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How did Newton come about with F=ma?

  1. Feb 5, 2008 #1
    If he's thought Force must be proportional to the mass and acceleration why didn't he write down the equation in:

    form , how did he cancel out the k constant?

    or did he come to the solution with Galileo's fallen objects experiments and reasoned :

    a=F/m (a equals to g here of course)

    examining the results and seeing that as m stays constant F must be changing so that g always the same??

    As far as I know Newton is not an experiment-man, he's more of a theoretical-physicist.
  2. jcsd
  3. Feb 5, 2008 #2


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    You can always choose a system of force, mass and acceleration units in which k equals 1.
  4. Feb 5, 2008 #3

    D H

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    In typical English units, one does have to use F=kma. Newton did not say F=ma. That statement is true in the metric system, but work on the metric system started in the late 18th century, 100 years after Newton published his Principia Mathematica. The metric system was explicitly formulated so as to make the constant of proportionality equal to one.
  5. Feb 5, 2008 #4

    Andy Resnick

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    I don't have a copy of the Principia with me (it's available online, IIRC), but Newton didn't have the same physical concepts of force and motion that we have today. As a point of fact, Newton never wrote "F = ma". He generally used geometry and geometrical arguments.

    Newton made several definitions:

    I) The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

    II) The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.

    III) The Innate force of matter (vis insita) is the power of resisting, by which every body, as much as in it lies,endeavors to perservere in its present state, whether it be of rest, or of moving uniformly forward in a straight line.

    IV) An impressed force (vis impressa) is an action exerted on a body, in order to change it's state, either of rest, or of moving uniformly forward in a straight line.

    V) A centripetal force is that by which bodies are drawn or impelled, or in any way tend, towards a point as to a center.

    It's important to note that definition II and III are now taken to refer to momentum, while IV would translate to F = dp/dt. Also, there is a distinction between the origins of various forces. AFAIK, Newton did not use the d/dt notation, either.

    Now, if you ask the question "How did Newton come up with these defintions?", the answer is simply that he was a scientist of the first rank. Or that he came up with correct definitions while his competitors came up with useless defintions.
  6. Feb 5, 2008 #5
    thank you
  7. Feb 5, 2008 #6
    Newton's second law states the following:

    Mathematical Physics, by Donald H.Menzel

    I see nothing in there that states that F=ma

    If you use mass pounds in the US customary system (also known as imperial) then the proportionality constant is not m. Of of course if you use slugs then it is m.

    Anyway, I presume in the metric system that the unit of mass was chosen to eliminate the constant of proportionality.
  8. Feb 5, 2008 #7
    You should probably read Newton's Principia to develop an insight into his method, although I think he was more about presenting results than he was about how he reached them.
  9. Feb 5, 2008 #8


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    In the metric (SI) system, the unit of force (the newton) is defined as the force that causes a 1-kg object to accelerate at 1 m/s^2.
  10. Feb 5, 2008 #9
    That's a good point. You can pick any unit of mass and then define the force to get rid of the constant or you can define any unit of Force and then choose then define the definition of the mass to get rid of the constant.

    It is interesting to think that given a certain set of units what other units can we derive from them. What to take as fundamental besides the speed of light?
  11. Feb 6, 2008 #10
    I think it is pretty amazing that the speed of light is 3*10^8 m/s, within many decimal places. One of the, if not the most fundamental quantities is just about an integer in metric. I suppose of all the quantities, chances are that one will be that way, but the fact it is the speed of light for metric is pretty cool.
  12. Feb 6, 2008 #11

    Andy Resnick

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    That's not an accident- because the speed of light and the definition of 'second' as a duration of time are related, c_0 was defined to be a nice even number, leaving the second to be some wacky number of electromagnetic oscillations. NIST has some documents online discussing precisely this point.
  13. Feb 6, 2008 #12
    Actually, the 3*10^8 figure is an approximation - just so students aren't frustrated using 2.99792458*10^8 every time something involving the speed of light needs to be calculated.
    Last edited: Feb 6, 2008
  14. Feb 6, 2008 #13
    [citation needed]

  15. Feb 6, 2008 #14

    Andy Resnick

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    OOps- my bad. You are right of course. The nice simple number resulting from the definition of c_0 is the permeability of free space: 4*pi*10^-7.
  16. Feb 6, 2008 #15


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    This isn't true. Newton actually carried out many important experiments in mechanics and optics. Some of these were amazingly precise given the available equipment. Their results were fundamental to his final synthesis.
  17. Feb 6, 2008 #16
    Dimensional analysis.
  18. Feb 9, 2008 #17

    Andrew Mason

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    Newton's Laws of Motion in their original form, translated into English from the original Latin, are http://members.tripod.com/~gravitee/axioms.htm#Law I". (press cancel if a password window comes up).

    Last edited by a moderator: Apr 23, 2017
  19. Feb 15, 2008 #18
    I think you're half-right to say the equation could have been F=kma. However, since the every object of mass m has a different characteristic eg. density, surface area, perhaps Newoton believed that k is already considered within m and thus decided to remove k as a redundant property?
  20. Feb 15, 2008 #19
    well,to know exacly how he came to this formula(and surely many other equations and quantities) u have to read the original textbooks writen by the past scientist if available,
    to get a half of what u want,the problem of today science is that no One knows how much effort is taken to get these information(i.e. did u know that scientist worked 50 years to prove that the Pythagorean theory :in a triangle: is not correct in every case and finaly it led to solid geometry!)
    and about your question Columb has done a similar experiment about electrical forces,i think some experiment are done in this case.
    Last edited: Feb 15, 2008
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