Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Ron Hughes on 02/11/2010 14:47:36
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The reason the equation F = ma was proposed by Newton is because F had been determined enough times in experiments to validate his equation so that no constant of proportionality was required. He fully understood all the parameters that made up the equation. Doesn't this suggest that an equation requiring a constant means that we don't fully understand the phenomenon we are trying to describe?
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I think I see where you're coming from:
You mean that a universe based on one Grand Unified Theory (GUT) is somehow incomplete if it has any actual values (ie. constants) in it, yes?
I think this is a misconception as even the simplest universe (one that could not even support complex structure, less life) would need some constant values for its description.
Eg. A 2D 'plane' universe where nothing happened and no 'particles' existed would still have a width and breadth.
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Isn't it just the case that F=ma doesn't need a constant of proportionality because F is defined in units that did not exist until Newton came up with the equation? The defined unit of Force is a Newton and is the force required to accelerate 1kg at 1m/s^2. I thought constants of proportionality appear when both sides of the equation have units which are defined already and maybe in common usage. It is quite common in GR to set c=1 for example, to simplify the equations, but to get back to real numbers you have to remember to put it back in a t some point.
I think I must be missing the point here so please explain. It's been a long day :-)
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Graham may be tired, but he's also spot on.
For those of us old enough to know what a poundal is -
"English units require re-scaling of either force or mass to eliminate a numerical proportionality constant in the equation F = ma."
http://en.wikipedia.org/wiki/Poundal
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Ron you appear to have a fundamental misconception about mathematics rather than any physics that you may be talking about here. If I find by experiment that anything is consistently related to anything for example acceleration and the force applied or the brightness of a light related to the inverse square of the distance of the light source this implies that there must be a constant of proportionality involved in the equation. Sometimes we chose our units conveniently to make this constant unity (one) but this constant is always there and has no implication of a lack of understanding of the process.
This often causes a problem with students in understanding the difference between the (invariant) mass of an object and its weight on the earth's surface. This is because we are so used to understanding "mass" in terms of the attraction of the earth on an object. This is something that changes according to where you are for example the weight of an object is smaller on the moon and zero in free fall whereas the true mass of an object is measured by the force required to accelerate it and is a fixed value. When measured in familiar units of mass length and time that we use on earth this force is not equal to its weight so this constant of proportionality is not unity.
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This was a source of constant confusion and frustration to me when I was at school in pre SI days as we were constantly quoted formulae but never told what the constants meant or anything about consistent units of measurement.
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Soul, Coulomb's law shows a complete understanding of what the electron is made of and what causes it's electric field? Newton's law shows a complete understanding of the mechanism that makes up gravity and it's relationship to the micro-Universe? I'm saying that if we completely understand a phenomenon we would be able to make equations that predict it's occurrence without a constant.
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I think it depends on what type of constant you're talking about, as unintuitive as that may sound. As noted above, in a simulation of even a simple universe, there are always some predefined constants; by-products of the very act of creation.
The constants you're referring to aren't so much constants, but rather more like the ID and placeholder referencing a particular set of conditions; call it a parameter-limited constant. In other word, if we're looking at something like PV=nRT, you can set PV/R=c for some constant provided that the pressure and volume are actually held constant. Another example would be the dielectric constant in the simplified equation for air (my memory regarding this is some what fuzzy). It isn't so much a constant as it is place holder referencing very common conditions (i.e. dielectric in air).
So given such "parametrized constants", I guess they *could* indicate incomplete understanding if the constant being used only appears to be constant because we haven't had the opportunity to test conditions which might otherwise change it. For example, if we never thought to test a dielectric in anything but air, then the constant could be considered a universal constant...except that we have and know that the "constant" itself changes based on other conditions and ultimately has it's own equation.
But having said that, just as matter isn't infinitely divisible, I don't think information is either. At some point, you will have simple, pure, constants...as far as I'm concerned.
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Newton's law shows a complete understanding of the mechanism that makes up gravity and it's relationship to the micro-Universe? I'm saying that if we completely understand a phenomenon we would be able to make equations that predict it's occurrence without a constant.
Newton only said that the acceleration of a body is proportional to the force applied and inversely proportional to the mass of the body. Hence,
F is proportional to ma
Therefore, F equals ma times a constant.
If you like, you can finagle the units so that the constant happens to equal one (as SI does), in which case you can suggest that F=ma, even though that is not generally the case.
The correct expression (if you don't specify any particular units) is F = m.a.k (where k is some constant)
EDIT:
BTW, the famous E=mc² (which I think is in itself a bit incomplete) is also meaningless if the units are not somehow specified or assumed.
The general expression is really E is proportional to mc², or if you must use an equals sign, E=mc².k (why can't I find the proportional symbol in the symbols?)
SI did the scientific community a big favour by making it a lot simpler to relate various quantities without many cumbersome conversions, but I'm beginning to think they simplified things to the point where they dumbed down the math so much that people might assume there are certain nonexistent rigid relationships in physics.
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Ron maybe I misunderstood you when you first posed the question I thought you were missing a bit if basic maths understanding but now I'm not so sure so lets open pandoras box and look at the question a bit more deeply. I don't agree with you when you suggest that Coulomb or Newton "completely understood" the mechanisms when they described the laws associated with them They just stated the way the maths worked but that is not really the point. The problem is all to do with our choices when we define fundamental units and dimensions. The currently preferred SI system used the kilogram mass the meter length and the second time. This is done because it is convenient not because it is totally fundamental. I grew up using the imperial foot pound second units and then went on to use the centimetre gram second units for A level. There are many other alternatives all the way from Planck units onwards.
It could well be worthwhile discussing alternative systems. One of my physics lecturers back in the early 60's suggested that mass was a redundant dimension anyway because gravitational mass is always expressed as an acceleration so everything can be expressed in terms of length and time
As far as standards are involved length and time can be expressed fundamentally using physical phenomena and the velocity of light however mass is a problem and is still related to a lump of metal although they are now trying hard to define precisely the number of atoms in that lump
To return to force mass and acceleration it would be quite possible to completely redefine our system of units based on the amount of force required to accelerate an object but this would mean changing everything and would be far less convenient and accurate when it came to setting standards.
It is always important to understand the constants are always there even if you have chosen to make them one.
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"I'm saying that if we completely understand a phenomenon we would be able to make equations that predict it's occurrence without a constant."
No we couldn't.
If we knew enough about the Universe we might be able (it would be a long shot) to predict my weight.
The trouble is which is the right answer?
about 70 Kgf or about 700 Newtons or about 154 pounds or about 11 stone.
You could get whichever answer you liked, provided that you put the right constant into the equation.
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I can imagine a scenario where by Newton calculates the gravitational constant. He realizes that the force between two masses must be related to the distance between them so he writes F = m1m2/r^2 and decides that does not work. He needs a constant of proportionality so he writes F = G(m1m2/r^2) and solves for G. Without understanding the actual mechanism of gravity he has an equation that works. My point is that, somewhere hidden from us,is an equation that will give more accurate solutions without the use of G.
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But then the dimensions wouldn't agree and the force would not be a vector (ie have a direction). So to the simple proportionality we add a constant that might change the proportion but doesnt have to, that corrects the units, and we add a unit vector to give the direction.
There is no underlying equation that will get much better. The Gravitational Constant G - is not a dimensionless constant that can be removed by changing units or systems of measurement - it is also necessary to get the RHS into units that look like force.
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Ron what you are trying to say is that the gravitational constant (and other non mathematical fundamental constants) must be dependant on some deeply buried property of our universe for example, say, some function of the total energy, particles and space in the total observable and unobservable universe. This may well be so and has been suggested in the past (Paul Dirac was particularly keen on this idea) some aspects of string theory can work towards this but there is no generally accepted model like this that I know of.
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Ron - I think Imatfaal is politely suggesting that you will have to invent an alternative math for your idea to work. Newton was quite creative, and he did invent quite a bit of new math, so I'm guessing that if he had observed a means to circumvent G, he would probably have latched on to it.
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I agree, he would have. It's a subject that needs to be thought about by me and I would hope others.
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Silly me. I see the "proportional" sign is there all the time - or did it just suddenly appear - hmmmm?
Anyway, this may eliminate the need for alternative math.
If you always write F[prop]ma, you never need a constant.
EDIT:
The proportional thingy is very handy, because it also lets you say stuff like;
F[prop]a
F[prop]m
a[prop]1/m
m[prop]1/a
There may be more
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this question seems related to the one i posed a while ago and the answer given:
http://www.thenakedscientists.com/forum/index.php?topic=28989.new#new