe where energy is not conserved.

We define many energy like kinetic energy, potential energy, thermal energy, electric energy. They may be defined by different mathematical form.
How do we make sure that kinetic energy calculated by work done is really related to electric energy or thermal energy? Or they are just describing different definition of "energy"

 

[CWatters]

Also, how do we know 1 energy calculated by work formula = 1 energy of other form...

It's not too hard to come up with experiments to prove it. For example you can heat a quantity of water using different energy sources (eg electricity vs mechanical stirring) and compare the result.

 

[mfb]

How do we make sure that kinetic energy calculated by work done is really related to electric energy or thermal energy? Or they are just describing different definition of "energy"

As we can convert them into each other, they are closely related and we call all of them "energy".

 

[Devil Moo]

Would you mind to tell me how people "define" like work done?
Is kinetic energy (1/2mv^2) defined or proved?

 

[mfb]

There is nothing to prove.
You define "kinetic energy" of an object as 1/2 m v^2 and so on. The 1/2 in that definition makes conversions to other types of energy more meaningful. Without it, energy conservation would be "the sum of [half the kinetic energy] plus [all other energy types] is conserved", which would be stupid.

 

[Devil Moo]

Is there any possibility that the formula defined is not correct for describing what we want.
For example, KE is defined as 1/2*m*v.

 

[mfb]

No. Hundreds of years of experiments showed that energy is related to the squared velocity.

m*v is another useful concept - momentum. You could use 1/2*m*v as definition of momentum, but that would be an unnecessary prefactor that does not help.

 

[CWatters]

Perhaps known since canons or fireworks were invented in China ? Someone must have noticed that to double the velocity you have to square the amount of gunpowder?

 

[Devil Moo]

But actually, do we have a method to know that our models are really describing the "real" world?
For example, if we push a box 1 F to move displacement 1m, work = F * s.
1J energy are transferred to the box.
Let me assume what the real world actually is.
KE = 1/4 m * v^2
The remaining energy is transferred to a unknown energy form of the box.
Since we do not know the unknown energy, we think all of them transfer to KE.
In this case, we never know the real model.

Perhaps known since canons or fireworks were invented in China ? Someone must have noticed that to double the velocity you have to square the amount of gunpowder?

This example...

When I see definitions in mathematical form, I am really curious of how people think to define them, and they do not have any proof!
Let go back to "work".
After a series of experiments, the equation is still true and successfully describe the energy transferred by force. So that's why we use and accept that equation!?
Does it mean definitions in physics can only be disproved but not be proved?

 

[CWatters]

Even when it works we have no way of knowing if our model of the universe is just a model or if it "really works" that way. Does it matter as long as it works?.

 

[mfb]

But actually, do we have a method to know that our models are really describing the "real" world?

That is philosophy, not physics. It doesn't matter if you think "energy" is "real" (whatever that means), it is an extremely useful tool to describe and predict physical systems.

Does it mean definitions in physics can only be disproved but not be proved?

You cannot even disprove a definition. You can disprove theories (and you cannot prove them) using those definitions to make predictions about the world.

 

[Devil Moo]

How about mathematics?
I think they are not just a model.

 

[sophiecentaur]

It's not too hard to come up with experiments to prove it. For example you can heat a quantity of water using different energy sources (eg electricity vs mechanical stirring) and compare the result.

Joule did experiments, drilling out gun barrels and comparing the heating effect with the mechanical work put in. He came up with 'The Mechanical Equivalent of Heat" which was about 4.2J / Calorie. We don't use that name any more but we still have 4.2.J/Cal

 

[nasu]

Are you mixing Rumford (the guy with the gun barrels) with Joule?
Joule's experiment was with paddles rotating in water.

 

[sophiecentaur]

You're right. They weren't even contemporaries. Rumford was 50 years before Joule.
The Mechanical Equivalent of Heat is right though. It seems to have lost favour, as a term but imo is very helpful with understanding the common theme of Energy.
I remember hearing a tale about Joule, in which he took enough food with him, on a mountain trip, which he calculated should be enough to get him up there ( Food Energy = mgh ). He forgot the additional need for food to keep warm, though and ended up in a bad way, as a result. Even if it's not true, it's a good dotty professor story.

 

[nasu]

Oh, I did not hear about the story with the food.
It may be made-up but the Italians have a nice saying: "Se non e vero, e bel trovato".

I don't think they had the calories content on food packages back then. From what I found, the first food tables were published in 1896, a few years after Joule's death.
So unless he did some earlier measurements himself, he did not know the caloric equivalent of his foods.
But the story made be look it up, so it's a good story anyway.

 

[sophiecentaur]

I'd bet he would have experimented with burning animal fat and other foods. He was a bit obsessed, so I believe. There is another story that he took his wife to the Alps on honeymoon and he attempted to find out if the water at the bottom of a high waterfall was any warmer than at the top. (No doubt she sat on a rug and polished her nails, whilst watching her hero.)
The numbers are not encouraging for that experiment.
PS My school Latin lessons just came in handy!

 

[mfb]

If ~1 gram out of 200 kg evaporates per meter of waterfall height, the water actually cools down.

I'd bet he would have experimented with burning animal fat and other foods.

Our bodies are not 100% efficient in that aspect. Actually, we are far away from 100% efficiency.