Can the Properties of the Lever be Deduced without Experimenting?

[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.