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Newton and approach to second law

  1. Dec 18, 2009 #1
    I am curious about the how Newton discovered his laws of motion and his law of universal gravitation. It seems in modern education, the laws are introduced to students somewhat 'magically', and then the students then do various practical experiments and calculations to verify them.

    If we step back into Newtons shoes, ignoring all the things we know about the world today, what was his actual chain of reasoning he used to discover his mechanical laws?

    In the second law, Newton introduced the idea of something we call 'force' and that when we apply such a force to a body, it will accelerate in inverse proportion to its mass. Yet wouldnt 'force' and 'mass' have been quite mysterious quantities to Newton.

    Assuming a completely blank knowledge about the world, an experimental approach to the second law would seem to require :

    (a) a consistent source of 'force', one that could be relied upon to produce the same mysterious amount of push or pull over and over again.
    (b) a scalable source of force, one that you can double/triple in size reliably. This requires defining the original 'force' as a 'base unit' of 'force'.
    (c) a scalable object to accelerate, one that you can double/triple reliably. Note that there is no notion of 'mass' available to us yet, so all objects would need made of the same material and we would be typically measuring the volume/length of the object.

    The experiment proceeds by applying the same 'force' to objects of various volumes, and comparing the accelerations. We deduce that the acceleration is inversely proportional to the volume of the object.
    We then scale the force, (i.e. double it), and apply it to our objects again and compare the accelerations. From this we can deduce that the 'force' is proportional to both the volume and the acceleration and we need to introduce a coefficient of inertia in the equation. i.e. F = kVa

    At this point we can calculate k for a given material. But at the moment k is only valid for the material of the object we have been accelerating (i.e. lead). We dont know that k is the same for other materials or not (i.e. iron), and we dont even know if other materials obey the same proportionality law. So we repeat our experiments and work out that k depends on the material that we accelerate.

    So we now have a concept of the inertia of a material (i.e. its resistance to acceleration), but crucially we still know nothing of its mass. We could then take objects of two materials which produce the same acceleration under the same force and compare them on a set of weighing scales and measure their 'weights' and notice that they balance.

    We now have two objects that appear to be 'equivalent' in some way. They accelerate the same, under the same 'force' and they balance on a set of scales. We dont really know if the thing that is pushing or pulling them to the floor (i.e. 'gravity' which we have no clue about yet) depends on the material of the objects, so we dont really know if the two objects contain the same 'mass' or 'stuff'. We simply know that they appear to act the same under the same force.

    I look back at my physics education (which was a long time ago) and I wonder if it would have benefited from a proper process of fundamental deduction rather than this reverse engineering approach. In particular, I have this common-sense notion on my head that 'mass' really is the quantity of 'stuff' that is there, and that it was Newton who discovered it. But it would seem that in truth Newton really introduced the notion of 'inertia' didnt he?, which is just a magical constant of proportionality. Indeed, we cant really talk about 'mass' actually being real 'stuff' until we know about the fundamental particles of the world which is not until the 20th century, then we can compare 1g of lead and 1g of iron contain the same number of protons/neutrons (neglecting small differences).

    How did Newton really approach this experimentation or did he deduce things from mathematics and his calculus? What did he choose for (a), (b), and (c) above. Did he dangle weights on bits of string as the source of his force and if so isnt there a built in assumption that local gravity is mg, which is like putting the cart before the horse.

    Also, did Newton discover his universal law of gravitation before his laws of motion or the other way around?

    Hope this makes sense. I know it is not advanced physics but it is interesting to approach things in such a fundamental way sometimes I think.
    Last edited: Dec 18, 2009
  2. jcsd
  3. Dec 18, 2009 #2
    I could very well be mistaken, but when I first learned of Newton, I recall having learned that by making the connection between gravity (at Earth's surface) and gravity (which holds the planets in their orbits) he was able to use kepler's laws to come up with the notion of some spooky abstract "action at a distance", which he played with in developing his laws. Since the motion of the planets was well studied for the time (I think..), that is.
  4. Dec 19, 2009 #3


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    In Newton's time concepts such as 'mass' and 'force' were straightforward. I think conservation of momentum as we know it today was generally accepted in Newton's time.

    In Newton's time, what we refer to today as Newton's second law of motion was a generally accepted law of motion. It is implicit in Galilei's mechanics. Galilei argued that under the influence of gravity objects will fall down with a constant acceleration, and that all objects fall with the same acceleration.

    Galilei argued:
    A) That the weight of an object is the force with which that object is attracted to the geometric center of the Earth
    B) That the inertia of any object is proportional to its weight.

    Of course Galilei phrasings were more convoluted, as these things were developing ideas at the time, but implicitly those ideas are there. Galilei argued that a heavy object will not accelerate harder than a lightweight object, because for the heavier object a proportionally larger force is necessary to accelerate it.

    The relation F=m*a may not have been formulated explicitly by Galilei (I don't know), but it is the logical implication of his reasoning.

    To scientists such as Christopher Wren, Huygens, Hadley, and Hooke it was known that if there would be an inverse square gravitational force from the Sun upon the planets then Kepler's third law would be accounted for. That is, others had independently come upon the idea of an inverse square gravitational force, and had explored its consequences. But they didn't have the means to prove that an inverse square law of gravitation is consistent with Kepler's _first_ law, that the planets move in ellipse-shaped orbits, with the Sun at one focus.

    Newton had developed mathematical tools (a branch of mathematics that today we call differential calculus) that allowed him to put everything in a mathematical framework. Newton could prove and derive what the others could only hypothesize.

  5. Dec 19, 2009 #4


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    For a consistent, scalable source of force Galilei advocated using an inclined plane.
    You can set up a inclined plane at a very shallow angle, and then roll down marbles. The shallower the angle, the smaller the acceleration.

  6. Dec 19, 2009 #5
    Thanks for the replies, quite interesting. It seems Newton did mathematical generalizations of existing knowledge than actual empirical discovery. It also seems that the laws of motion were more driven by the universal gravitation than vice versa...

    In ancient times, I can imagine people lifting a rock and thinking "this is being sucked down (call this weight) and I need to strain my muscles to lift it (call this manual effort a 'force') ". They then invented the weighing scales to conclude that objects could be compared in weight and that they would seem to take the same amount of physical force to lift. This would seem to give them a clear intuition about the notion of 'force' I guess.

    Then using the scales, they could see that a volume of a given material weighs the same as two lumps of the same material of half the volume. The relation between weight and the quantity/volume of actual 'stuff' becomes clear for any single substance.

    But comparing different materials is more of a problem. We can take an arbitrary object (a lump of iron say) and call it an ounce, and create copies of it using the balance. This object becomes our base unit of weight. We can weigh out an ounce of gold but there is no logical deduction that there is the same amount of 'stuff' in each pan, they are qualitatively different things to us, gold and iron. All we can say is that they seem to be sucked to the floor in the same fashion, that their 'weights' are the same. Indeed we could even consider the notion that the force of gravity operates on different types of material with different strengths, i.e. that gravity depends on the particular type of the material(*).

    It seems to me that if we are deducing our laws of motion by using gravity as our consistent supply of force and also our means of determining weight, that F=ma can only really apply to the force of gravity. Did they experiment on other forces to prove the relation in a general manner, what else could they use in a consistent manner other than gravity, springs maybe?

    Looking up 'mass' on wikipedia reveals some interesting subtleties that I had never really considered before. It states there are at least 7 descriptions of what mass is. The two that seem most interesting to early classical physics are :

    Inertial mass - resistance to motion under a force
    Gravitational mass - weight divided by freefall accelleration

    These are not necessarily the same thing, but experimentally they seem to be...

    (*) Incidentally, it would seem to me, that if they saw that the same volume of two different substances did not balance on a weighing scale that either :
    (i) the objects are made up of different types of 'basic stuff', and gravity operates with different strengths on each type of substance.
    (ii) the objects are made up of different quantities of the same 'basic stuff' and that in some objects the stuff is more closely packed than in others.

    (i) one might seem intuitively a messy and non-beautiful law, but (ii) would point the way to thinking about fundamental particles
  7. Dec 19, 2009 #6


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    Newton made groundbreaking empirical discoveries in the field of optics. For example, on the basis of his experiments Newton argued that white light is always a mixture of a spectrum of colors.

    As to mechanics, yeah, Newton's formulation of three laws of motion was in itself not more advanced than what was already in circulation at the time.

    In developing new mathematics Newton was superior to his contemporaries. However, Newton was extremely reluctant to share his discoveries with others. It's actually remarkable that Hadley was able to persuade Newton to publish at all about his celestial mechanics. And in the Principia Newton did not even mention his primary tool: fluxion reckoning (differential calculus) In the Principia Newton did prove all this theorems, but not with his fluxion reckoning.

    So in the Principia Newton's contemporaries could see that a vastly superior science was possible, but they could not learn methods to do that superior science from the Principia. Urged by the prospects revealed by the Principia others developed ways to perform differential calculus, and science could develop to a higher level.

    The point you raise here was very relevant for Newton. If inertial mass mass is always equivalent to gravitational mass then to calculate the orbit of a planet you don't need to know the mass of that planet. If not then the planet's orbits can't be calculated.

    So how plausible is it that inertial mass is always equivalent to gravitational mass?
    Newton pointed out that pendulum swing is sensitive to the ratio of inertial mass to gravitational mass. In a derivation of the period of a simple pendulum the mass of the pendulum bob drops out. (More precisely, inertial mass and gravitational mass drop away against each other.)
    Newton prepared a box in such a way that he could place different lumps of some substance in it, so that the center of mass would always coincide with the geometric center of the box. That box served as pendulum bob. Newton writes that he established to within the accuracy attainable with his setup he did not find any anomaly.
    See also Kevin Brown's discussion of http://www.mathpages.com/home/kmath582/kmath582.htm" [Broken]

    Given the confirming evidence he had obtained Newton decided it was justified to assume that no matter the composition of each of the planets there would always be equivalence of inertial and gravitational mass.

    Once the strength of the the laws of motion and the inverse square law of gravitation are abundantly clear you can turn that around and present it as evidence in itself for the equivalence of inertial and gravitational mass. If inertial mass would not alway be equilvalent to gravitational mass then you would need a different gravitational constant for each and every celestial body. But a single value for G suffices; that is the case only when inertial mass and gravitational mass are equivalent.

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