# F=ma -- How did Newton find this?

I did some googling but only found nothing to answer my question. How did Newton come up with this formula, literally/physically? Did he have chunks of stuff and moved them around, then logged various results. Then maybe he played around with various formulas and saw which fit the data?

Andrew Mason
Homework Helper
I did some googling but only found nothing to answer my question. How did Newton come up with this formula, literally/physically? Did he have chunks of stuff and moved them around, then logged various results. Then maybe he played around with various formulas and saw which fit the data?
It is difficult to know whether Newton developed his second law as an empirical law or whether he worked it out from existing principles. It can be shown to be a consequence of Galilean principles.

Newton observed, as did Galileo that the laws of motion are the same for bodies in all frames of reference of bodies undergoing no change in motion.

If acceleration was not proportional to force and inversely proportional to mass two very important principles would be violated: 1. the additive nature of forces and mass and 2. Galilean relativity.

See my posts on this thread:
https://www.physicsforums.com/threa...ntuitively-obvious.711442/page-3#post-4514684

"It is essentially a symmetry argument based on the first law and Galilean relativity (ie. absolute time and the principle that all inertial frames if reference (IFR) are equivalent: that the laws of motion are the same in all IFRs) and a certain concept of mass: that matter is made up of fundamental units of matter and that mass is simply a measure of the quantity or number of such units of matter.

A force F acting on a body m over a time interval δτ will cause a certain change in velocity of δv, as measured in the initial rest frame of m (IFR1). Moving now to the rest frame of m after the change in motion occurs (IFR2 ie. the IFR whose origin is moving at velocity δv with respect to IFR1) we measure the change in velocity of the same force F over a second identical time interval δτ. The equivalence of IFRs requires that there be the same change in velocity δv. If it were not the case, IFR1 and IFR2 would not be equivalent because the same force acting for the same time interval would effect different changes in motion. As measured in IFR1, the change in velocity is 2δv over time interval 2δτ. Repeating that you can see that Δv/Δt is constant. So a constant force causes constant acceleration.

Now, if you repeat the experiment by applying the same force to each of 2 identical bodies (each having the same quantity of matter) alternately for very brief intervals of δτ, the bodies accelerate at exactly half the rate in the first experiment. Because the force is only applied to them every other interval of δτ, it takes 2δτ to achieve a change in velocity of δv. So acceleration is inversely proportional to mass.

So that is pretty much the second law.

AM

• DaveE
Thanks but you've just restated the math and derivation based on math and theory. I'm asking about how, physically, he did the experiments. Did he place a chunk of metal on a table. Then did he move it with his hand, pull it with a spring, with a clock on the desk? Did he write down the amount of mass, the velocity, how hard he pushed it, etc. I'm talking nitty gritty.

Andrew Mason
Homework Helper
Thanks but you've just restated the math and derivation based on math and theory. I'm asking about how, physically, he did the experiments. Did he place a chunk of metal on a table. Then did he move it with his hand, pull it with a spring, with a clock on the desk? Did he write down the amount of mass, the velocity, how hard he pushed it, etc. I'm talking nitty gritty.
Galileo is known to have done experiments. Newton may have but little is known about them.

AM

berkeman
Mentor
I did some googling but only found nothing to answer my question.
It's hard to believe that some quality time with Google didn't help you find this. What search terms did you use?

• Dr. Courtney
russ_watters
Mentor
I agree with @berkeman; you should work on your google skills. Google tells me Newton didn't even discover the 2nd law, Galileo or his contemporaries did. Galileo did experiments, and some famous ones are often performed in high school and freshman physics courses, such as rolling balls down an inclined plane and dropping objects of different masses.
http://muse.tau.ac.il/museum/galileo/galileo_low_of_fall.html

My understanding is that Newton was mostly a mathematician and he added mathematical rigor to the experimental work done before him.

• pinball1970 and berkeman
Take a look at what Newton wrote:

https://en.wikisource.org/wiki/The_...l_Philosophy_(1846)/Axioms,_or_Laws_of_Motion

Mass is not mentioned anywhere in Law II (nor is acceleration for that matter). So, what you have attributed to Newton is a formulation of the law which developed over a long (depending on how you consider it) span of time.

Short answer to your question: a collaborative effort of induction, deduction, and careful testing led to what we call "Newton's" second law. I hazard to guess this is true for any type of axiomatic law you can name (as opposed to, say, empirical laws, which can more easily be attributed to individuals).

• berkeman
I googled just fine, but most of everything I found simply explained WHAT it was, not how he arrived at it, the nitty gritty. I knew of Galileo's experiments, but didn't know enough to draw the connections. I just assumed Newton did something similar but better, more refined. But the responses in this post were very helpful, much appreciated. And they ring true too, how history books are filled with "executive summaries" that make it seem like scientists are a bunch of lone wolves when in fact like any normal person they use whatever information they have available.

Andrew Mason
Homework Helper
Mass is not mentioned anywhere in Law II (nor is acceleration for that matter). So, what you have attributed to Newton is a formulation of the law which developed over a long (depending on how you consider it) span of time.
Mass is defined by Newton in the very first definition at the beginning. He says:
Isaac Newton said:
"DEFINITION I.
The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. ..... It is this quantity that I mean hereafter everywhere under the name of body or mass."
Newton's concept of matter was that all matter consists of finite indivisible units of matter and that the quantity of these finite indivisible units gave the body its mass, or quantity of inertia. Newton had long experimented with alchemy and this is likely Newton's own original idea.

In the next definition he defines "quantity of motion" as the product of mass and velocity or momentum:

Isaac Newton said:
"DEFINITION II. The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.
The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple."

In the second law, Newton spoke about the "alteration of motion", which by definition is the change in momentum (mass x velocity), being "ever proportional to the motive force impressed":
Isaac Newton said:
"LAW II. The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed."

Newton used the term "impressed force" and "action" as synonymous terms:
Isaac Newton said:
"DEFINITION IV. An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.
This force consists in the action only; and remains no longer in the body when the action is over.
Thus "action" or "impressed force" carries a magnitude, direction and duration. The concept of duration in the second law is alluded to by use of the word "ever". We can conclude that he is saying that the alteration of motion, ie. the change in mv, is in the direction of the motive force impressed and is proportional to the magnitude and duration of the impressed force (the action). Thus the continuing alteration of motion proportional to the motive force (F) impressed (i.e. for a duration Δt) can be stated: FΔt ∝ Δp which is equivalent to FΔ(mv)/Δt or (if m is constant): F ∝ mΔv/Δt = ma. So, with the right units, F = ma

AM

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• vela, tech99, brainpushups and 2 others
Really nice to see that concise explanation of translating Newton's phrasing into the common symbolic representation @Andrew Mason. I didn't mean for my response to come across as saying that Newton's explanation was somehow incomplete. My intention was just to point out that Newton didn't come up with the 'formula' (since this is how the initial inquiry was stated) and that the way it is stated now is because of the efforts of many.

Newton's concept of matter was that all matter consists of finite indivisible units of matter and that the quantity of these finite indivisible units gave the body its mass, or quantity of inertia. Newton had long experimented with alchemy and this is likely Newton's own original idea.

Somewhat off topic: I think that saying Newton's concept of matter as finite units as an original idea of his is giving him too much credit. Descartes had proposed a theory of matter akin to this and many scholars of this era were starting to take a more mechanical worldview than had been offered previously. It might be interesting to the OP to look up Descartes' laws of motion and to note the similarities to that of Newton. Much of the experimental work that occurred during the mid to late 17th century was related to testing (and in most cases refuting) what Descartes wrote. Much of Book II of Newton's Principia was dedicated to fluids. Newton wrote about it in the way he did to provide clear arguments against Descartes' vortex theory of gravitation.

For example, in the Scholium after Proposition LII. Theorem XL:

I have endeavoured in this Proposition to investigate the properties of vortices, that I might find whether the celestial phenomena can be explained by them; for the phenomenon is this, that the periodic times of the planets revolving about Jupiter are in the sesquiplicate ratio of their distances from Jupiter's centre; and the same rule obtains also among the planets that revolve about the sun. And these rules obtain also with the greatest accuracy, as far as has been yet discovered by astronomical observation. Therefore if those planets are carried round in vortices revolving about Jupiter and the sun, the vortices must revolve according to that law. But here we found the periodic times of the parts of the vortex to be in the duplicate ratio of the distances from the centre of motion; and this ratio cannot be diminished and reduced to the sesquiplicate, unless either the matter of the vortex be more fluid the farther it is from the centre, or the resistance arising from the want of lubricity in the parts of the fluid should, as the velocity with which the parts of the fluid are separated goes on increasing, be augmented with it in a greater ratio than that in which the velocity increases. But neither of these suppositions seem reasonable. The more gross and less fluid parts will tend to the circumference, unless they are heavy towards the centre. And though, for the sake of demonstration, I proposed, at the beginning of this Section, an Hypothesis that the resistance is proportional to the velocity, nevertheless, it is in truth probable that the resistance is in a less ratio than that of the velocity; which granted, the periodic times of the parts of the vortex will be in a greater than the duplicate ratio of the distances from its centre. If, as some think, the vortices move more swiftly near the centre, then slower to a certain limit, then again swifter near the circumference, certainty neither the sesquiplicate, nor any other certain and determinate ratio, can obtain in them. Let philosophers then see how that phenomenon of the sesquiplicate ratio can be accounted for by vortices.

Andrew Mason
Homework Helper
Really nice to see that concise explanation of translating Newton's phrasing into the common symbolic representation @Andrew Mason. I didn't mean for my response to come across as saying that Newton's explanation was somehow incomplete. My intention was just to point out that Newton didn't come up with the 'formula' (since this is how the initial inquiry was stated) and that the way it is stated now is because of the efforts of many.
Point well taken. Granted that Newton's expression of the law and explanation is not particularly clear. It may have lost something in the translation (from his native English to Latin back to English).
Somewhat off topic: I think that saying Newton's concept of matter as finite units as an original idea of his is giving him too much credit. Descartes had proposed a theory of matter akin to this and many scholars of this era were starting to take a more mechanical worldview than had been offered previously. It might be interesting to the OP to look up Descartes' laws of motion and to note the similarities to that of Newton.
I would not credit Newton with coming up with the concept of matter as consisting of finite indivisible units. The Greeks were the first to think of atoms and others before Newton had renewed the concept (although I am not sure about Descartes being one of them). Rather it his theory that the units were all the same and that it was the number of these units that was the measure of the body's inertia. He was remarkably prescient in this because if you think of the number of neutrons or proton + electron pairs as the units, he was pretty accurate (to within a relatively small margin of error) .

AM

• brainpushups
RPinPA
Homework Helper
In fact, Newton did not say F = ma. He said (in words, not equations), F = dp/dt.

"LAW II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed."
I took that translation from here, but I do have images of the Latin original on my hard disk.

If I recall correctly, the word "motion" here was motus in the Latin, and elsewhere Newton defines motus as the product of mass and velocity, or perhaps "inertia" and velocity. Actually I don't think he even defines it as a product, but just says (again going by memory) that it is proportional to both of those things. Principia is more verbal than mathematical so you sometimes have to do a little work to extract the equation.

Another way of describing Newton's three Laws is that of: The Law of Conservation of Momentum. In the absence of outside force the momentum of a system remains constant. This is linear momentum: the motion ascribes a line. Sometime the lines are in circles: as pendulum motion.

This means that one kilogram moving 5 m/sec can combine with 4 kilograms (at rest) and the new velocity will be 1 m/sec. This can be reversed; and 5 kilograms moving 1 m/sec can transfer all its motion to 1 kilograms that will then be moving 5 m/sec.

There was a reason why Leibniz and Newton were antagonists.

Klystron
Gold Member
Remember from biographies that Newton conducted experiments with optics. The hand illustrations of prisms and spectra in Optiks imply experiments by the author but whether Newton directly collected his data requires verification from primary sources.

This wiki source states Newton built and experimented with optical and electrostatic devices:
https://en.wikipedia.org/wiki/Isaac_Newton

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hilbert2
Gold Member
Even if the laws of nature were something different than what they are, it would always be possible to define a force ##F## by saying that ##F = ma## is what it means. The only requirement is that it must be possible to measure masses and accelerations. The validation of this kind of definition comes from the fact that the theory of mechanics built around this concept of force is found to be useful, simple and logical.

It is the most useful and simple and logical formula of all; but I would like to point out that it is also ignored.

The Dawn Mission satellite despin cable has 3 kilograms that stops the spinning of 1420 kilograms. The F in the cable is equal in both directions (third Law) therefore the momentum change of the 1420 kilograms equals the momentum change of the 3 kilograms. The final velocity of the 3 kilograms has to be around 400 m/sec to equal the spinning momentum of the 1420 kilograms spinning at about 1 m/sec.

This 400 m/sec appears to be correct with a violent throw of the weights. But NASA predicts it is only 20 m/sec. This 20 m/sec is a loss of 95% of the Newtonian momentum: so NASA predicts (without proof: no data is given) that F = ma is false.

The cylinder and spheres prove that Newton was correct; because all the motion can be returned from the spheres back to the cylinder. See delburtphend.com

Andrew Mason
Homework Helper
Even if the laws of nature were something different than what they are, it would always be possible to define a force ##F## by saying that ##F = ma## is what it means.
While you could define force in this way, the result would be that the laws of motion would depend on the inertial frame of reference one chooses.

So, a mass being pulled by a spring stretched a constant amount would not experience constant acceleration i.e. where there is no observable physical change in the source of whatever it is that is causing the change in motion of a body, the change in motion would not be constant with time.

Consequently, a body M of mass m in IFR1 at rest (an inertial frame which defines a rest state) subjected to such a spring would experience a change in velocity of δv1 in time δt and a different change of velocity of δv2 over a subsequent equal time interval δt.

Now we try the first part of the experiment in IFR2 travelling at δv1 relative to IFR1. We would do the first part of the identical experiment on an identical body M' of mass m commencing after the first δt interval.

Question: would we observe:

1. a change in the velocity of M' of δv1 over the time interval δt? or
2. a change in velocity of M' of δv2 over that time interval δt?

If the frames of reference IFR1 and IFR2 were equivalent, since the experiment in IFR2 is identical to the first part of the experiment in IFR1, we should see a change of velocity of M' in IFR2 of δv1.

But the problem is that IFR2 is identical to the frame of reference of the body M after having its velocity changed by δv1. And the change in velocity of M in the second interval δt is δv2, not δv1.

So, we must conclude that the inertial frames of M after δt and IFR1 are not equivalent, despite there being no distinguishing feature between them.

Therefore, the laws of physics would depend on the inertial frame of reference one chooses.

In other words, if all inertial frames measure time the same and if the laws of motion are the same in all inertial frames of reference, an unchanging agent acting effecting a change in motion of a body of a given mass will produce equal changes in velocity in equal times. And that is why F=ma is more than just an arbitrary definition.

AM

CWatters
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Gold Member
Delburt, you need to explain that bit about the Dawn sat if the mods will let you.

It doesn't matter how fast the weights unwind. All that matters is the change in moment of inertia. Where did you find the moment of inertia for the spacecraft with the weights still wound in? You need to know that do do the calculation.

I estimated that the distance to the center of rotational mass of the despun satellite is about a .5 meter.

The extended cable length is given to be 12 meters. So the radius difference is 24 to one. So if you are going to use angular momentum as your formula for finding the speed of the 3 kilogram weight you would have to divide the Newtonian momentum by 24. Angular momentum is L = r m v.

So The initial L is .5 m * 1420 kg * 1 m/sec = 710

The final L would have to be equal to the initial L therefore: 710 = 12 m * 3 kg * 19.72 m/sec

The initial Newtonian momentum is 1420 kg time one meter per second, 1420 units. The final Newtonian momentum (if the angular momentum formula is used) would be 3 kg time 19.72 m/sec = 59 .167 units.

If the angular momentum formula is used only 1/24 (59.167 / 1420) of the original Newtonian momentum remains.

But the spheres in the cylinder and spheres experiments can return all of the motion back to the cylinder (satellite).

Ballistic pendulum experiments prove that only Newtonian momentum is given from small masses to large masses. The three kilograms have to have all the Newtonian Momentum because that is the only momentum that can give all the motion back to the satellite.

The three kilograms must be moving closer to 400 m/sec if F = ma is correct. The Newtonian momentum change (mv) of the 3 kilogram weights must equal the Newtonian momentum change of the cylinder (satellite).

CWatters
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I estimated that the distance to the center of rotational mass of the despun satellite is about a .5 meter.

The extended cable length is given to be 12 meters. So the radius difference is 24 to one. So if you are going to use angular momentum as your formula for finding the speed of the 3 kilogram weight you would have to divide the Newtonian momentum by 24. Angular momentum is L = r m v.

So The initial L is .5 m * 1420 kg * 1 m/sec = 710

The final L would have to be equal to the initial L therefore: 710 = 12 m * 3 kg * 19.72 m/sec

Which seems close to the 20m/s you say NASA predict.

The initial Newtonian momentum is 1420 kg time one meter per second, 1420 units. The final Newtonian momentum (if the angular momentum formula is used) would be 3 kg time 19.72 m/sec = 59 .167 units.

Now you appear to be talking about linear momentum. That's not how you calculate the linear momentum of the spacecraft.

But the spheres in the cylinder and spheres experiments can return all of the motion back to the cylinder (satellite).

Never heard of the "cylinder and spheres experiment". Nor it seem has google. I'm not sure any of this is relevant to the OP so perhaps best to start another thread in the physics forum if you want to discuss something different.

Ballistic pendulum experiments prove that only Newtonian momentum is given from small masses to large masses.

If you are saying that small masses cannot transfer angular momentum to large ones that's totally incorrect.

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Try searching delburtphend despin

The opening question was: did Newton collect data and see which formula worked. I would say 'Yes' and there were three formulas available. There was one formula each from: Kepler, Leibniz, and Galileo. Newton chose to refine Galileo's

NASA used the angular momentum formulas: (Kepler) so that is what they predicted (20 m/sec). But note that they present no data to confirm their predictions.

You will never find an event that conserves angular momentum, in motion transfers, in the Lab; where data is collected. Angular momentum can not pass the simplest of tests.

Tie a soft ball on the end of a string. Swing the ball around on the end of the string in a large circle. You should be able to get the radius out to 3 meters. Then have the string come in contact with an immovable post; so that the radius of the swing is reduced to .25 meter or so. I am guessing that the speed of the ball should work at about 3 m/sec. And lets have the ball have a mass of .250 kilograms.

The original angular momentum would be; L = r m v = 3 meters * .250 kg * 3 m/sec = 2.25

We all know that when the string contacts the immovable post the speed of the ball around the arc of the circle remains the same. This is Newton's mv and this constant speed has been photographed many time.

So the final angular momentum is: L = .25 m * .250 kilograms * 3 m/sec = .1875

2.25 does not equal .1875: angular momentum always fails, in the lab, if you collect data.

Kepler formulated angular momentum conservation for space: planets; comets; and other satellites were gravity changes the velocity.

jbriggs444
Homework Helper
We all know that when the string contacts the immovable post the speed of the ball around the arc of the circle remains the same. This is Newton's mv and this constant speed has been photographed many time.

So the final angular momentum is: L = .25 m * .250 kilograms * 3 m/sec = .1875
You are mistaken about what conservation of angular momentum means. For a system free of external forces, angular momentum is conserved about any stationary (or inertially moving) reference point you choose. If you choose to change reference points, however, you can change angular momentum.

• CWatters
CWatters
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You will never find an event that conserves angular momentum, in motion transfers, in the Lab; where data is collected. Angular momentum can not pass the simplest of tests.

There are several experiments that demonstrate conservation of angular momentum. One involves two horizontal discs rotating on a vertical axle. The top one is free to move up and down the axle. The bottom one is set rotating and the top stationary disc is allowed to drop down onto it. The two disc then spin together. You can measure the rpm before an after and confirm conservation of angular momentum.

CWatters
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Gold Member
Tie a soft ball on the end of a string. Swing the ball around on the end of the string in a large circle. You should be able to get the radius out to 3 meters. Then have the string come in contact with an immovable post....

What jbriggs said.

Angular momentum is only conserved in a closed system. Your immovable post and the planet it is fixed to are outside your system of ball and rope so its not closed.

If you wanted to do that experiment properly you would have to take into account the change in angular momentum of the planet when the rope hits the post. You may think the post is immovable but its not. The planet is so massive that the change in angular momentum has an immeasurably small effect on the planets angular velocity but it does change and angular momentum is conserved.

Very good CWATTERS; if you keep the radius the same and only change the mass then the velocity will adjust to keep the angular momentum the same. I did not think of only changing the mass; the new challenge is to find one where the radius is changed and angular momentum remains the same. It is important to note that the Newtonian momentum remains the same as well in this experiment.

Each particle of mass times its arc motion gives you a total mv. That total mv will remain the same when the second disk is placed on the first disk.

The equation for the disk experiment would be this: L = r * m * v = r *2m * ½ v it is the same r so it can be removed from both side of the equation and you would get mv = 2m *1/2v. Which is a true statement.

CWatters
Homework Helper
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the new challenge is to find one where the radius is changed and angular momentum remains the same.

jbriggs444
Homework Helper
the new challenge is to find one where the radius is changed and angular momentum remains the same
When you speak about "changing the radius", you will have to be more specific. Are you:

1. Changing the reference axis (or point) about which angular momentum is judged.
2. Pulling in an orbiting object without changing the pivot (like an ice scater pulling in the arms).
3. Changing the pivot point around which near-circular motion is occurring (as when a ball on a string hits a vertical pole at the midpoint on the string)

1 will change angular momentum. But that does not count since you would be frame-jumping. Angular momentum is conserved but not invariant.
2 will not change angular momentum.
3 will not change angular momentum. Though it does invite you to frame-jump.

Dale
Mentor
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Since the OP has been answered and since the thread has taken a bad turn, it is now closed

• Bystander and jbriggs444