# Which way did Newton find F = ma?

1. Sep 12, 2011

### prosteve037

I was under the impression that, by experiment, Newton deduced

$\textit{F}\propto{m}$ $\rightarrow$ $\textit{F = k}_{1}\textit{m}$

(where $\textit{k}_{1}$ is some constant)

and

$\textit{F}\propto{a}$ $\rightarrow$ $\textit{F = k}_{2}\textit{a}$

(where $\textit{k}_{2}$ is some constant)

and then found that either/both

$\textit{k}_{1}\propto{a}$ $\rightarrow$ $\textit{k}_{1}\textit{ = c}_{1}\textit{a}$

(where $\textit{c}_{1}$ is some constant)

and/or

$\textit{k}_{2}\propto{m}$ $\rightarrow$ $\textit{k}_{2}\textit{ = c}_{2}\textit{m}$

(where $\textit{c}_{2}$ is some constant)

thus creating

$\textit{F = c}_{1}\textit{ma}$

and/or

$\textit{F = c}_{2}\textit{ma}$

where in SI Units they would be in the form

$\textit{F = ma}$

However, I've read in some other forums how Newton actually meant

$\textit{F}\propto{ma}$ $\rightarrow$ $\textit{F = kma}$

Which is the case? Did he use the first method or did he simply state the second?

If he did use the first method, how did he resolve that
$\textit{k}_{1}$ is dependent on acceleration and/or that $\textit{k}_{2}$ is dependent on mass?

2. Sep 12, 2011

### xts

Actually Newton defined 'force' as $F\propto ma$ paying little attention to proportionality constant. Before Newton the concept of "force" was only intuitive and had no precise, well defined meaning.

Newton's original works (Principia...) are pretty hard to read and understand - he used the formalism based on geometrical relations rather than on algebraic ones. A formalism we are used to, itroduced in 18th century, defined it without any proportionality constant, as F=ma.

3. Sep 12, 2011

### prosteve037

Hmm okay. I always wondered what those geometric methods were. I'm assuming he plotted or graphed the parameters as coordinates. And perhaps took $\textit{ma}$ as the area and $\textit{F}$ to be some coordinate. That's my guess at least.

Does anyone know exactly what he did?

4. Sep 12, 2011

### xts

You may try to read "Principia..." just to see yourself the kind of argumentation Newton used.
There are several English translations available on-line - I recommend (as the easiest to read) - American translation from mid 19th century: http://rack1.ul.cs.cmu.edu/is/newton/ [Broken]
You may want to skip all the introduction, and start from page 84 - there are corollaries to the principles - explained and illustrated geometrically.

Last edited by a moderator: May 5, 2017
5. Sep 12, 2011

### prosteve037

Thanks xts.

I'm also curious as to what you said about Newton's view on proportionality, being that he payed little attention to the proportionality constant. Mind if I ask where you read/heard this?

This gives me a different perspective as to how proportionality was defined. Was it that proportionality was crudely defined in Newton's time? Or was it just Newton that disregarded this?

6. Sep 12, 2011

### xts

I read "Principia..." - long time ago I assisted my professor with his work on history of science. I even read part of it in nasty Newton's latin...

At Newton's times proportionality was perfectly known and understood - it was used even more frequently than nowadays. Just contrary - those times the numerical values, measurement units, etc. were rather neglected. It was the effect of geometrical representation. As in Euclid "Elements" - the line segment represent the value, and twice longer section represent twice bigger value, but no one cares to say that 1 inch on the drawing represents 1 pound of mass. You may make the same Euclidean construction in different scale, and all conclusions will be the same.
Such approach is really difficult to understand for modern people, who learn on numbers, rather than on Euclodean constructions.
Newton was one of the very last scientists using such geometrical representations (but it was common till his times, Copernicus did the same). It was 18th century when numerical approach (started by Rene Descartes even a bit earlier than Newton worked) finally won popularity.

7. Sep 12, 2011

### xts

You should notice that Newton worked on astronomical data. He knew precisely what were angular positions of celestial bodies. But he could only roughly estimate their absolute positions. He had no idea about mass of Earth or Moon - but he could calculate their proportion. He did not know the exact size of Earth and Mars orbits, but he could very precisely determine their proportion.

That lack of absolute scale is another reason why those times proportionality was treated as very meaningful, while nobody cared about exact value of proportionality constant. Newton had no idea what gravitational constant value may be and he was not bothered by it - it had to pass over 100 years till H.Cavendish, who lived in 'numerical era' rather than in 'geometrical times', estimated its value.

8. Sep 12, 2011

### prosteve037

I'm confused here. How was Newton able to calculate the proportions between objects/parameters if he didn't have values for either object?

And specifically what experiment did he perform to find the relation $\textit{F}\propto{ma}$?

EDIT:

Wait, did you mean he realized that the mass of the Earth and Moon would have to be proportional to each other and not that he could "calculate their proportion"? And similarly for the orbits?

I could understand how he would have thought the Earth's mass/orbit and Moon's mass/orbit would have to increase/decrease as the gravitational force of the Sun/Earth increased/decreased. But other than that, I can't see how he would've calculated their proportions without having values for each parameter.

Last edited: Sep 12, 2011
9. Sep 13, 2011

### xts

Try to look at it from geometrical point of view. I hope you had a course of Euclidean geometry (ruler+compass) at school. Euclidean ruler has no marks. Euclid says about proportions between legths of line segments (e.g. Thales' theorem), but he never use any absolute measure of them. It was a proportion, which made a sense of geometry. Exact measures, expressed in inches or pounds could be important for merchants and for other practical purposes, but not for reasoning.

If you have two segments A and B in proportion A/B=3/2 you may check it such, that using a compass you markA three times on some line, and you mark B 2 times, and finally those constructed 2A and 3B will be equal. You don't need to know how many mm or inches any of them has.

Please note that Newtonian mechanics do not depend on units of measure we chose. We may use any unit for time, distance and mass - they will be bounded by only one constant (gravitational), which was totally unknown to Newton (it got estimated by Cavendish hundred years later). All the mechanics is expressed in terms of proportions: e.g. centre of mass of two bodies is a point such that distances to both of them are in proportion reverse to proportion of their masses. That's a law true independently from units and from actual configuration of bodies.

He didn't any. Once again - it is a definition of force, not an experimental knowledge.
You could however ask, what experiment he did to find the symmetry of such defined forces (3rd principle) - he utilised Galileo's observations and astronomical observations. But he also performed some laboratory experiments with elastic and half-elastic collisions. He also made a gedanken-experiment with a planet cut into two parts - and concluded, that if 3rd principle was not true, than the planet should accelerate on itself, which is absurd.

Honestly, I am not sure if Newton knew it - it could require a bit more accurate observations than he was able to perform. But he surely could know the proportion of masses Jupiter/Saturn, as it may be calculated from proportion of periods of their moons and the proportion between sizes of orbits of these moons. And only the periods might be measured in absolute units. Orbital sizes might be only expressed as a proportions to each other.

Please note that most of pre-Newtonian science, especially astronomy, had always been expressed in terms of proportions, never quoting the actual values of proportional constants. Take for example Kepler's laws (they inspired Newton to formulate his mechanics).

Last edited: Sep 13, 2011
10. Sep 13, 2011

### harrylin

According to Motte's translation*, it was not a definition but an axiom (a generally accepted law that is however open to disproof). I suppose that Newton did not invent the relationship but expressed what others already had experienced.

http://gravitee.tripod.com/axioms.htm

together with his definition 2:
http://gravitee.tripod.com/definitions.htm

then it appears (or he suggests) that he first deduced the existence of a momentum (m*v) which he defined as quantity of motion (p), and about which he next claimed the force law.
That seems indeed to correspond to your second case.

More precisely (although not clearly formulated as such, I interpret "alteration" as d/dt):
F ~ dp/dt

Harald

* I find http://gravitee.tripod.com/toc.htm handy: just press "cancel".

Last edited: Sep 13, 2011
11. Sep 13, 2011

### xts

Yes and no ;)
'Principia...' are not perfectly consistent structure, even as for 17th century standards and often mix definitions with axioms (e.g. 3rd law is used in def. III)

Yes - force is introduced as a 'law of motion' (axiom), but:
No - the term 'force' is also defined quantitatively by Definitions III-VIII - III defines it as proportional to mass, and the next ones about centripetal forces are equivalent to proportionality to acceleration (compare with geometrical representation of calculus in further parts of the Book I)

I fully agree that the definition of momentum (quantity of motion) is a key point for the reasoning. I also admit that further, in the calculations, Newton equals (or rather makes proportional) the derivative of momentum with the force without deeper justification - so it may be taken as an implicit definition of both 'force' and 'alteration'.

So to answer Prosteve's question - regardless the 'force' is introduced as definition or as an axiom, it is not a subject to be tested alone. It is rather a foundation part of the structure, which must be tested against experiment (mostly against astronomical observations) as a whole.

PS. - thanks for Motte's translation better readable than the one I used!

Last edited: Sep 13, 2011
12. Sep 13, 2011

### harrylin

- First of all, I distinguish the elaborations from the definitions. Anyway, I checked those definitions (in italics) for "proportional" and found them starting from definition VI (which I could not apply to the law of motion). Definition VII seems to say that a certain acceleration corresponds to a certain force, and that F ~ v_t. Definition VIII is unclear to me; from the elaboration it appears to mean that F ~ p_t.
Thus indeed, I now also think that the difference between definitions and axioms is very blurred!

- About the translation: regretfully the site that I use is only partial (although the most important part), so I'm also happy with yours!

13. Sep 13, 2011

### harrylin

As you can read in http://gravitee.tripod.com/axioms.htm ,
he knew from experience (other people's for sure!) that "If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively."

In particular, in his elaboration on the definitions, he related to the experience with weapons:
"If a leaden ball, projected from the top of a mountain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far."
- http://gravitee.tripod.com/definitions.htm
He probably knew that double the amount of powder has double as much force; and also that can be tested, with depth of penetration.

Note that air resistance is important for bullets but less so for canon balls.

Moreover, the same "impressive" force gives the same impression (deformation) of deformable objects such as springs, and to a certain degree the impression is even proportional to the force - as his competitor Hooke observed.

Last edited: Sep 13, 2011
14. Sep 13, 2011

### xts

Don't take it as a serious experience with weapons! It is rather gedanken-experiment like, exaggerated argument. Cannons exceeding in range one mile were not available even during Napoleon Wars.

15. Sep 13, 2011

### sankalpmittal

Always remember in physics "that to prove a theory first there is experimental observation and then mathematical deductions"

Newton's laws are proved experimentally and then deducted .
He first gave his 1st law of motion which states "that every body remains in its genuine state of rest or uniform motion in a straight line unless acted upon by some external force (assuming body to be displaced linearly)"

Have you ever heard of the mathematical proof of his 1st law ? No , but it is proved by his 2nd law! But his 1st law was given prior to his 2nd law ! However his 1st law is logically explanatory. Inertia. Mass resists . Each molecule each atom of it !

Proof :
By his 2nd law we know ,
F=ma
If body remains in its original state then there is no force being applied so,
0=ma
or
a=0
So it will have no displacement.

Now lets come on to his 2nd law . What does it state ? It states "that rate of change of momentum in a body is directly proportional to the impressed force acting on it and it takes place in direction of that force assuming body to be linearly displaced in straight line "

F∝m → F = k1m

(where k1 is some constant)

?

How did Newton get that ?:rofl:
Ans: Experimentation.

Precisely yes, this k1 is the constant acceleration in a body.
It will be
F∝k1m
You are messing the constants but precisely yes , this k2 is the constant mass of body considering same body.

It will be
F∝k2a
This is where you are messing it all up and you did it wrong.

No 'twill be k1=a
if c1 is some constant then c1=1

Again I would rather do same , you are messing it up .
No 'twill be k2=m
if c2 is some constant then c2=1

where c1=c2=k=1
Exactly.

Exactly , this is pretty cent percent correct representation of 2nd law .

2nd equation is correct . He simply stated the second after deriving it mathematically.
I repeat , he never used the first method. It is absurd yet I have corrected it in my "this" post above.
k1 is itself the value of acceleration if it is kept constant in a body.
k2 is itself the value of mass if it is kept constant in a body.(taking same body)

____________________________________________________________________

Proof of 2nd law

Experimentally Newton found
F∝dp/dt
F∝m(v-u)/t
F∝ma
F=kma

By his experiments we know that 1 kgms-2 force on 1 kg of body produces in it 1 ms-2 of acceleration.

So
1=1 x k
or k=1
So
F=ma

Note : k is a universal constant which is always 1 for all the conditions. However logically his 2nd law is self explanatory.

Hope this helps.

:)

Last edited: Sep 13, 2011
16. Sep 13, 2011

### harrylin

17. Sep 13, 2011

### xts

Chapeau bas!

On a quick search I found only field artillery of mid 19th century exceeding 1 mile range ;(

Anyway, I would really never expect that in 1702 Spaniards were able to kill four and wound six men on 3000 yards distance.

I must read "History of Cyprus" - I hardly believe in 3 miles range of 16th century Venetian culverines... But gentlemen don't dispute the facts... I must accept it, at least till checking the relation...

18. Sep 14, 2011

### prosteve037

Okay. I think I know where you're going with this.

See before I asked this question, I assumed some things:

1 - Newton used experiments to deduce two separate proportionality statements to arrive at the conclusion $\textit{F}\propto{ma}$.

2 - Subsequently, Newton assigned and used units of measurement in his experiments to arrive at this relationship.

Now from what I've read, the former ("1") is partly correct. Correct in that he arrived at the conclusion $\textit{F}\propto{ma}$. But according to what I understand xts to have said,

that assumption was incorrect in that he didn't deduce that relationship from experiments. Specifically, 2 different experiments ($\textit{F}\propto{m}$ and $\textit{F}\propto{a}$). I'm confused by this.

How was Newton able to deduce, let alone define, the relationship $\textit{F}\propto{ma}$ without having used data to notice that the product of mass and acceleration of an object is proportional to the force applied? I'm having a tough time understanding how Newton could have made such a bold definition without math leading up to its definition.

I never meant to imply that Newton was concerned about the value of proportionality constants between parameters, but what I wanted to address was how he realized that $\textit{F}$ is proportional to the product of mass and acceleration; not $\textit{F}\propto{m}$ and $\textit{F}\propto{a}$ individually of each other.

Last edited: Sep 15, 2011
19. Sep 15, 2011

### harrylin

See the discussion of the same in this parallel thread (almost the same topic, how did that happen?):

20. Sep 15, 2011

### xts

That is simplest of all - pure algebraic deduction!
Any value proportional to two other values, must be also proportional to their product.

21. Sep 15, 2011

### harrylin

What I suppose prosteve meant: if that's indeed the logical way of development, as we all seem to think, then - despite the way Newton summarized it - the first approach of post #1 is what perhaps really happened, well before Newton wrote his Principia.

22. Sep 15, 2011

### xts

OK. Let me try this way: I agree with one of Harrylin's posts, pointing that central concept of Newton's dynamics is momentum (quantity of motion), and observed phenomenon of preservation of momentum in closed systems (or - if you prefer: uniform motion/rest of centre of mass of closed system). Newton did some experiments to test the last: colliding penduli of different masses made of different materials (to get partially inelastic collisions).
All the rest of his dynamics (esp. F~dp/dt; F~ma; Fab=-Fba) is a deduction from momentum preservation principle of the closed system combined with convenient definition of force (which had no strict meaning before, so Newton was free to define it as he liked).

Of course - his theory of gravity goes beyond this - it required additional justification. And here, I believe, it was pure deduction from astronomical observations. Or even not directly from observations, but rather from Kepler's laws.

We may only regret taht Principia are not so consistent axiomatic structure as Euclid's Elements - so we have mixed definitions and axioms, and the main concept of momentum is not stressed enough - it is not even present in any of axioms.

23. Sep 15, 2011

### harrylin

A few little remarks:
Not completely free, he was bound by observations: see my post #21 on https://www.physicsforums.com/showthread.php?t=383019&page=2

In general, definitions in physics only make sense (are useful) if they are compatible with the theory for which they are used.

I now come to think that perhaps Newton's Principia only appears a little inconsistent because he simply grouped together his basic definitions, which in part refer to some of the laws that he presents next. If so, that's a matter of taste, not of inconsistency.

Note also that he apparently calls descriptions of identities "definitions", while he calls descriptions of physical behaviour "laws" or "axioms"; and I now also come to agree with that.

Best,
Harald

Last edited: Sep 15, 2011
24. Sep 15, 2011

### sankalpmittal

Will you have a look at my post #15 ? I think , it carries out the answer of your question and confusion ?

See the first paragraph in the post 15 and lay emphasis on it. The first derivation is correct but partly and the second representation is cent percent correct.

F∝m and F∝a , then F∝ma is the simple concept of proportionality.

Let me try this simple explanation :
suppose,
x=ab
Now imagine that a is a constant and a=2 , say.
then
x=2b

Now
if b=1 , x=2 . if b=2 , x=4 .if b=100 , x=200.

So we notice that b increases x also increases. if its 3b the result is x=6b

So we can say that
x∝b if a is kept constant .
Now repeat that with a ,
x∝a if b is kept constant.

so , x∝ab
or if k=1, then x=ab !

I once again adjure you to see the post #15 .
He deduced the relation by several experiments of mass and acceleration by keeping one constant at a time , and then established that F∝ma.

:)

"Again I recommend that thou shalt see post #15"

25. Sep 15, 2011

### xts

So I'll then say it as he was free to call "the alteration of amount of movement" as he liked - and he fortunately chose the common language word 'force', which was consistent enough with it to be used.
I agree - the proportionality to mass is a part of common meaning of 'force' (two horses have twice bigger force than one horse, and they may pull two carts, or one twice bigger cart). But common meaning relation of force to acceleration is limited to monotonicity (two-horse cart starts quicker than the same cart pulled by one horse).

Actually, the 'force' is the only of Newton's terms which survived to modern times: 'alteration' got replaced by Leibnizian 'derivative over time', and 'amount of movement' occured to be so important to earn its own name of 'momentum'.

I disagree - definitions are used in axioms. So if axioms are in turn used in definitions (3rd law is used explicitely in definitions of force), he comes to circularily-self-referencing statements or idem per idem explanations. You may recover from these problems, but the structure, as presented in Principia, is quite inferior to Euclid's clarity and logical order.