Can the Properties of the Lever be Deduced without Experimenting?

[pmclough]

I have written a short paper that addresses the question posed in the title.
You can find a copy of the paper here: http://www.math.csusb.edu/faculty/pmclough/LP.pdf.
I think the paper may have some pedagogical value. If you have any comments or suggestions regarding the paper or any different ideas about answering the question posed in the title please feel free to share.

 

[voko]

There is on old proof by Archimedes that uses only geometry and the principle that equal masses at equal lengths from the fulcrum are in equilibrium.

Lagrange in his Mécanique analytique, in the first section on Statics, discusses this and a few other elementary principles that can be used to deduce the law of the lever.

 

[pmclough]

Thanks for the reply and the references to Archimedes and Lagrange.

I am aware of the proof by Archimedes that you mentioned. However, your description of it is not completely accurate. In particular, Archimedes uses the following statement as an axiom (this is T.L. Heath translation of the axiom): ''equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance''. The last part of this axiom presupposes properties of the lever that were observed from experimenting with the lever. Therefore, Archimedes' work does not answer the question that was posed in the title of this thread.

I am not aware of what axioms Lagrange takes when he deduces the law of the lever. I would appreciate it if you or someone else could give me a precise list of these axioms.

 

[voko]

Thanks for the reply and the references to Archimedes and Lagrange.

I am aware of the proof by Archimedes that you mentioned. However, your description of it is not completely accurate. In particular, Archimedes uses the following statement as an axiom (this is T.L. Heath translation of the axiom): ''equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance''. The last part of this axiom presupposes properties of the lever that were observed from experimenting with the lever. Therefore, Archimedes work does not answer the question that was posed in the title of this thread.

If I remember correctly, the last part is not used to prove the law of the lever. What he does is (in modern terms) distribute the masses symmetrically about their attachment points so that the density of the distributed masses is equal; then, if the distances are inversely proportional to the masses, the central portion of the lever is covered symmetrically with the distributed masses, so that for each bit on a side of the fulcrum there is an equal bit on the other side, which implies equilibrium.

But more generally, I do not think I understand your intent. Something has to come from experience. Even pure geometry has some axioms that are based solely on our experience. Your use of conservation of energy is no different: it is either hoisted directly from experience as a fundamental principle, or is arrived at indirectly, from another experimental principle such as Newton's second law.

I am not aware of what axioms Lagrange takes when he deduces the law of the lever. I would appreciate it if you or someone else could give me a precise list of these axioms.

He builds his statics upon the principle of the virtual work, but before that he conducts a review of principles of statics, and that has an interesting commentary to the law of the lever.

 

[A.T.]

http://www.math.csusb.edu/faculty/pmclough/LP.pdf.

Assuming conservation of energy makes it rather trivial. Try to explain it for a static case without invoking virtual work or energy. We had some threads on this here.

 

[pmclough]

If I remember correctly, the last part is not used to prove the law of the lever. What he does is (in modern terms) distribute the masses symmetrically about their attachment points so that the density of the distributed masses is equal; then, if the distances are inversely proportional to the masses, the central portion of the lever is covered symmetrically with the distributed masses, so that for each bit on a side of the fulcrum there is an equal bit on the other side, which implies equilibrium.

I respectfully disagree. I do not think the law of the lever can be proven using only geometry and the first part of the axiom. If you think you can do this I would love to see your proof.

But more generally, I do not think I understand your intent. Something has to come from experience. Even pure geometry has some axioms that are based solely on our experience. Your use of conservation of energy is no different: it is either hoisted directly from experience as a fundamental principle, or is arrived at indirectly, from another experimental principle such as Newton's second law.

Let me clarify my intent. I think my question may be a little ambiguous the way it is currently worded. Here is what I mean: "Can the Properties of the Lever be Deduced without Experimenting with the Lever?". I agree that any axioms that we choose to use to prove the law of the lever will be based on things that were observed in the physical world. However, the question is asking if it is possible to choose a set of axioms that are not based on any observations that were made while experimenting with a lever. I hope this better clarifies things.

 

[pmclough]

Assuming conservation of energy makes it rather trivial. Try to explain it for a static case without invoking virtual work or energy. We had some threads on this here.

I would say being able to make a concept trivial, from a pedagogical point of view at the very least, is a good thing.

 

[voko]

I respectfully disagree. I do not think the law of the lever can be proven using only geometry and the first part of the axiom. If you think you can do this I would love to see your proof.

I believe that the explanation that you quoted proves that.

Let me clarify my intent. I think my question may be a little ambiguous the way it is currently worded. Here is what I mean: "Can the Properties of the Lever be Deduced without Experimenting with the Lever?".

Hmm. Instead of a stated (and, hopefully, small) set of admissible experimental knowledge, we are given a rather vague set of inadmissible knowledge.

 

[pmclough]

I believe that the explanation that you quoted proves that..

If you are referring to my paper then you are mistaken. In the paper I take gravity and the conservation of energy as my axioms and then prove the law of the lever.

Hmm. Instead of a stated (and, hopefully, small) set of admissible experimental knowledge, we are given a rather vague set of inadmissible knowledge.

There still seems to be some confusion about what I am asking. To avoid any miscommunications between us let my try to state the question again in an isomorphic way: "Without ever experimenting with a lever is it possible to deduce the law of the lever?".

 

[voko]

If you are referring to my paper then you are mistaken. In the paper I take gravity and the conservation of energy as my axioms and then prove the law of the lever.

I was referring to my comment in #4.

There still seems to be some confusion about what I am asking. To avoid any miscommunications between us let my try to state the question again in an isomorphic way: "Without ever experimenting with a lever is it possible to deduce the law of the lever?".

Mathematically, yes. You can postulate just about any powerful principle, such as the conservation of energy, and ignore the fact that physically we have obtained this principle via a very long chain of development, where fitting the theory with the experiment - with the lever in particular - has played a role.

 

[chingel]

For the static case you could use angular momentum. The rod with the two masses is connected by a hinge to something and angular momentum with respect to the hinge is calculated. If the rod would start moving, the angular momentum would change. But the rate of change of the angular momentum is the torque and for the rod not to start moving the torque has to be zero. Torques of the forces from the hinge are 0, because the lever arm is 0. From calculating the torque you would get that d1F1=d2F2

 

[sophiecentaur]

When I first learned about 'Machines', we were told that Velocity Ratio and Mechanical Advantage were two different things. VR is governed by the geometry of the machine system and can be derived without experiment. MA involves knowledge of the masses of the parts of the machine and the friction (practical aspects, which need to be measured) . Then
Efficiency = MA/VR
and you can't ignore efficiency.

 

[AlephZero]

"Without ever experimenting with a lever is it possible to deduce the law of the lever?".

That is obviously true, if you assume enough "axioms" of Newtonian mechanics. Real-world engineers "deduce" how new design concepts will behave all the time, before they build anything to "experiment" with!

But I'm still not really sure what you are trying to achieve here, since "gravity" is not a very obvious concept. For example if you make the pre-experimental "common sense" assumption that objects with different masses fall with different accelerations, you would come to a different conclusion about levers (or, you wouldn't come to any conclusion at all if you didn't have some clear ideas about "force" that were probably based on experiment).

And "conservation of energy" is VERY non-obvious concept - it wasn't nailed down till hundreds of years after Newton and Galileo, let alone Archimedes.

 

[mfb]

Assuming conservation of energy makes it rather trivial. Try to explain it for a static case without invoking virtual work or energy. We had some threads on this here.

Conservation of angular momentum works fine in the static case, and you can derive it from Newton's laws.

 

[pmclough]

But I'm still not really sure what you are trying to achieve here, since "gravity" is not a very obvious concept. For example if you make the pre-experimental "common sense" assumption that objects with different masses fall with different accelerations, you would come to a different conclusion about levers (or, you wouldn't come to any conclusion at all if you didn't have some clear ideas about "force" that were probably based on experiment). .

The law of the lever being a simple consequence of gravity and the conservation of energy I thought maybe of some value to physics education (some students find the lever mysterious for example see: True Explanation of a Lever...please. ).

And "conservation of energy" is VERY non-obvious concept - it wasn't nailed down till hundreds of years after Newton and Galileo, let alone Archimedes.

Maybe I am mistaken but I was under the impression that gravity and the conservation of energy are both topics that are covered in a beginning physics course.

 

[sophiecentaur]

Levers work with or without the help of Gravity, though.
Virtual Work, otoh, can be invoked anywhere and is an excellent, valid, approach. Why would one want to use only the intellectual tools that were available to 'the Ancients'?

 

[pmclough]

Levers work with or without the help of Gravity, though.
Virtual Work, otoh, can be invoked anywhere and is an excellent, valid, approach. Why would one want to use only the intellectual tools that were available to 'the Ancients'?

Could you please explain what you mean by a lever working without the help of gravity.

 

[sophiecentaur]

How about a pair of scissors?

 

[pmclough]

Mathematically, yes. You can postulate just about any powerful principle, such as the conservation of energy, and ignore the fact that physically we have obtained this principle via a very long chain of development, where fitting the theory with the experiment - with the lever in particular - has played a role.

If I understand you correctly, it seems that you are suggesting that the conservation of energy notion is dependent on the law of the lever. If this is so then I would appreciate if you could please explain how you came to this conclusion.

 

[pmclough]

How about a pair of scissors?

Thanks for the example. However, outside of a constant force field the law of the lever does not hold.

 

[sophiecentaur]

Are you referring to what I would know as the 'Principle of Moments'? If not, then you can make any statement you like in the context of ancient Science and I would not necessarily disagree. But how relevant would it be?

 

[voko]

If I understand you correctly, it seems that you are suggesting that the conservation of energy notion is dependent on the law of the lever. If this is so then I would appreciate if you could please explain how you came to this conclusion.

This is not what I said. You require independence of experimentation with the lever. You deem conservation of energy so independent. However, as a matter of fact, conservation of energy was historically deduced from certain principles of mechanics, all of which were based on experimentation, with the lever in a particular. Besides, this principle, even if very powerful, is not as general as the principles it is usually derived from. So, to me, accepting this principle but not some others seems quite arbitrary.

 

[pmclough]

Are you referring to what I would know as the 'Principle of Moments'? If not, then you can make any statement you like in the context of ancient Science and I would not necessarily disagree. But how relevant would it be?

I am referring to the Theorem 1 from the paper here: http://www.math.csusb.edu/faculty/pmclough/LP.pdf.

 

[sophiecentaur]

I am referring to the Theorem 1 from the paper here: http://www.math.csusb.edu/faculty/pmclough/LP.pdf.

I read it but does the result mean more than the principle of moments? You wrote it, so you presumably can tell me what you meant.
A lever is a lever, wherever it's used and the same calculations apply, surely. You chose to derive a relationship, using a particular scenario but a lever will work the same in all conditions, surely (?).

 

[pmclough]

This is not what I said. You require independence of experimentation with the lever. You deem conservation of energy so independent. However, as a matter of fact, conservation of energy was historically deduced from certain principles of mechanics, all of which were based on experimentation, with the lever in a particular. Besides, this principle, even if very powerful, is not as general as the principles it is usually derived from. So, to me, accepting this principle but not some others seems quite arbitrary.

I agree that what we choose to take as our axioms is quite arbitrary. I chose to prove the law of the lever by taking gravity and the conservation of energy as the axioms for the following reasons:

1.) Gravity and the conservation of energy are both topics that are covered in a beginning physics course;
2.) The proof is a simple consequence of these axioms;
3.) Some students find the lever mysterious and may benefit by being exposed to the proof.

 

[pmclough]

I read it but does the result mean more than the principle of moments? You wrote it, so you presumably can tell me what you meant.
A lever is a lever, wherever it's used and the same calculations apply, surely. You chose to derive a relationship, using a particular scenario but a lever will work the same in all conditions, surely (?).

It would seem that if I removed gravity then any two masses placed anywhere on the lever would balance the lever.

 

[A.T.]

Some students find the lever mysterious

More mysterious than the conservation of an abstract quantity called energy? Usually, when someone ask why a lever works, he seeks an explanation based on static forces. Not on conservation laws, which are non-obvious themselves.

 

[Dale]

Thanks for the example. However, outside of a constant force field the law of the lever does not hold.

You could use a lever to open a stuck door on the space station.

 

[sophiecentaur]

It would seem that if I removed gravity then any two masses placed anywhere on the lever would balance the lever.

How about forces from two springs, pulling on the lever? Or two rocket engines? etc. etc.? It is surely the Forces and not their origin that counts here.
You still haven't said whether you are referring to the Principle of Moments (very / universally known to apply to levers and many other arrangements). I imagine you know what I mean? Your analysis isn't 'wrong', it's just that I can't see anything special about that paper. Could you point it out if there is anything?

 

[voko]

1.) Gravity and the conservation of energy are both topics that are covered in a beginning physics course;

But then your proof is only valid for a lever in a uniform gravitational field. Which, to me, is too restrictive. While this is an important case, levers work just fine with arbitrary forces, not just gravity, and those forces are not always applied at the right angle to it.

 

[pmclough]

You could use a lever to open a stuck door on the space station.

True, however, the law of the lever (i.e. Theorem 1 from the paper here: http://www.math.csusb.edu/faculty/pmclough/LP.pdf) does not hold.

 

[pmclough]

But then your proof is only valid for a lever in a uniform gravitational field. Which, to me, is too restrictive. While this is an important case, levers work just fine with arbitrary forces, not just gravity, and those forces are not always applied at the right angle to it.

The law of the lever (i.e. Theorem 1 from the paper here: http://www.math.csusb.edu/faculty/pmclough/LP.pdf) does not hold outside a constant force field.

 

[mfb]

True, however, the law of the lever (i.e. Theorem 1 from the paper here: http://www.math.csusb.edu/faculty/pmclough/LP.pdf) does not hold.

There is nothing to balance without gravity, so this is kind of obvious. The modified law "the lever does not accelerate if and only if $\vec{d_1}\vec{F_1} = -\vec{d_2}\vec{ F_2}$" remains true.

 

[voko]

The law of the lever (i.e. Theorem 1 from the paper here: http://www.math.csusb.edu/faculty/pmclough/LP.pdf) does not hold outside a constant force field.

The law of the lever does not require any force fields to begin with. Nor does it require that the force be potential. You are making the solution laden with advanced concepts yet ridiculously narrow in applicability. I am not convinced this is a sound strategy pedagogically.

 

[pmclough]

How about forces from two springs, pulling on the lever? Or two rocket engines? etc. etc.? It is surely the Forces and not their origin that counts here.
You still haven't said whether you are referring to the Principle of Moments (very / universally known to apply to levers and many other arrangements). I imagine you know what I mean?
?

I agree that it is the forces and not their origin that counts. I am referring to Theorem 1 in the paper I referenced. Theorem 1 does not hold outside of a constant force field (please note this does not imply that the lever no longer works there).

Your analysis isn't 'wrong', it's just that I can't see anything special about that paper. Could you point it out if there is anything?

I thought the paper may have some pedagogical value.

 

[pmclough]

There is nothing to balance without gravity, so this is kind of obvious. The modified law "the lever does not accelerate if and only if $\vec{d_1}\vec{F_1} = -\vec{d_2}\vec{ F_2}$" remains true.

I agree.

 

[sophiecentaur]

I agree that it is the forces and not their origin that counts. I am referring to Theorem 1 in the paper I referenced. Theorem 1 does not hold outside of a constant force field (please note this does not imply that the lever no longer works there).


I thought the paper may have some pedagogical value.

But you have to question its value if it leads students to think that levers only work when there's gravity involved. This is such a specialised instance of levers that it could be seriously misleading for someone not familiar with the basics of machines.
What is the point of it, as an alternative to the conventional analysis?

 

[pmclough]

But you have to question its value if it leads students to think that levers only work when there's gravity involved. This is such a specialised instance of levers that it could be seriously misleading for someone not familiar with the basics of machines.
What is the point of it, as an alternative to the conventional analysis?

It seems the proof in the paper could be extended to include the modified lever law that mfb proposed above. The law of the lever would then become a simple corollary to this. I think this may cover the concerns you raised about misleading students.

 

[Dale]

True, however, the law of the lever (i.e. Theorem 1 from the paper here: http://www.math.csusb.edu/faculty/pmclough/LP.pdf) does not hold.

Hmm. Yes. Strictly speaking, theorem 1 is wrong. If there is no force field, if the force field is non-uniform, or if the force is proportional to something besides mass then you can get a balance when the equation is false or you can get an imbalance when the equation is true. IMO, theorem 1 requires quite a bit more to make it true.

I think that it is true that you can use energy principles to derive the operation of a lever. However, if the goal is to find the most basic set of principles needed I would say that the most general, useful, and basic principle is the static equilibrium condition, i.e. that the sum of the external forces and torques are 0.

 

[Dale]

You are making the solution laden with advanced concepts yet ridiculously narrow in applicability. I am not convinced this is a sound strategy pedagogically.

I agree.

 

[pmclough]

Hmm. Yes. Strictly speaking, theorem 1 is wrong. If there is no force field, if the force field is non-uniform, or if the force is proportional to something besides mass then you can get a balance when the equation is false or you can get an imbalance when the equation is true. IMO, theorem 1 requires quite a bit more to make it true.
.

If you look at the paper you will see that gravity and the conservation of energy are assumptions that are both implied in the statement of Theorem 1. If you want to replace the gravity assumption with a more general assumption about forces that is fine but Theorem 1 would then need to be modified (like the modification that mfb proposed in an early post). It seems, however, that the proof would still follow in a similar way.

 

[A.T.]

Usually, when someone ask why a lever works, he seeks an explanation based on static forces. Not on conservation laws, which are non-obvious themselves.

Here is what I mean, in a related thread:
https://www.physicsforums.com/showthread.php?p=4485252

 

[sophiecentaur]

The law of the lever does not require any force fields to begin with. Nor does it require that the force be potential. You are making the solution laden with advanced concepts yet ridiculously narrow in applicability. I am not convinced this is a sound strategy pedagogically.

This is well put.
That paper is not a complete waste of time, though. It was clearly a useful exercise for its author and has the status of something that results from an alternative and personal way through a well known problem. As such, it has merit but it breaks no boundaries.

 

[Dale]

If you look at the paper you will see that gravity and the conservation of energy are assumptions that are both implied in the statement of Theorem 1.

This may be personal preference, but I believe that the statements and assumptions of theorems should always be completely explicit never implicit. The statement, as is, refers to the figure for the definitions of the distances and the masses. But nowhere do you explicitly mention that the lever is massless (or balanced at the fulcrum) and rigid, that the gravitational field is uniform, nor that there are no other forces acting on the masses or the lever (besides the forces at the fulcrum). All of those assumptions are snuck in without justification or explanation.

All of those can be fixed, making the derivation clear. However, a bigger question is the pedagogical value of this approach in general. I am strongly of the opinion that physics, properly taught, is a small collection of powerful general principles, not a large list of disconnected trivia. The underlying principal is the static equilibrium condition. That is what should be taught. The mass times the distance becomes a special case of the general principle, but in itself it is merely a coincidence rather than the essential physics.

 

[A.T.]

Philip Wood has posted a simpler proof for the lever function based only on static linear forces. It doesn't rely on torque, virtual work or conservation laws:

https://www.physicsforums.com/showthread.php?p=4486117

 

[sophiecentaur]

Philip Wood has posted a simpler proof for the lever function based only on static linear forces. It doesn't rely on torque, virtual work or conservation laws:

https://www.physicsforums.com/showthread.php?p=4486117

It's debatable whether Vectors are an easier concept than torque, though.
I still don't see where the need arises for proofs which deliberately avoid the tools we all know and love, apart from the fact that they can show some nice connections between different ideas. And personal satisfaction, perhaps.
There are many ways of killing a cat and, if they're legal and lead to a dead cat, take your pick.

 

[A.T.]

It's debatable whether Vectors are an easier concept than torque,

It's not supposed to be easier to use. It’s supposed to lead from linear mechanics (forces) to angular mechanics (torque).

 

[pmclough]

I am strongly of the opinion that physics, properly taught, is a small collection of powerful general principles, not a large list of disconnected trivia.

I agree. Should any principle in this collection of principles be independent of the others (i.e. should the collection of principles be as small as possible)?

 

[pmclough]

Philip Wood has posted a simpler proof for the lever function based only on static linear forces. It doesn't rely on torque, virtual work or conservation laws:

https://www.physicsforums.com/showthread.php?p=4486117

Very nice proof.