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True Explanation of a Lever please.

  1. May 11, 2005 #1
    I have asked and I have asked, an no one has yet answered. I'm pleased to have discovered this free meeting place where I can ask so many educated people: how does a lever work? What's going on with the forces involved?

    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. jcsd
  3. May 11, 2005 #2


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    It's just Newton's third law. If you apply a force to one end of a lever, you're exerting a torque, which is defined as the product of two quantities: the length of the arm you're pushing on, and the amount of force you're applying.

    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
  4. May 11, 2005 #3
    Force-Distance Quantity


    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?
  5. May 11, 2005 #4
    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? Have I simply set up a handy model to describe what's going on, with prediction ability, but not actually explained what the forces are doing?

    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.
  6. May 11, 2005 #5


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    There's no such thing as a "torque force." There's torque, and there's force. Torque is the angular analogue of force, if you'd like to think of it as such.

    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
  7. May 11, 2005 #6


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    That's a good observation. If you apply a force F1 to one end of a lever, and move it a distance d1, you've done work W1 = F1 * d1 on it. You'll find that the same amount of work is done at the far end. If the other arm is longer, the force will be larger, but the distance smaller, for example.
    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 work that way, instead of some other way.

    - Warren
  8. May 11, 2005 #7
    I guess so. I saw a book by Feynman on amazon, dealing with QED...highly recommended. I'll have to give it a look.

    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.
  9. May 11, 2005 #8


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    You're exerting a constant torque on the entire lever. In other words, the product of the distance from the fulcrum and the force is everywhere constant. 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.

    - Warren
  10. May 11, 2005 #9
    >You're exerting a constant torque on the entire lever. In other words, the product of the distance from the fulcrum and the force is everywhere constant.

    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?
  11. May 11, 2005 #10


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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.
    Angular momentum and torque are closely related. Angular momentum is defined by

    [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
  12. May 11, 2005 #11


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    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.
    Last edited: May 12, 2005
  13. May 11, 2005 #12


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    The question is like asking Why does a mass gain speed as it's falling? or What is a magnetic field really? or What is inertia and how does it work? Physics is simply a consistent set of rules that nature is found to obey. At the fundamental level, physics does not tell you why anything. I can give you an explanation of how a lever works, but I will always use other terms that you have to accept.

    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...?
  14. May 12, 2005 #13
    is there any formal proof on the conservation of angular momentum ... if so is it complete in itself or based on conservation of energy?
    Last edited: May 12, 2005
  15. May 12, 2005 #14


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    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.
  16. May 12, 2005 #15


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    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.

  17. May 12, 2005 #16


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    The following pairs come out of Noether's Theorem :

    {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
  18. May 12, 2005 #17


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    One thing that I'm not sure was made clear enough about the lever is that force is conserved (or rather, is balanced). While it may appear that you have two forces with two different magnitudes acting on the lever, you actually have three: don't forget that the two downward forces are acting in the same direction and are opposed by an upward force provided by the fulcrum.

    Three forces, two torques. Both the torques and the forces are in equilibrium.
  19. May 12, 2005 #18

    N's theorem proves that corresponding to every continuous symmetry of the hamiltonian , there exists a conserved (not invariant) "charge" - i.e. a quantity that satisfies a continuity equation. (4 divergence = 0). It also explicitly constructs this charge for a given symmetry.

    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.
  20. May 12, 2005 #19
    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.
  21. May 13, 2005 #20
    I think that your question really isn't about a lever, babysteps. You're asking a very philisophical question. That is really "why is the universe the way it is" which at present scientists can still not really understand. The closest answer you can get to your lever question really is that the lever is a stiff rod ok? as the molecules and atoms and held together by strong and weak atomic forces(what are these i wonder?) lol. Where else is your exerted movement going to go? other than the direction in that the lever should act?
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