Want to understand ke formula without having to accept any assumptions

[speedingelf]

To prepare for a future physics course, I googled Newton’s Laws and then found about the Work Energy Theorem. I can see how Newton’s Laws makes sense but I don’t get why the kinetic energy formula is correct. The only experiment for the Work Energy Theorem I can find does not meet the requirements of the scientific method unless you make an assumption. Can someone show me why the work energy theorem is valid without making any assumptions.

 

[Dale]

No. You have to make assumptions. That is part of science. You make assumptions and then you test those assumptions against experimental evidence.

 

[Doc Al]

To prepare for a future physics course, I googled Newton’s Laws and then found about the Work Energy Theorem. I can see how Newton’s Laws makes sense but I don’t get why the kinetic energy formula is correct.

Typically, the work energy theorem is derived from Newton's laws. Do you have questions about the derivation?

 

[paisiello2]

The only experiment for the Work Energy Theorem I can find does not meet the requirements of the scientific method unless you make an assumption.

What assumption is that?

 

[speedingelf]

What assumption is that?

The experiment I'm referring to is the one that uses a table with a pulley and an air track. Basically, I see that this experiment demands that you must accept that work (force x distance) is already a proven fact. If you assume work is scientifically valid, the experiment works (no pun intended) as advertised. However, if you do the exact same experiment and make a different assumption (measuring the time force acts instead of distance force acts through) it proves something else entirely. It shows that ft =m|v|. (I know that momentum cannot be used to represent energy but the scalar version of impulse and momentum could.)

To me, this means the experiment I see discussed on the internet requires that I take the validity of work on faith; and that does not sound very scientific. Is there an experiment that does not require that particular assumption? Maybe, there is an experiment that converts, say, electrical energy into kinetic.

By the way, my thanks to everyone willing to help me out.

 

[UltrafastPED]

work (force x distance) is already a proven fact.

This is a definition of what we mean by work!

See http://en.wikipedia.org/wiki/Work_(physics )

 

[mattt]

Work: $$W = \int_{t_0}^{t_1}\vec{F}(t)\cdot\vec{v}(t)dt$$ that is a definition.


$$\int_{t_0}^{t_1}\vec{F}(t)\cdot\vec{v}(t)dt = \frac{1}{2}mv^2(t_1)-\frac{1}{2}mv^2(t_0)$$

that is a mathematical theorem, applicable when $$\vec{F}(t) = m\vec{a}(t)$$, i.e. relative to an inertial frame.


Also, Impulse: $$\vec{I} = \int_{t_0}^{t_1}\vec{F}(t)dt$$ that is a definition.

$$\int_{t_0}^{t_1}\vec{F}(t)dt = \vec{P}(t_1) - \vec{P}(t_0)$$

that is a mathematical theorem, applicable when $$\vec{F}(t) = \frac{d}{dt}\vec{P}(t)$$, i.e. relative to an inertial frame.


These are just mathematical definitions and mathematical theorems, nothing weird here.

 

[A.T.]

Maybe, there is an experiment that converts, say, electrical energy into kinetic.

That would be based on the definition of electrical energy, instead on the definition of work. How would that be any different?

 

[speedingelf]

That would be based on the definition of electrical energy, instead on the definition of work. How would that be any different?

I might be confused but, isn't one of the ideas about energy that it changes form and it's supposed to be conserved. Last year when I was fooling around with solenoids, I found that I could change electrical energy into what I now know as mechanical energy. After remembering I had done that, I figured that somebody must have confirmed the work energy theorem in a laboratory using precise measuring equipment doing something like that.


I don't want to be a pain about this but if I'm going to study a science, I figure I should be allowed to see the experimental proof for something as simple as the work energy theorem. I don't want to take work or the kinetic energy formula on faith. I don't think I should be forced to accept something just because every website that I found on google tells me it is true.

 

[UltrafastPED]

This is usually covered in your lab courses; for example:
"Conservation of mechanical and electric energy: simple experimental verification"

http://iopscience.iop.org/0143-0807/30/1/005/pdf/ejp9_1_005.pdf

If you continue with physics into graduate school you will have the opportunity to re-derive most of the common equations during your coursework - often as part of your homework.

 

[A.T.]

I found that I could change electrical energy into what I now know as mechanical energy.

To confirm this quantitatively you had to use some formulas to calculate electrical energy and mechanical energy, didn't you? If you had no problem assuming these formulas, then why do you have a problem assuming the work formula?

 

[CWatters]

I figure I should be allowed to see the experimental proof for something as simple as the work energy theorem.

One experiment using an air track..

http://www.umsl.edu/~physics/files/pdfs/Mechanics Lab/Exp6.WorkEnergy.pdf

"The objective of this experiment is to examine the conversion of work into kinetic
energy, specifically work done by the force of gravity. The work-kinetic energy theorem equates
the net force (gravity, friction, air resistance, etc.) acting on a particle with the kinetic energy
gained or lost by that particle."

"I don't want to take work or the kinetic energy formula on faith."

I'm not sure what you mean by "I don't want to take work on faith"?

We define work as the ability to do something useful. What we mean by useful depends on the system. For example in a mechanical system we might define it as the ability to move an object a distance against a force. In an electrical system it might be defined as an amount of charge raised a voltage.

 

[speedingelf]

There is a big difference in assuming work is scientifically valid compared to how we look at electrical energy.

It is easy to prove electrical energy is correct. I watched an electrician hook up a motor and he had the option of wiring it using 120 volts or 240 volts. At 120 volts the motor used about 10 amperes and he showed me that by rewiring the same motor to run at 240 volts, the motor used about 5 amperes. The motor worked just the same in both situations running at 1200 watts.

My concern with work (force x distance) is that Newton's 3rd law seems to indicate there is a natural link between force and time, not force and distance. If an astronaut floating in space throws a baseball, he and the ball can only accelerate for the same amount of time. The distance each accelerates through is different.

The more I think about it, the more and more it looks like somebody goofed a long time ago in physics. Everyone since has just gone with the flow but I am a bit of a rebel and I want to see real proof that work is the right way to measure mechanical energy. Maybe it is a good thing I started studying Newton's Laws and the work energy theorem on my own. It allows me to ask questions and spend time getting them answered. In school, we have to keep up or risk failing.

 

[A.T.]

If an astronaut floating in space throws a baseball, he and the ball can only accelerate for the same amount of time.

That's why both gain equal but opposite momentum.

The distance each accelerates through is different.

That's why both gain different amounts of kinetic energy.

So yes, momentum and energy are different. That's why they have different names. But what is your point?

 

[mattt]

I don't get what you mean. You can test the "work-energy" theorem, experimentally, in thousands of ways. You'll see it is always satisfied.

If an astronaut floating in space throws a baseball, he and the ball can only accelerate for the same amount of time. The distance each accelerates through is different.

And?

They both are accelerating during the period of time the astronaut is pushing the ball and they both are in physical contact, call it

$$[t_0, t_1]$$

As soon as the ball leaves the astronaut hand, ball and astronaut will be moving with uniform rectilinear motion.

They get different accelerations during the time interval [t_0, t_1] because they have different masses. So obvioulsy the astronaut will have moved a shorter distance during that time interval [t_0, t_1] than the ball.

You see something strange here?


EDIT:

For example, suppose F is the force the astronaut exerts upon the ball (and also the force the ball exerts upon the astronaut, but with different sign) during the time interval [t_0, t_1] and suppose F is constant (during that interval of time). Suppose M is the astronaut mass, m is the ball mass, a_{a} is the astronaut constant acceleration and a_{b} is the ball constant acceleration (during the time interval). You'll have:

Astronaut:

$$F = M a_{a}$$

so

$$a_{a} = \frac{F}{M}$$

is the constant acceleration of the astronaut during the time interval [t_0, t_1].

The distance the astronaut moves during that time interval will be:

$$x(t_1)-x(t_0) = \frac{1}{2}\frac{F}{M}(t_1-t_0)^2$$


Ball:

$$F = m a_{b}$$

so

$$a_{b} = \frac{F}{m}$$

is the constant acceleration of the ball during the time interval [t_0, t_1].

The distance the ball moves during that time interval will be:

$$x(t_1)-x(t_0) = \frac{1}{2}\frac{F}{m}(t_1-t_0)^2$$

which is longer than the astronaut distance just because m<M.


The "work" done on the astronaut by the force (the force the ball exerts on the astronaut) along the trajectory the astronaut follows during that time interval, is:


$$\int_{t_0}^{t_1}F(t)v(t)dt = \int_{t_0}^{t_1}F a_{a}(t-t_0)dt= \int_{t_0}^{t_1}F\frac{F}{M}(t-t_0)dt=\frac{F^2}{M}\frac{(t_1-t_0)^2}{2}$$


Which is exactly equal to:

$$\frac{1}{2}Mv(t_1)^2-\frac{1}{2}Mv(t_0)^2$$

given that

$$v(t_0)=0$$

and

$$v(t_1) = a_{a}(t_1-t_0) = \frac{F}{M}(t_1-t_0)$$


The "work" done on the ball by the force (the force the astronaut exerts on the ball) along the trajectory the ball follows during that time interval, is:


$$\int_{t_0}^{t_1}F(t)v(t)dt = \int_{t_0}^{t_1}F a_{b}(t-t_0)dt= \int_{t_0}^{t_1}F\frac{F}{m}(t-t_0)dt=\frac{F^2}{m}\frac{(t_1-t_0)^2}{2}$$


Which is exactly equal to:

$$\frac{1}{2}mv(t_1)^2-\frac{1}{2}mv(t_0)^2$$

given that

$$v(t_0)=0$$

and

$$v(t_1) = a_{b}(t_1-t_0) = \frac{F}{m}(t_1-t_0)$$


As you can see, the work done by F on the ball along its trajectory is greater than the work done by F on the astronaut along his trajectory, which is the same as saying that the ball gets more kinetic energy than the astronaut.


Do you see something weird in all this?

 

[UltrafastPED]

There is a big difference in assuming work is scientifically valid compared to how we look at electrical energy.

It is easy to prove electrical energy is correct. I watched an electrician hook up a motor and he had the option of wiring it using 120 volts or 240 volts. At 120 volts the motor used about 10 amperes and he showed me that by rewiring the same motor to run at 240 volts, the motor used about 5 amperes. The motor worked just the same in both situations running at 1200 watts.

The same experiment can be done with mechanics; can you construct an example using levers, pulleys, or gears? I can lift the same load with half the force applied over twice the distance ...

My concern with work (force x distance) is that Newton's 3rd law seems to indicate there is a natural link between force and time, not force and distance. If an astronaut floating in space throws a baseball, he and the ball can only accelerate for the same amount of time. The distance each accelerates through is different.

Newton's Third Law of Motion is certainly symmetric in time ... so applying Newton's Second Law of Motion, F = dP/dt, to our opposite but equal pair, F12 = -F21, and integrating over equal time periods we have:

∫dP12/dt * dt = ∫dP12= ΔP12 which is equal to -∫dP212/dt * dt = -∫dP21= -ΔP21.

So if the bodies started at rest we have P12 = -P21: equal but opposite momentum; but in every case the changes in momentum are equal but opposite.

You are correct that the distances covered by the two bodies are different. The Third Law of Motion implies the conservation of momentum. The Second Law of Motion is the source of the Work-Energy theorem: it's about changes in kinetic energy.

The more I think about it, the more and more it looks like somebody goofed a long time ago in physics. Everyone since has just gone with the flow but I am a bit of a rebel and I want to see real proof that work is the right way to measure mechanical energy. Maybe it is a good thing I started studying Newton's Laws and the work energy theorem on my own. It allows me to ask questions and spend time getting them answered. In school, we have to keep up or risk failing.

You are correct - Isaac Newton goofed by writing his great work in Latin, and by using geometry to prove the theorems instead of the calculus. The work went through three editions during his lifetime, so you can be certain that he also felt the need for more than a few updates.

If it was a bit easier to read beginning students might understand it more easily! :-)

 

[speedingelf]

I do see something very strange and I'm very confused as to why no one else saw this before.

The force acting on the astronaut is exactly equal to the force acting on the ball and yet, one of those two things winds up with more kinetic energy. That makes zero sense since both receive the same ACTION and to show why I say that, imagine that same astronaut throwing a different ball that is twice as massive.

If he throws the more massive ball exactly like the other one, he will accelerate exactly the same as before but the ball's speed will be less. Compare the work done in those situations and you get the following:

work done on astronaut with small ball EQUALS work done on astronaut with bigger ball

work done on small ball is GREATER than work done on bigger ball


The thing I see is that when force acts, it acts both over time and distance. The astronaut/ball situation seems to indicate that the time force acts is FAR more important than the distance force acts through. I got the idea that just because it is possible to measure the distance force acts through, it does not necessarily mean it is really that important. That should explain why I want to see the experiment that decided physics should use force x distance instead of force x time to describe energy.

 

[A.T.]

The force acting on the astronaut is exactly equal to the force acting on the ball and yet, one of those two things winds up with more kinetic energy. That makes zero sense since both receive the same ACTION

Define "ACTION".

If he throws the more massive ball exactly like the other one,

What is exactly the same? Force? Time? Distance?

That should explain why I want to see the experiment that decided physics should use force x distance instead of force x time to describe energy.

Pull something up against gravity at constant speed and hence constant force. The energy gained depends on the height gained, not the time you need to lift it.

 

[mattt]

I do see something very strange and I'm very confused as to why no one else saw this before.

The force acting on the astronaut is exactly equal to the force acting on the ball and yet, one of those two things winds up with more kinetic energy. That makes zero sense

What?

A mathematical theorem is just a mathematical theorem. You could also say that 2 + 2 = 4 (in ordinary arithmetic) does not make sense to you, but that would only imply that there is something you are not understanding and/or something you are not interpreting the correct way.

since both receive the same ACTION

You can easily get misleaded the moment you start using vague terms. Define "action" with mathematical precision.

and to show why I say that, imagine that same astronaut throwing a different ball that is twice as massive.

If he throws the more massive ball exactly like the other one, he will accelerate exactly the same as before but the ball's speed will be less.

Yes, in my previous example you just have to change m to m'=2m and leave F as F. Then the new ball speed will be exactly half the previous ball speed and the same with the new ball distance during the time interval [t_0, t_1] (will be half the previous distance).

All this not only does make complete mathematical sense, but also is what actually happens in nature, as you can experimentally test in multiple ways.

Compare the work done in those situations and you get the following:

work done on astronaut with small ball EQUALS work done on astronaut with bigger ball

It does make complete sense if you understand what is the mathematical definition of "work". If the force F is the same in both cases (which it is in our example) and the mass M of the astronaut is the same in both cases (which it is) and the time interval this force is being exerted is the same (which it is in our example), then obviously the work will be the same in both cases (if you don't see it, then you don't understand the mathematical definition of work).

work done on small ball is GREATER than work done on bigger ball

Again, this does make complete sense if you understand the mathematical definition of work. Work done (by the same force F, acting during the same time interval [t_0, t_1] ) on the smaller ball will be obviously greater than work done (by that same F which acts during that same time interval [t_0, t_1] ) on the bigger ball just because the smaller ball (under these circumstances) travels a longer distance during that time interval, than the bigger ball.


In the astronaut case, F is the same, M is the same, [t_0, t_1] is the same, so obviously the work done on the astronaut by those two different balls will be the same.

In the two balls case, F is the same, [t_0, t_1] is the same, but the mass is different, m' = 2 m, so obviously the work done will be different (greater on the smaller ball, than on the bigger ball).

Again, if you don't see all this, if it does not make complete sense to you, then you are not understanding the definition of work.

The thing I see is that when force acts, it acts both over time and distance. The astronaut/ball situation seems to indicate that the time force acts is FAR more important than the distance force acts through. I got the idea that just because it is possible to measure the distance force acts through, it does not necessarily mean it is really that important.

Both are important concepts, but different concepts. Work and Impulse are different concepts, but very important and useful.

Work is equal to increment of kinetic energy.

Impulse is equal to increment of linear momentum vector.

That should explain why I want to see the experiment that decided physics should use force x distance instead of force x time to describe energy.

"Energy" is what we decide to define as energy (just as it is true with any other concept). We call "energy" a given entity that is conserved under concrete circumstances and that has dimension M.L^2/t^2.

F.t=M.L/t is different, and we call it "linear momentum".

If you start changing names, this can get messy very fast :-).

 

[speedingelf]

I appreciate the effort everyone is giving but the only thing I need is the experimental proof that scientists used to verify the idea that work (force x distance) should represent energy.

I understand that energy is thing that changes forms and so on. There is no argument with that.

The scientific method requires that a hypothesis be independently tested. Someone must have come up with the idea of energy and other scientists confirmed it. I know the air track experiment well but it does not prove the work energy theorem is valid unless you assume work is valid. If you substitute force x time (a scalar expression), the air track experiment "proves" force time should represent energy. Obviously, or it should be, the air track experiment does not really test whether force x distance should represent mechanical energy.

Please see that all I am asking for is the sort of experimental proof physicists would demand of me if I came up with something. For example, if I said that gravity is related not to mass but to the physical size of an object but offered no proof, no one would take me seriously. The work energy theorem came from somewhere and somebody showed it was true. By the way, if my last statement is proven true, I want credit. LOL

Seriously, what is the experimental proof that work should be used instead of force x time? As far as I know, work has been around for at least 100 years. Is it too much to ask to see it?

 

[A.T.]

Seriously, what is the experimental proof that work should be used instead of force x time?

Used to do what? Compute energy? Force x time doesn’t even have the same dimensionality as energy, so your idea is obviously nonsense.

 

[speedingelf]

I am well aware that the dimensionality would be different for force x time. When I ask for the scientific evidence for work, I am also asking for the dimensionality of energy. Where did mass x velocity x velocity come from? What is the proof that energy should be that and not something else?

You may think that I am being annoying but all I am doing is asking for the scientific evidence that the scientists of the past must have accepted.

Based on the responses I have been getting, I'm beginning to get the idea that maybe someone said one day there has to be this thing, let's call it energy, and it has the units of kilograms x meters per second x meters per second. I know that can't be true so there has to be an experiment that showed scientists that work was force x distance. What was it? I'd like to examine it so I don't have to have faith in my physics lessons.

 

[UltrafastPED]

The force acting on the astronaut is exactly equal to the force acting on the ball and yet, one of those two things winds up with more kinetic energy. That makes zero sense since both receive the same ACTION and to show why I say that, imagine that same astronaut throwing a different ball that is twice as massive.

The kinetic energy of the astronaut is P12^2/2M; that of the ball is P21^2/2m. Since the magnitudes of the momenta are the same, any differences are due to the masses.

Newton's Second Law of Motion for constant mass is F=ma; so F12=M*A12 = -F21=-m*a21. Note that the accelerations are in opposite directions, just as the final momenta and velocities are in opposite directions.

Rearranging we have A12 (astronaut) = - (m/M)*a21 (ball). If the masses were equal, the accelerations would have been the same. If the astronaut is much heavier than the ball then she will have a much lower acceleration.

Isn't this what you would expect if you were standing on ice and threw a pebble?

If he throws the more massive ball exactly like the other one, he will accelerate exactly the same as before but the ball's speed will be less. Compare the work done in those situations and you get the following:

work done on astronaut with small ball EQUALS work done on astronaut with bigger ball

work done on small ball is GREATER than work done on bigger ball

You are confounding the work done _on the ball_ with the work done _on the astronaut_; in fact all of the work is done _by the astronaut_, but we have no accounting for the details of how this work was accomplished, since people are not ideal machines, and have a lot of internal losses.

The result of the work done by the astronaut is the motion of both the astronaut and the ball - so if the astronaut was a perfect machine we would expect that the work done _by the astronaut_ is equal to the sum of the kinetic energies of the astronaut plus the ball. There is no reason to believe that they the kinetic energy of the two bodies should be the same! Newton's Third Law of Motion only says that the forces will be equal but opposite.

If you follow the ball, and watch it strike a second ball, once again the total momentum will be the same before and after the impact; but now we also have a complete accounting of the energy budget of the system: the sum of the kinetic energies before and after the collision will be the same, but neither the kinetic energy nor the momentum needs to be parcelled out equally to the two balls!

The thing I see is that when force acts, it acts both over time and distance. The astronaut/ball situation seems to indicate that the time force acts is FAR more important than the distance force acts through. I got the idea that just because it is possible to measure the distance force acts through, it does not necessarily mean it is really that important. That should explain why I want to see the experiment that decided physics should use force x distance instead of force x time to describe energy.

Your "intuition" is misleading you. Please review the analysis above - you need to follow through on the details - and see what happens when you obey Newton's Laws of Motion step by step.

 

[UltrafastPED]

Based on the responses I have been getting, I'm beginning to get the idea that maybe someone said one day there has to be this thing, let's call it energy, and it has the units of kilograms x meters per second x meters per second. I know that can't be true so there has to be an experiment that showed scientists that work was force x distance. What was it? I'd like to examine it so I don't have to have faith in my physics lessons.

When you have time you can read the long, convoluted history of "vis viva" and the other terms which come to us from Aristotle via the middle ages, Simon Stevin, Galileo, Newton, Leibniz, and others ... there were many tentative advances, missteps, and arguments ... what you read in a modern textbook is a bare and schematic history of science.

And no, you don't have to have any faith in your physics lessons; physics is an experimental science, and the "proof" should be learned by experimentation in your lab sessions.

 

[Nugatory]

Based on the responses I have been getting, I'm beginning to get the idea that maybe someone said one day there has to be this thing, let's call it energy, and it has the units of kilograms x meters per second x meters per second. I know that can't be true so there has to be an experiment that showed scientists that work was force x distance. What was it? I'd like to examine it so I don't have to have faith in my physics lessons.

In elastic collisions there are two conserved quantities: A vector whose magnitude is $mv$; and a scalar $mv^2/2$. That's the experimental fact that we have to start with.

We label one of them momentum (because we have to name it something) and the other kinetic energy; and we define force to be the time derivative of momentum because that's convenient and agrees with the experimentally confirmed result $F=ma$. It follows from this definition that we apply a given force for a particular length of time to change the momentum by a given amount - that's just doing math on the relationships we already have.

Now, if we turn our attention to that other conserved quantity, the scalar $mv^2/2$, and ask the question "What application of a given force will change the quantity $mv^2/2$ by a given amount... a bit of algebra and an easy integration will give you force times distance.

The discovery of kinetic energy and momentum, and the relationship of kinetic energy to potential energy (which comes directly from $W=Fd$) didn't follow exactly this path, of course. The history is full of misconceptions, false turns, and a even people expressing confusions similar to yours. It is only when we arrive at the end of the journey and look back that we are able to say "Oh - so that's the shortest route with the fewest wrong turns", and start writing the textbooks.

 

[A.T.]

I'm beginning to get the idea that maybe someone said one day there has to be this thing, let's call it energy,

Yep, energy is a human made concept. It is useful, because it is conserved (under certain conditions), as it is defined right now.

If you want to change the definition, then you should also change the name to something else than "energy", to avoid confusion. I propose "momentum".

 

[speedingelf]

Has anyone noticed that I have not gotten my question answered? What is the experimental proof scientists used when they accepted the work energy theorem?

The astronaut and ball example I mentioned earlier was only given to show why there may be reason to question the many websites I visited that, basically, want me to take the work energy theorem on faith. Per Newton's 3rd Law, there seems to be this very natural link between force and time but we measure energy by force x distance. There has to be a reason that work is defined as it is and thus energy having the units kilograms x meters per second x per second.

I know enough about physics to know that at one time, the universe was thought to exist in a steady state. Then one day someone said, no that is not right and they provided some sort of observational proof. (kinda hard to test it experimentally) The rest of the physics community did not take someone's word, did they? So, why it is so hard for someone to answer a reasonable question? Please, what is the experimental experiment that proved the work energy theorem? Who did it? When?

Am I wrong to ask this?

 

[russ_watters]

When I ask for the scientific evidence for work, I am also asking for the dimensionality of energy. Where did mass x velocity x velocity come from? What is the proof that energy should be that and not something else?

Those questions aren't really equivalent - the derivation and experimental evidence are different things - but the experimental evidence is all around you, in all sorts of mechanical devices: cars, elevators, fans, power plants, roller coasters, dropping an object off a building, etc. Pick one and we'll help you understand how to analyze it using the concept of work.

Also, note that the history and derivation is probably not as useful/relevant as you think it is: ultimately, "work" and "energy" are simply names for mathematical relationships that were found to be useful enough and used often enough to be worthy of naming.

Am I wrong to ask this?

Not wrong per se, but the question is so basic that it is basically insulting to ask it and tough to answer without insulting you. But if you want to start off that basic, we can do that. But we also aren't going to spoon feed you either. The question is easy enough that we'll make you do it yourself.

Let's try something easy like dropping a ball off a building. 1 kg ball, 5 meter height. How much work did gravity do on the ball before it hit the ground? How much kinetic energy does it have?

 

[mattt]

I appreciate the effort everyone is giving but the only thing I need is the experimental proof that scientists used to verify the idea that work (force x distance) should represent energy.

As I said before, "energy", "linear momentum", "force", "angular momentum" are exactly what we decide them to be, and we make it by means of a mathematical definition and then an experimental setup to be able to interpret it and measure it.


It just happen that in many mathematical structures we use in Physics, there are some mathematical objects (some functions in Classical Mechanics, Self-Adjoint Operators in Quantum Mechanics, but it does not matter now for you, just some mathematical objects) that are constant during the evolution of that system. One of them, that is mathematically related to the time-translation symmetry of that mathematical structure, we decide to call it "Energy".

It happens that that concrete mathematical object we decide to call "Energy" has dimension ML^2/T^2.

For example:

System = one point-particle with mass m that moves in one dimension under the force $$F(x) = -kx$$


In this mathematical model, the function

$$f(x(t),v(t)) = \frac{1}{2}mv^2(t) + \frac{k}{2}x^2(t)$$

is constant, does not vary with time.

There are other functions in this model that are also constant, for example

$$g(x(t),v(t)) = \frac{1}{2}mv^2(t) + \frac{k}{2}x^2(t) + 42$$

does not vary with time.

We call the first one expression "Mechanical Energy of the particle", the second one expression is just as good, and it is exactly just as useful as the first one. Why useful?

Because if you call

$$\frac{1}{2}mv^2(t)$$

"kinetic energy of the particle" and you call

$$\frac{k}{2}x^2(t)$$

"potential energy of the particle", then you can know a lot of useful things about this system just by those two quantities.

For many important questions, you don't have to solve the (usually) difficult system of differential equations that would let you know what happens (the value of each physical magnitude of the system at each and every moment of time), but simply doing some arithmetic computations (addition and subtraction) with those two "energy terms" between some initial and final state, you can know important things about how the system changes between those initial and final states. (when you are interested only in those initial and final states, but not in every intermediate state).

I have put you the simplest model, but it is exactly the same in any other mechanical model. So now you know why those mathematical quantities we decide to call "Energy" are extremely important.


So I hope you now undertand why it is very useful and important, in a "real" experiment or observation (for example, when analyzing/observing the trajectory of a planet in the solar system, or when observing/studing this real tennis ball during a tennis match, etc) to calculate those quantities we call "energy". It simplifies the computation of many important things.

I understand that energy is thing that changes forms and so on. There is no argument with that.

The scientific method requires that a hypothesis be independently tested.

You create a mathematical model (a mathematical structure plus an interpretation of some of the mathematical objects in terms of some experimental setup and measure).

Then what we test is the mathematical predictions of that mathematical model (and it is possible to do it precisely because of the interpretation in terms of experimental/observational setup and measure process).

Someone must have come up with the idea of energy and other scientists confirmed it I know the air track experiment well but it does not prove the work energy theorem is valid unless you assume work is valid.

I really don't understand what you mean here. To test a model you just have to do the following:

1) Compute the mathematical expressions of the relevant mathematical objects of the model.
2) Measure, in the real experiment, those magnitudes that correspond by means of the interpretation of the model, to these mathematical objects.
3) See if it match.

For example, if you are going to do the following real experiment:

A tennis ball of mass m falling from h meters to the ground.

One useful mathematical model will be a point particle of mass m and a force F= -mg (along the vertical axis).

You compute whatever you want in the mathematical model that has an interpretation in terms of real measures (final speed of the tennis ball, for example, any other observable).

You measure the real final speed of the tennis ball.

You see if it match.

If you substitute force x time (a scalar expression), the air track experiment "proves" force time should represent energy.

Seriously, I don't have the slightest idea what you mean there. If you measure (in the real experiment) F.(t_1-t_0) you will get a number, if you also measure (in the real experiment) P(t_1) and P(t_0) you will get two more numbers. You can check that those three numbers satisfy

F.(t_1-t_0) = P(t_1)-P(t_0)

and you can mathematically prove in the mathematical model that for any two time points t_0 and t_1, the model implies F.(t_1-t_0) = P(t_1)-P(t-0)

So you have just witnessed that the model captures reality ( = is a good/useful model).

Obviously, or it should be, the air track experiment does not really test whether force x distance should represent mechanical energy.

I think you have a serious mess about how things work. There is no "should" anything.

"Mechanical energy" is what we decide it to be, and we have ALREADY decided that Mechanical Energy is "X-expression" depending on the mathematical model we are using (to modelize some real experiment) and I already told you the reasons why it is useful to put that name to those expressions of those models.

Please see that all I am asking for is the sort of experimental proof physicists would demand of me if I came up with something. For example, if I said that gravity is related not to mass but to the physical size of an object but offered no proof, no one would take me seriously. The work energy theorem came from somewhere and somebody showed it was true. By the way, if my last statement is proven true, I want credit. LOL

No, the more I read you, the more I see you have big problems understanding how it works. But I really not know what more to say to you (apart of all I have just written).

Seriously, what is the experimental proof that work should be used instead of force x time? As far as I know, work has been around for at least 100 years. Is it too much to ask to see it?

That question does not even make sense. "the work should be used instead of force x time"

I don't really know what you are asking, but something is seriously lacking in your understanding of physics.

I hope some other person gets what you mean.

 

[mattt]

I am well aware that the dimensionality would be different for force x time. When I ask for the scientific evidence for work, I am also asking for the dimensionality of energy. Where did mass x velocity x velocity come from?

I already answered this question in my previous post. In those mathematical structures we use in Physics, one of those mathematical objects that are constant during the evolution of the system, has a mathematical expression whose dimension is ML^2/T^2. This mathematical object (and that is true also of other mathematical objects whose values also remain constant) is very useful for the reasons I cited in my previous post (it allow us to compute many important things without having to solve a difficult system of differential equations).

We decide to call it "energy". To some other mathematical objects that also remain constant during the evolution of the (mathematical) system we decide to use other names (linear momentum vector, angular momentum vector...)

What is the proof that energy should be that and not something else?

This question really does not make any sense.

That is like saying "What is the proof that "natural number 2" should be that (the natural number 2) and not something else?".

Well, the natural numbers are mathematical concepts that are useful for so many things, just like the concepts "force, vector, function, mass, energy, work, integral, etc" are useful for so many things.

You may think that I am being annoying but all I am doing is asking for the scientific evidence that the scientists of the past must have accepted.

No. This has nothing to do with that. What you are asking is exactly the same as asking for a "experimental proof" that the natural number 2 should be the natural number 2 and nothing else.

Obviously the question is absurd, and it shows you don't understand some basic things about how science (Physics in this case) works.

Based on the responses I have been getting, I'm beginning to get the idea that maybe someone said one day there has to be this thing, let's call it energy, and it has the units of kilograms x meters per second x meters per second. I know that can't be true so there has to be an experiment that showed scientists that work was force x distance. What was it? I'd like to examine it so I don't have to have faith in my physics lessons.

I already answered this.

We define mathematical concepts of all kind and we put them "names" and we use some of them (many of them) to modelize physical systems, and they are very useful.

 

[russ_watters]

That question does not even make sense. "the work should be used instead of force x time"

I don't really know what you are asking, but something is seriously lacking in your understanding of physics.

I hope some other person gets what you mean.

It kinda looks like the OP doesn't really get, on a basic level, what physics is for. Force times time is another relationship that is useful enough to have a name. Which gets used depends on what kind of problem you are trying to solve! That's what physics is for: solving problems that quantitatively describe how things in the natural world work.

 

[speedingelf]

I am continuously surprised by the inability to understand and answer a very simple question.

I just want to examine the experiment that scientists used to decide that work is force x distance.

I know that every website and book on intro physics says it is true but they do not show why it is scientifically true.

Why is work defined as force x distance?

I am getting the feeling that everyone who has studied physics just accepts what they are told and are very good at explaining everything except why it is true experimentally. I have been told that I'm a strange person. I guess Galileo was strange too because he asked questions like mine. I'll bet some of the other notable physicists of the past also asked "strange" questions. Does anyone know the answer to mine?

 

[russ_watters]

Perhaps answering this will help:

The experiment I'm referring to is the one that uses a table with a pulley and an air track. Basically, I see that this experiment demands that you must accept that work (force x distance) is already a proven fact. If you assume work is scientifically valid, the experiment works (no pun intended) as advertised. However, if you do the exact same experiment and make a different assumption (measuring the time force acts instead of distance force acts through) it proves something else entirely. It shows that ft =m|v|. (I know that momentum cannot be used to represent energy but the scalar version of impulse and momentum could.)

And what does it prove? It proves that impulse and momentum are related. So what? I thought you wanted to know about work? Analyzing the experiment a different way, using different parameters doesn't do anything beyond proving that particular analysis "works". It says nothing about any other type of analysis. I mean: you didn't look at friction or temperature with either of those analyses - does that prove they don't exist/work?

 

[mattt]

Has anyone noticed that I have not gotten my question answered? What is the experimental proof scientists used when they accepted the work energy theorem?

Already answered in my previous two posts.

The astronaut and ball example I mentioned earlier was only given to show why there may be reason to question the many websites I visited that, basically, want me to take the work energy theorem on faith.

A mathematical theorem has its mathematical proof (which is this case is very simple).

"Taking the work-energy theorem by faith" is just like saying "taking a + b = b + a in ordinary arithmetic (which is another mathematical theorem) by faith". It is just an absurd statement, because mathematical theorems are some of the very few things in this life that you can check its validity by means of its mathematical proof.

Per Newton's 3rd Law, there seems to be this very natural link between force and time but we measure energy by force x distance. There has to be a reason that work is defined as it is and thus energy having the units kilograms x meters per second x per second.

I already told you how and why we define the "energy concept" in previuos posts.

I know enough about physics

Sorry but you have already showed you don't have a clue about physics, and this is not something bad at all, just describing a fact.

to know that at one time, the universe was thought to exist in a steady state. Then one day someone said, no that is not right and they provided some sort of observational proof. (kinda hard to test it experimentally) The rest of the physics community did not take someone's word, did they? So, why it is so hard for someone to answer a reasonable question? Please, what is the experimental experiment that proved the work energy theorem? Who did it? When?

Am I wrong to ask this?

As I told you before, what is not reasonable is your "question", which is exactly as saying "what is the experimental proof that "the natural number 2" is that (the natural number 2) and nothing else?" (when you ask about the energy concept) or "what is the experimental proof that "a + b = b + a in ordinary arithmetic" (when you ask about the work-energy theorem, not realizing that a mathematical theorem has a mathematical proof, not an "experimental anything").

 

[russ_watters]

I am continuously surprised by the inability to understand and answer a very simple question.

We understand it fine, you just don't seem to like/understand the answer. We get that you are young, but you are going to need to try harder.

I just want to examine the experiment that scientists used to decide that work is force x distance.

I know that every website and book on intro physics says it is true but they do not show why it is scientifically true.

Why is work defined as force x distance?

Again: no such experiment exists. It is true because it was chosen to be true. That's all W=FD is: a definition of a name. It can't be false any more than my mother can be wrong for naming me "Russ".

I am getting the feeling that everyone who has studied physics just accepts what they are told and are very good at explaining everything except why it is true experimentally. I have been told that I'm a strange person. I guess Galileo was strange too because he asked questions like mine. I'll bet some of the other notable physicists of the past also asked "strange" questions. Does anyone know the answer to mine?

I had the same problem you had: I thought I was smarter than I was. The fact that I was smarter than some of my teachers just made it worse. It was an obstacle I had to get over because it made learning more difficult. You will have to too. One day you may look back at this thread and shake your head or laugh at yourself over it.