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What does F=ma really means?

  1. Oct 17, 2009 #1
    How do we define m and F? If their definition both come from this equation, then the equation doesn't really mean anything...We can also definite a F', and say F'=ma^2, and there isn't anything wrong... However, we have F=ma as something true, then F and m must be something true, indentifiable and measureable. So what does F and m really means??
  2. jcsd
  3. Oct 17, 2009 #2
    Er, where to even start.

    F=ma is a mathematical fit for the force experienced by an accelerating object. It isn't arbritraty as it can be found experiementally.

    F = force m = mass a = acceleration.

    You can assign any letter you want to any mathematical value. So it's perfectly valid mathematically to say that F' = Ma^2, but it dosent show anything in reality, its purely a mathemetical construct.

    F=MA shows something.
  4. Oct 17, 2009 #3
    In classical physics F and m are phenomenological quantities which are specified by other relations. So, you can take some amount of matter and on the basis of that assign to m some value. You can postulate some force law and then make predictions for the motion.

    Deeper explanations for m and F can only be obtained from fundamental physics (quantum field theory).
  5. Oct 17, 2009 #4


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    Sort if true. But the question asked by the OP is by no means trivial. There were whole books written about this back when classical physics was still "state of the art"( as far as I remember the "The Science of Mechanics" by Ernst Mach discusses this at great lenght) and and far as I know they never reached a generally accepted conclusion.
    I guess one could say that back then they were having the same sort of issues with classical mechanics (and thermodynamics) as we are having now with quantum mechanics.

    Also, some of the questions that come out of this are still very much "hot" topics in physics, e.g. why "heavy" (gravitational) mass is the same as inertial (the m in F=ma) mass. Meaning it touches upon topics such as the Higg's boson etc.
  6. Oct 17, 2009 #5


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    This is totally wrong.

    cocosisi's question is very meaningful, and goes, for example, deep into the philosophical underpinnings of Einstein's General Relativity.

    Einstein noted that we are using essentially two distinct definitions of mass, namely inertial mass and gravitational mass (a fundamental assumption within G.R is that these have the same value).

    To take the case of inertial mass:


    1. Suppose you have a mechanism that can be observed to impart accelerations to objects in contact with it (note that ACCELERATIONS are measurable (not that easy, but it CAN be done!))

    2. We say that this mechanism, BY WAY OF A FORCE, imparts acceleration to various objects.

    3. Let us ASSUME that the mechanism gives the SAME force to different objects; for example that our mechanism is a spring that we have compressed 1cm prior to bringing it into contact with the object.
    (That assumption can be given empirical verification that by performing NUMEROUS experiments on some set of objects, each of those objects gains the same acceleration it did on a prior trial)

    4. We observe, however, that different objects gains DIFFERENT acceleration from what we can arguably say is the same force.

    We may therefore construct, within this particular experimental setting, an object-connected scale of ratios F/a, i.e, a "mass scale" We can pick out one object, O_{0}, whose acceleration equals [itex]a_{0}[/itex], and we set the value of the common force, [itex]F_{0}[/itex], equal to a_{0}.

    Thus, that object is given "unit mass", and the rest will be assigned various mass values directly related to the acceleration they experience.

    5. Now, some non-trivial results can be observed within the setting:
    If we take two objects, and makes them stick to each other so that they move as one object, then that compound object's ACCELERATION will be found as if we could add the masses of the two objects together and find the compound mass (i.e, degree of resistance to acceleration) from that.
    Thus, mass is seen to be an ADDITIVE property, that certainly is consistent with the (as yet unproven) idea that mass is some constant property belonging to the object itself, rather than some meaningless, accidental correlation!

    6. By fiddling with two objects, 1 and 2, we find that in the above experiment, the relation:

    Now, SUPPOSE that masses are something intrinsic to objects, "forces" something that this mass-property is independent of, something that merely "happens to" the object.

    What would this entail?

    It would entail that if we did ENTIRELY DIFFERENT experiments (than the spring example, compression of 1cm), then the ratio of observed accelerations would REMAIN CONSTANT, also when we have absolutely no reason of thinking that the forces are generated by the same mechanism as before, and for that matter, have another value than the one in the spring example.

    7. And this is indeed what we see!
    The ratio [tex]\frac{m_{1}}{m_{2}}[/tex] (and therefore acceleration ratio) remains CONSTANT, irrespective of what type of force is applied to both objects.

    8. The accelerations are seen to change, but we are fully entitled to regard the masses as being constant properties of the objects themselves.

    9. Now, we can utilize this discovered property to gauge forces of a variety of forms, and thus to create force scales as well.
  7. Oct 17, 2009 #6
    You are talking to an engineer here not a physicist. I run with arms flailing from GR, Newtonian mechanics all the way.

    On reflection it does seem that the OP's question could be asking something deeper than what is 'F' and 'M' in the most basic context.
  8. Oct 17, 2009 #7


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    And an engineer ought to know everything about how to construct mass and force scales, and WHY they can be regarded as meaningful constructions that most likely have some important relation to actual reality.
  9. Oct 17, 2009 #8
    All I need to know from F=MA is; shove something and it moves. Shove it twice as hard it accelerates twice as fast.
    F=ma can be regarded as meaningful becuase... thats what happens in reality. We know this to be true.
    For practical purposes it really doesnt need to be more complicated than that.
    Last edited: Oct 17, 2009
  10. Oct 17, 2009 #9


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    This equation is the definition of F. m is just a constant of proportionality, defined to make the equation true.

    The goal of classical physics is to find how properties of objects (such as their colour, texture, size etc) cause them to produce or react to forces. For example, the the electric force is produced by a property called "electric charge", and F=(q1.q2)/(r.r).
  11. Oct 17, 2009 #10


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    And on what basis do you really define what it means to "shove twice as hard"??

    cocosisi's question is highly apposite.

    IF, for example, we REALLY had a relation:


    then shoving "twice as hard" wouldn't necessarily result in twice the acceleration.
  12. Oct 17, 2009 #11
    There is really no point in going into this deeper. As what 'shove it twice as hard' means is pretty much what you put a few posts above (post 5), albeit in a very wordy fashion.

    The fact that you can explain it better than I can makes no odds and is the reason I would make a crap teacher.
  13. Oct 17, 2009 #12


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    In most consistent systems of units (including the SI system) F=ma would define force, and mass would be defined in some other way (e.g. by reference to a standard mass).
  14. Oct 17, 2009 #13
    Mass is the total energy content of an object. The (generalized) force is defined as minus the derivative of the energy content w.r.t. the external variable.

    If you have two atoms at some distance, then the force is given by minus the derivative of the total energy of the system w.r.t. the distance. And the energy can be obtained by solving the Schrödinger equation. A fully relativistic treatment will give you the total energy and thus the mass.
  15. Oct 17, 2009 #14


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    Well, you could for example use Newton's 3.law to postulate the equality of two forces within a couple, and then use the ratio of observed accelerations to define a scale for inertial mass.
  16. Oct 17, 2009 #15
    SI units are not consistent. In SI units we use inconsistent units for Time, Length, Mass, Temperature etc. etc. This comes at the price of having to introduce extraneous conversion factors (c, G, hbar, k_b, etc.) in formulae.

    From a metrological point of view, however, it is advantageous to have different physical standards for physical quantities, even if they are related by fundamental physics. As technology advances, it can happen that we can rely on fundamental physics to perform accurate measurements. If that enhances the acuracy whit which you can perform measurements, the SI system will be revised. That happened in 1983 when the meter was redefined in terms of the second by spcifying a value for the speed of light.

    For historical reasons we still use different units for lengths and time intervals, but we could just as well measure time in meters.
  17. Oct 17, 2009 #16


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    The word you are looking for is "natural"; the SI system of units is consistent but not natural.

    Consistent simply means that the base units and the derived units in the system don't have any conversion factors between them. For example, in the SI system the unit of power is 1 W = 1 kg m²/s³, but in the US system the unit of power is 1 hp = 550 ft lb/s. The US system is inconsistent because it needs a conversion between a derived unit and the base units with the same dimensions.

    If a system of units is "natural" then some of the dimensionful universal constants are numerically equal to 1. You can think of these as conversion factors between units of different dimensions. A "geometrized" system of units takes this one step further and considers the universal constants to be not just numerically equal to 1, but also dimensionally equal to 1 (i.e. c is unitless). This means that all of the fundamental units have dimensions of length.
  18. Oct 17, 2009 #17
    I see! I prefer to use geometrized units. :approve:
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