Can you explain the lever using Newton's Laws?

[wheels123]

I have always thought that all of "Classical Mechanics" can be explained with Newton's Laws and a few force equations (e.g., gravitational and Coulomb). In particular, the concepts of energy and torque are convenient but not necessary.

If this is true, then can you explain with Newton's Laws - without talking about energy or torque - how a lever "multiplies a force"? For example, consider a see-saw that is balanced (and not rotating) because the child who weighs half as much sits twice as far from the fulcrum. Is there a model that shows how the intermolecular forces are transmitted down the board in such a way that the board pushes upward with twice the force on the heavier child?

 

[Dr. Courtney]

Good question. I don't think you can, but it may be that I've just forgotten how. Torque/work/energy are in all the standard explanations that I remember.

 

[brainpushups]

Yes. It can be done. If you have access to a library you can access the article from The American Journal of Physics in which David Cross derives the rotational second law from first principles: American Journal of Physics 83, 121 (2015); doi: 10.1119/1.4896574

 

[rootone]

Newton's laws DO involve the idea of energy though.

 

[brainpushups]

Newton's laws DO involve the idea of energy though.

I think I might argue otherwise. No doubt that energy principles can achieve the same results, but energy is not part of Newton's laws nor does he mention (or even know about) what we now call energy.

 

[rootone]

I think I might argue otherwise. No doubt that energy principles can achieve the same results, but energy is not part of Newton's laws nor does he mention (or even know about) what we now call energy.

That's fair enough, but the second law has pretty much become our present day definition of 'energy'.
(for most practical purposes anyway, it works fine without needing elaborations like relativity and QM) ,

 

[pbuk]

In particular, the concepts of energy and torque are convenient but not necessary.

No, you can't do without (conservation of) energy - without it you can't explain what happens in a collision.

Once you have energy, the lever can be explained by equating the change in potential energy when one end of the beam moves down and the other moves up by twice as much.

 

[brainpushups]

No, you can't do without (conservation of) energy - without it you can't explain what happens in a collision.

Once you have energy, the lever can be explained by equating the change in potential energy when one end of the beam moves down and the other moves up by twice as much.

No. It can be done without energy. See post #3. The article is called The physical origin of torque and the rotational second law.

 

[Dr. Courtney]

http://scholarship.haverford.edu/cgi/viewcontent.cgi?article=1494&context=physics_facpubs

Right, it can be done without energy, but not without torque.

Or you can do it with energy without torque.

But you cannot do it without energy or torque.

 

[brainpushups]

But you cannot do it without energy or torque.

Um... If I'm not mistaken the author makes no use of torque or energy. He simply uses Newton's third law and a non rigid, steady state model of two connected masses. Indeed he shows that, by applying this model, the expression that we identify as torque emerges as a useful quantity, but he does not presuppose that it is.

 

[A.T.]

For example, consider a see-saw that is balanced (and not rotating) because the child who weighs half as much sits twice as far from the fulcrum. Is there a model that shows how the intermolecular forces are transmitted down the board in such a way that the board pushes upward with twice the force on the heavier child?

Yes, by using static force balance analysis of a truss model. The simplest rigid truss structure is a triangle. We had several threads on this here, try to search for them.

 

[pbuk]

No. It can be done without energy. See post #3. The article is called The physical origin of torque and the rotational second law.

Levers maybe, but not "all of classical mechanics" as asserted by the OP.

 

[brainpushups]

Levers maybe, but not "all of classical mechanics" as asserted by the OP.

I took that statement as a little introduction to the OP's actual question which was about levers. But, since you brought it up, are you claiming that there phenomena in classical mechanics that simply cannot be explained by Newton's laws of motion or are you saying that there are situations in which it is impractical to attempt an analysis with his laws? I certainly agree with the latter, but I think I need some convincing to get me to accept the former.

 

[pbuk]

are you claiming that there phenomena in classical mechanics that simply cannot be explained by Newton's laws of motion?

Yes - I don't think you can explain collisions without conservation of energy.

 

[Philip Wood]

Try this treatment of balancing. It's not original, but I got to it when I pondered the problem: how does a weight on a see-saw know how far from the fulcrum the other weight is? My puzzlement was because I'd been thinking of the see-saw plank as thin and structureless. It isn't. The balancing can take place only because of the system of stresses in the plank. At first I replaced the plank by a pin-jointed lattice structure and calculated the tensions and compressions in the members laboriously, step-by-step. But this treatment did deliver the law of balancing, usually derived from the law of moments. Then someone pointed out to me that a much simpler structure would deliver the law in a couple of steps. See thumbnail.

 

[atyy]

Yes. It can be done. If you have access to a library you can access the article from The American Journal of Physics in which David Cross derives the rotational second law from first principles: American Journal of Physics 83, 121 (2015); doi: 10.1119/1.4896574

http://scholarship.haverford.edu/cgi/viewcontent.cgi?article=1494&context=physics_facpubs

That's an interesting article, and it actually seems to say that the law cannot be derived from Newton's laws, because the rigid body model fails. It goes on to say how a non-rigid model is ok.

However, it is the rigid body case that we really want to consider in elementary physics, and we believe the rigid body is fine within the Newtonian framework, although it may need additional postulates.

IIRC, Kleppner and Kolenkow have very interesting remarks on this, which I cannot remember off the top of my head.

I'm pretty sure this is also why Spivak wrote his terrible physics book http://alpha.math.uga.edu/~shifrin/Spivak_physics.pdf. But he is right that this is a difficult point, so there is a great chance that he will also get the logic right.

 

[brainpushups]

Thanks for the Spivak link. I hadn't seen that before.

That's an interesting article, and it actually seems to say that the law cannot be derived from Newton's laws, because the rigid body model fails. It goes on to say how a non-rigid model is ok.

However, it is the rigid body case that we really want to consider in elementary physics, and we believe the rigid body is fine within the Newtonian framework, although it may need additional postulates.

Right. The rotational second law can't be derived by assuming perfectly rigid bodies. I interpreted this to be a failure of the rigid body model not a failure of Newton's axioms. It would be interesting to go back and read Euler's treatise in which he introduces the rigid body model: Decouverte d'un nouveau principe de Mecanique (available in the Euler Archive). I didn't find a translation in a quick search online and my French isn't really up to par.

 

[wheels123]

Thanks to all for interesting feedback. I'm not surprised that the bigger question - Are the concepts of energy and torque necessary in Classical Mechanics? - captured your attention. I've pondered that for a long time.

Regarding the more specific lever question, the solution posted by Philip Wood (and perhaps the same truss model mentioned by A.T.) seems to directly address the situation. I haven't searched for the previous truss thread yet, but will. My first reaction to Philip Wood's attachment is that it seems to do exactly what I want - to show that the lever law can be derived from Newton's Laws, if you imagine the see-saw as being replaced by a 3-particle system with inter-particle forces and 3 external forces. However, the repulsive force between the particles at points A and B is shown as horizontal, and it doesn't seem like that would necessarily be true. Am I missing something?

 

[wheels123]

Thanks to brainpushups and Dr.Courtney for pointing me to the Physical Origin of Torque article. I'm still digesting that article, but it convinces me that you can indeed derive the lever principle from Newton's Laws, without mentioning energy or torque. Of course, you cannot assume a perfectly rigid body because it is the small displacement of particles within the body that allow forces to be "propagated" from one particle to another. But you can just think of a rigid body as a body with very small particle displacements.

With regard to my original question - How does a lever "multiply a force"? - it seems to me that it has to do with the geometry of the particle chain that comprises the lever. In particular, the particle chain is bent - not straight - and a particle that's far from the fulcrum is capable of transmitting more tangential force (perpendicular to the lever) than a particle close to the fulcrum. For an initially horizontal lever that's subjected to a downward force far from the fulcrum, the inter-particle forces become less horizontal as the distance from the fulcrum increases.

For me, this is an exciting "personal discovery", because it means that we don't have to think of torque as another fundamental variable in Classical Mechanics. I'd like to see Classical Mechanics textbooks specify exactly which variables and equations are fundamental and which are derived.

 

[ViolentCorpse]

I've always wondered about the necessity of the concept of energy as well. I have a habit of trying to understand most daily life things from a mechanical standpoint. It's more intuitive. I've never seen a purely mechanical explanation of why a bouncy ball bounces less after each bounce, or how in the Earth a carjack helps us lift so huge a weight with so little exertion.

 

[A.T.]

I've always wondered about the necessity of the concept of energy as well. I have a habit of trying to understand most daily life things from a mechanical standpoint. It's more intuitive.

Isn't the concept of energy a part of mechanics?

I've never seen a purely mechanical explanation of why a bouncy ball bounces less after each bounce, or how in the Earth a carjack helps us lift so huge a weight with so little exertion.

Then maybe the concept of energy is quite useful.

 

[wheels123]

Isn't the concept of energy a part of mechanics?

Then maybe the concept of energy is quite useful.

Clearly, the concept of energy is a useful part of Classical Mechanics. In particular, systems with many particles (like bouncy balls and carjacks) are often so complex that analysis with Newton's Laws is very difficult, perhaps even impossible.

But to me an important question is: Is the concept of energy necessary in Classical Mechanics? In other words, is there some information in energy equations that is not present in or derivable from Newton's Laws? To me, to clearly express the theory of Classical Mechanics you have to do two things: (1) decide on your representation - for example, Newton's Laws or energy - and (2) present all the laws/equations in that representation. All textbooks I'm familiar with jump between the two representations. I'd think the ideal approach would be to present the entire theory first in Newton's Laws representation, then repeat the entire theory in energy representation.

 

[Let'sthink]

Newton's laws as such are applicable to point particles only. For broader application we have to take recourse to modelling or sister topic such as elasticity etc. The tension in the balanced plank will be asymmetric when the weights of the riding boys are different. The point particle ideas breakdown even when we consider the contact and frictional forces which are self adjusting.

 

[brainpushups]

Newton's laws as such are applicable to point particles only. For broader application we have to take recourse to modelling or sister topic such as elasticity etc.

Well sure. Newton's law aren't of much use if you don't have a mathematical expression for the forces between particles. I would still say that Newton's laws explain gravity (for example) even though we need to model the gravitational force.