# Can the Properties of the Lever be Deduced without Experimenting?

1. Aug 24, 2013

### 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.

2. Aug 25, 2013

### 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.

3. Aug 25, 2013

### 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.

Last edited: Aug 25, 2013
4. Aug 25, 2013

### voko

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.

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.

5. Aug 25, 2013

### A.T.

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.

6. Aug 25, 2013

### pmclough

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.

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.

7. Aug 26, 2013

### pmclough

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

8. Aug 26, 2013

### voko

I believe that the explanation that you quoted proves that.

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

9. Aug 26, 2013

### pmclough

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.

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?".

10. Aug 26, 2013

### voko

I was referring to my comment in #4.

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.

11. Aug 26, 2013

### 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

12. Aug 26, 2013

### 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.

13. Aug 26, 2013

### AlephZero

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.

14. Aug 26, 2013

### Staff: Mentor

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

15. Aug 26, 2013

### pmclough

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. ).
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.

Last edited: Aug 26, 2013
16. Aug 26, 2013

### 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'?

17. Aug 26, 2013

### pmclough

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

18. Aug 26, 2013

### sophiecentaur

How about a pair of scissors?

19. Aug 27, 2013

### pmclough

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.

20. Aug 27, 2013

### pmclough

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