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I have read the typical formulas that are good at predicting the behavior of a lever. But very smart people have been unable to explain to me how a lever works.

I look forward to your answers.

Thank you! This has been bugging me for years.

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- Thread starter BabySteps
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I have read the typical formulas that are good at predicting the behavior of a lever. But very smart people have been unable to explain to me how a lever works.

I look forward to your answers.

Thank you! This has been bugging me for years.

- #2

chroot

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The same torque acts on the far end of the lever. The force generated at the far end is scaled by the ratio of the lengths of the arms of the lever.

- Warren

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Warren,

Thanks for the response.

I can observe the output of a lever...and it does seem that the force on the far end of the lever is scaled by the ratio of the lengths of the lever arms.

But...I can't yet see how that explains what's going on in a lever. Newton's third law. So what we are saying here is that a lever creates a torque force that is conserved.

Looks like my question would then be: what is a torque force and how exactly does it work? How is distance part of a force equation when it comes to describing kinetic energy?

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Thanks again, everyone, for thinking about this with me. I'm very puzzled...very confused by this one. It seems so hard to find an actual explanation to describe what's going on with the forces in a lever.

- #5

chroot

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Torque, [itex]\vec \tau[/itex] is defined as:

[tex]\vec \tau = \vec r \times \vec F[/tex]

It is a (pseudo-)vector quantity, but you can often ignore its direction.

Torque is thus commonly described as the product of the applied force (F) and the distance from the point of rotation at which the force is applied (r).

See http://hyperphysics.phy-astr.gsu.edu/hbase/torq.html#torq for an illustration.

We are not considering kinetic energy at all here, since we're talking about static, motionless levers, and just considering the forces.

- Warren

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chroot

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That's a good observation. If you apply a force FBabySteps said:If I observe a lever and feel disturbed by the apparent amplification/inability to explain what's going on with the forces...I could notice that the distance has a direct relationship to the forces applied. If I define energy in this situation as length x distance, then energy will be conserved, of course, but what have I understood?

It sounds like you want an answer to why Newton's third law works in the first place. If you'd like to understand that, you can descend down to atomic theory and consider the interatomic electromagnetic forces that allow a push on one end of a lever to be transmitted to the far end. If you'd like to understand electromagnetics in full detail, you can study QED. If you'd like to know why QED works, there will no further answers forthcoming from anyone: the universe just happens to workThanks again, everyone, for thinking about this with me. I'm very puzzled...very confused by this one. It seems so hard to find an actual explanation to describe what's going on with the forces in a lever.

- Warren

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I still can't wrap my head around *why* the length of the lever arm should matter. Obviously it does matter...but I can't see any explanation as to why it does matter yet. I see formulas the repeat what I can see: it does matter...and it matters "this much."

But I've not yet heard (or not yet realized I have heard it) any reason as to why a the input force is amplified the longer the physical length of the input lever arm.

Equal and opposite reaction...I can assume that for now (though I'd like to understand that as well). But as for why torque works...or why a longer arm can produce more force.

Argh. I wish Feynman were alive.

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chroot

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- Warren

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What do you mean by this? You mean the force is the same everywhere? or that whatever the force is on a given point along the lever...that force is not changing?

>If you're further from the fulcrum, you feel less force. It works that way because angular momentum is conserved. Angular momentum is conserved because physics is invariant to rotations in space. There's no other way it could work.

Can you explain how conservation of angular momentum results in feeling less force as distance from fulcrum increases?

- #10

chroot

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No. I mean the product of two variables is constant. If one variable goes up (the distance), the other must go down (the force), and vice versa.What do you mean by this? You mean the force is the same everywhere? or that whatever the force is on a given point along the lever...that force is not changing?

Angular momentum and torque are closely related. Angular momentum is defined byCan you explain how conservation of angular momentum results in feeling less force as distance from fulcrum increases?

[tex]L = r \times p[/tex]

where torque is defined by

[tex]\tau = r \times F[/tex]

Of course, force is nothing more than the rate of change of momentum:

[tex]F = \frac{dp}{dt}[/tex]

This leads to the familiar form

[tex]\frac{dL}{dt} = \tau[/tex]

- Warren

- #11

learningphysics

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BabySteps said:

I have read the typical formulas that are good at predicting the behavior of a lever. But very smart people have been unable to explain to me how a lever works.

I look forward to your answers.

Thank you! This has been bugging me for years.

Firstly we'll start with a rigid body (any shape)... that is attached to a pivot. So it can rotate about that pivot. Suppose it is rotating. Now this rigid body can be divided up into infintesimal masses dm... The force acting on dm is [tex]dF=a dm[/tex] (where a is the acceleration). But [tex]a=r\alpha[/tex] where [tex]\alpha[/tex] is the angular acceleration.

So we have [tex]dF=r\alpha dm[/tex]. No multiply both sides by r. r is the distance from the pivot point to dm.

[tex](dF)r=r^2\alpha dm[/tex]

Now integrate both sides of the equation... this is the important part... unfortunately I'm going to be a little sloppy here. Inside a rigid body forces occur in equal opposite pairs due to newton's third law. Suppose you have a dm1 next to a dm2... dm1 exerts a force dF on dm2.... that means that dm2 exerts a force -dF on dm1... So the torque due to both of these forces is 0 (rdF - rdF =0). So on the left side... all the internal torques cancel to zero, and all we are left with is the external torque due to external forces outside the body. (I realize this is not very rigorous... hopefully it will help for now).

[tex]\tau_{net} = \alpha\int r^2dm [/tex]

[tex]\int r^2 dm[/tex] is the moment of inertia I.

[tex]\tau_{net} = I\alpha[/tex]

What this says is that the net external torque determines the angular acceleration of a rigid body about a fixed point. Torque = Force * distance (just like the dF * r, I used before). So only the product matters. This is all the consequence of newton's laws, and having a rigid body.

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krab

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Here's one based on energy:

Work done is force times distance. It requires a fixed amount of work (energy) to lift a given rock a distance. You can do it directly, or with a lever. Let's say your lever has a ratio of 2. That means you move your end twice as far as the rock moves. Since the work done is the same whether you use the lever or not, and that work is force times distance, and the distance is twice as large, the force must be only half as large.

But now you ask What is energy and why is it conserved...?

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krab.

is there any formal proof on the conservation of angular momentum ... if so is it complete in itself or based on conservation of energy?

is there any formal proof on the conservation of angular momentum ... if so is it complete in itself or based on conservation of energy?

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HallsofIvy

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govinda said:krab.

is there any formal proof on the conservation of angular momentum ... if so is it complete in itself or based on conservation of energy?

No, conservation of momentum (angular or not) is not based on conservation of energy- you can have situations in which momentum is conserved but energy is not (inelastic collisions for example).

I'm not sure what you mean by "formal proof". This is physics, not mathematics, after all. All physics laws are based on experimental evidence.

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govinda said:

is there any formal proof on the conservation of angular momentum ... if so is it complete in itself or based on conservation of energy?

Take note that via the Noether Theorem, ANY conservation laws is in fact a result of an underlying symmetry principle. The conservation of angular momentum is directly a manifestation of the isotropic symmetry nature of the classical empty space. A spinning or rotating object has now broken that symmetry because there is now, for that system, a well-defined direction in space. Similarly, there are other symmetry principles accompanying the conservation of linear momentum, and the conservation of energy.

As with any fundamental conservation laws and symmetry principle, these are NOT derived via First Principles. These are deduced via a consistent observation on how our universe behave. That is what makes physics different than mathematics.

Zz.

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Gokul43201

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{Conserved quantity, quantity under whose change the system is invariant - popular name for symmetry)

{linear momentum, linear position - spatial symmetry}

{angular momentum, angular position - isotropy}

{energy, time - temporal symmetry}

Note : temporal symmetry is not with respect to time reversal

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russ_watters

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Three forces, two torques. Both the torques and the forces are in equilibrium.

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Gokul43201 said:

{Conserved quantity, quantity under whose change the system is invariant - popular name for symmetry)

{linear momentum, linear position - spatial symmetry}

{angular momentum, angular position - isotropy}

{energy, time - temporal symmetry}

Note : temporal symmetry is not with respect to time reversal

comments:

N's theorem proves that corresponding to every

once that is stated clearly, then the symmetries in question, for the problem, are evidently, continuous spatial translation, continuous rotations about an axis and continuous time translation.

There is a reason (and it is not to make things obscure) why the terminology is important - clearly, from N's theorem, now one can tell that time reversal symmetry, being a discrete symmetry, as any reflection is, has no corresponding conserved "charge"

hope that helps.

adi

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BabySteps said:I have read the typical formulas that are good at predicting the behavior of a lever.

The "typical" formulas regarding a lever at not "good at predicting".

They EXACTLY describe a perfect lever. Flex of the bar and fulcrum contact greater than a line explain the difference between perfection and reality.

As arguably the simplest of the "simple" machines, it takes nothing more than play to understand the interaction of distance and mass AND why perfection is impossible.

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learningphysics

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The operation of a lever can be rigorously explained from Newton's laws.

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I was thinking about living systems and energy utilization/conservation, and arrived at this same basic question - how does the lever really work?

To summarize what makes sense to me - to lift something, some work needs to be done. One way of looking at it is how can i do this work without as much 'effort' so to speak. Essentially the way the lever works is to reduce the force that you need to apply, but increase the duration for which you apply it. That is, you are moving your end of the lever over a longer arc (greater distance ~ more time) than the arc traced by the object itself. So clearly there is a compromise to be struck between how long I am willing to apply the force for, and how much force i am willing to apply. This to me is the essence of the lever mechanism.

As Archimedes said - give me a lever long enough and i will move the world. He forgot to add that it could take a long, long time.

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Jesus what an old thread you dug up!

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I have the same question in the mind too... Is it possible to have sub-atomic levers ?

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