# Why doesn't Newton define F=m(a^2)

• AlonsoMcLaren

#### AlonsoMcLaren

Why did Newton define force as F=ma instead of stuff like F=m*(a^2) or F=(m^2)*a?

Newton did not simply "define" F=ma, he made observations and based his results on those observations.

Newton did not simply "define" F=ma, he made observations and based his results on those observations.

Then what is the definition of force?

Then what is the definition of force?

Force is defined by Newton's laws of motion.

Then what is the definition of force?

The resultant force is proportional to the rate at which the momentum of an object changes with respect to time. i.e.
$F = k \stackrel{ \underline {dp} }{dt}$
where k is a numerical constant equal to 1 in S.I. units and p is given by the product of the mass of the object and it's velocity.

Why did Newton define force as F=ma instead of stuff like F=m*(a^2) or F=(m^2)*a?
Nature defined it that way, not Newton.

We have an understanding of force apart from F=ma. Force is the thing that causes a body's motion to change. We understand that a stretched spring exerts a force, for example.

We observe that a constant force (eg. pulling with a spring of a certain length of stretch) will cause a mass m to experience constant acceleration a or mass 2m to have acceleration a/2 or a mass m/2 to accelerate at the rate of 2a.

If we add twice the force (pulling with 2 springs each with the same stretch as before) the mass m has acceleration 2a; 2m has acceleration a; etc.

So it appears from these observations that $F \propto ma$ (taking forces as being additive). We then just defined the units of force so that F = ma.

AM

Nature defined it that way, not Newton.

Nature doesn't define anything. In fact there were some confusions about force:

Force is the thing that causes a body's motion to change.

That's what Newton says, but according to Aristoteles force is the thing that causes a body's motion not to change. Because using the same word with different sense is not acceptable in science a clear definition was necessary. This definition is given by Newton's laws of motion.

I'm not sure if this works, would love comments:

1) From the first law and the principle of relativity, the initial position and momentum of an isolated object determine its future trajectory.

2) If we assume that we don't need more initial conditions for non-isolated objects, then the force law should not contain more than the second derivative of position.

3) The form F=ma is partly determined by the principle of Galilean relativity, however, it is not completely clear why it should not contain dx/dt. The remaining ambiguity is absorbed by saying we choose the LHS to make the RHS true. This choice makes the cosmic force of gravity velocity independent, but makes the earthly drag force velocity dependent.

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Nature doesn't define anything.
http://dictionary.reference.com/browse/define" [Broken]: to explain or identify the nature or essential qualities of;

Do humans define the speed of light and make it constant for all inertial observers? Do humans define Galilean relativity? Or does nature do this and leave humans to just work out the units? Perhaps it is just a different way of looking at it.

That's what Newton says, but according to Aristoteles force is the thing that causes a body's motion not to change. Because using the same word with different sense is not acceptable in science a clear definition was necessary. This definition is given by Newton's laws of motion.
The question was why Newton defined force as mass x acceleration (second law), not why he defined it as something that changes the motion of a body (first law).

Newton's first law implies that "force" is something that changes the motion of a body: in the absence of force, there is no change in a body's motion. So a body at rest in one inertial frame of reference moves at constant speed relative to another inertial frame of reference.

To derive the second law one need only accept:

#1: that two forces are equal if they effect equal changes in the motion of a body in equal times; and

#2: all inertial frames of reference are equivalent (Galilean relativity)

Suppose you have a body, A, at rest and you subject it to force F for a time t giving it a speed v. The change in motion is v-0 = v. Now it is at rest relative to a frame of reference moving at velocity v relative to the initial frame of reference. I then consider the new frame of reference as the "rest" frame and I repeat the same experiment: i.e. I apply the same force F again for a time t. Again the change in motion (principle #1) must be the same, so the velocity again changes by v, but this time it is relative to the new inertial reference frame.

So by applying the force F for 2t I get a change in motion of 2v. I can repeat this experiment over and over and we can see that Δv/Δt = constant if the force is constant and mass is constant (ie. same body).

Then I add another identical body to the first (ie. I double the mass) and I apply the same force F to each body simultaneously, so I have a force of 2F. This is identical to the first experiment except that I have two bodies and two forces. So the change in motion in time t has to be the same as the first: v. Pretty soon we realize that force varies as mass and as Δv/Δt. And that is pretty much the second law.

Now was that the handiwork of nature or of Newton?

AM

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http://dictionary.reference.com/browse/define": to explain or identify the nature or essential qualities of;

Do humans define the speed of light and make it constant for all inertial observers? Do humans define Galilean relativity? Or does nature do this and leave humans to just work out the units? Perhaps it is just a different way of looking at it.

The question was why Newton defined force as mass x acceleration (second law), not why he defined it as something that changes the motion of a body (first law).

Newton's first law implies that "force" is something that changes the motion of a body: in the absence of force, there is no change in a body's motion. So a body at rest in one inertial frame of reference moves at constant speed relative to another inertial frame of reference.

To derive the second law one need only accept:

#1: that two forces are equal if they effect equal changes in the motion of a body in equal times; and

#2: all inertial frames of reference are equivalent (Galilean relativity)

Suppose you have a body, A, at rest and you subject it to force F for a time t giving it a speed v. The change in motion is v-0 = v. Now it is at rest relative to a frame of reference moving at velocity v relative to the initial frame of reference. I then consider the new frame of reference as the "rest" frame and I repeat the same experiment: i.e. I apply the same force F again for a time t. Again the change in motion (principle #1) must be the same, so the velocity again changes by v, but this time it is relative to the new inertial reference frame.

So by applying the force F for 2t I get a change in motion of 2v. I can repeat this experiment over and over and we can see that Δv/Δt = constant if the force is constant and mass is constant (ie. same body).

Then I add another identical body to the first (ie. I double the mass) and I apply the same force F to each body simultaneously, so I have a force of 2F. This is identical to the first experiment except that I have two bodies and two forces. So the change in motion in time t has to be the same as the first: v. Pretty soon we realize that force varies as mass and as Δv/Δt. And that is pretty much the second law.

Now was that the handiwork of nature or of Newton?

AM

But how do you define the mass, ie. how do you know you've doubled the mass?

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Why did Newton define force as F=ma instead of stuff like F=m*(a^2) or F=(m^2)*a?
Newton did not define force as F=ma. That is an after-the-fact refinement of Newton's laws that took about 200 years to develop.

To understand Newton, a good place to start is with his definitions. Otherwise his laws of motion (which are not definitions) are a bit incomprehensible. So, his definitions in English; those who want to use the original Latin are free to do so. My sources are http://www.Newtonproject.sussex.ac.uk/view/texts/normalized/NATP00075, the original Latin, various translations of the Principia Mathematica that are freely available on the net, and google translator, which now understands Latin.

Definition 1: Quantity of Matter
The measure of the quantity of matter is the product of density and volume. It is this quantity that is referenced by the term 'mass'. This quantity has been found to be proportional to weight by experiments with accurate pendulums, as shall be shown later.

Commentary: Newton leaves 'density' and 'volume' as undefined terms with this definition. However, he does provide a means of measuring mass without knowing either density or volume.

Definition 2: Quantity of Motion
The measure of the quantity of motion is the product of quantity of matter and velocity. The motion of the whole is the sum of the motion of all the parts.

Commentary: Here Newton is defining what 'quantity of motion' as what we now call momentum. Note very well: When Newton uses the term 'motion' he means 'quantity of motion', or momentum, not velocity.

Definition 3: Innate Force of Matter (Vis Insita)
The innate force of matter is a characteristic of all matter that makes a body continue in its state of rest, or continue moving uniformly in a straight line.

Commentary: Nowadays we do not think of this vis insita as a force. It is just momentum. It is very important to put Newton's writings in the right temporal context. Even though Galileo and Kepler preceded Newton by 70 years or so, Newton was still very much fighting to overthrow Aristotelian physics. This concept hearkens back to Aristotle, who thought that some external force was needed to keep a body in motion. Here Newton is turning the table on Aristotelian physics, saying that whatever it is that keeps a body in uniform motion is an intrinsic characteristic of matter rather than something applied externally.

Definition 4: Impressed Force (Vis Impressa)
An impressed force is an action on a body that acts to change the body's state of rest or moving uniformly in a right line.

Commentary: This is an intentionally vague definition. It does not say what an impressed force is, nor does it say what an impressed force does other than to somehow act to change a body's state. Newton's explanatory text explicitly says that impressed forces can be of different origins.

Newton proceeds to give four more definitions with a much larger amount of explanatory text, but these first four should suffice for understanding Newton's laws of motion.

Law 1
Every body perseveres in its state of rest or of moving uniformly in a right line, unless it is compelled to change that state by impressed forces acting on the body.

Commentary: Members of PhysicsForums regularly ask why this law even exists. First off, it is once again important to place this law in its correct temporal context. Newton felt compelled to once and for all overthrow Aristotelian physics. This law says in no uncertain terms that Aristotelian physics is wrong. It also establishes a null effect, a concept that is still important in medical science. Finally (but this is historical revisionism) it implicitly defines the concept of an inertial frame of reference. This law is only valid in an inertial frame; in a non-inertial frame a body can undergo an apparent change in state when no external forces act on the body.

Law 2
The alteration of motion is proportional to the motive force impressed, and is made in a straight line which that force is impressed.

Commentary: An algebraic interpretation of this law is $\Delta p \propto F$. This is not quite historical revisionism. Had I used expressed momentum and force as vectors it would be historical revisionism. Had I used $F\propto ma$ instead of $\Delta p \propto F$ it would also be a case of historical revisionism. The impressed force in Newton's second law has units of momentum, not force. That this is the case becomes even clearer on reading the scholium.

Response to the original question
This let's us answer one of the questions raised by the OP: "Why did Newton define force as F=ma": The answer is that he never did. F=ma, or better stated, $\vec F = m\vec a$ is a 200-year after-the-fact expression of Newton's second law. Between Newton's time and that modern version
• Newtonian physics was rewritten from the ground up multiple times;
• Newton's calculus was pretty much discarded, with Leibniz' formulation and notation winning the day;
• Leibniz calculus was rewritten from the ground up;
• A consistent set of units was developed, enabling the elimination of the constant of proportionality;
• The clumsy and very verbose geometric representations employed by Newton were discarded in favor of a more compact algebraic notation; and
• Those still clumsy and verbose algebraic notations were discarded in favor of our modern vector notation.

As to the second part of the OP's question, "instead of stuff like F=m*(a^2) or F=(m^2)*a?" The answer is that Newton's second law is not, as others have said, a definition of force. Newton already had a somewhat vague definition of force, and he ways to measure forces independent of acceleration.

Newton attributed his first two laws to Galileo. Galileo's experiments showed that force is proportional to (modern view) the product of mass and acceleration. Newton did even more experiments in this regard. Newton's second law is a law of physics. That F=ma (modern version) is not a definition of force. It is an experimentally-verified scientific law.

We don't need Newton! Who needs the idea of a force? The Lagrangian can provide you with everything you need to know about anything. If we knew the Lagrange density of the universe, we could determine all the symmetry principles (or vis versa) and know how all the dynamics worked, without ever using the idea of a force. To bad the standard model is only a small part of the real Lagrangian of the universe...

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But how do you define the mass, ie. how do you know you've doubled the mass?
If I physically add another identical body, I have doubled the quantity of matter. If I weigh the two together, they have double the weight of one alone. So I pick a standard body (eg. a gram). I compare the quantity of matter in an object to the quantity of matter in the standard body (using, say, a scale) and I do an experiment to show whether they behave the same way in response to the same force. I see that they do. So I call this property mass.

It appears from experiment that masses are additive. This, of course, is only true within the limits of my measurements. The difference in mass between two identical bodies separated by some distance and the same two bodies close together appears to be 0. So I conclude that masses are additive.

AM

Who needs the idea of a force?
Engineers.

I challenge you to come up with a Lagrangian or Hamiltonian formulation for a rotating turbine blade that is just on the verge of cavitating, for modeling the lift and drag on a supersonic jet, or for modeling the behavior of a space vehicle with rockets that have thrust (thruster on/off, and when on, magnitude and direction) controlled by some automated control system.

Engineers.

I challenge you to come up with a Lagrangian or Hamiltonian formulation for a rotating turbine blade that is just on the verge of cavitating, for modeling the lift and drag on a supersonic jet, or for modeling the behavior of a space vehicle with rockets that have thrust (thruster on/off, and when on, magnitude and direction) controlled by some automated control system.
Yeah. I don't think I would want an office on the 10th floor of a building designed by an engineer who did all the calculations using the Hamiltonian formulation of mechanics.

AM

What a really mean to say is that the idea of a force doesn't really provide any new physical insight. All the physics is contained in the Lagrangian. Of course, from a practical point of view, engineers need to know about forces. But this is a physics forum, not an engineering forum. From a physicist point of view, forces don't add much to our physical understanding of the world.

First thought:
I've read many threads of the form: "Why F=ma anf not F=m/a or V=IR and not R=IV?"
It amazes me how frequently people forget Physics is about experiments and not mere definitions.

Second thought

What a really mean to say is that the idea of a force doesn't really provide any new physical insight. All the physics is contained in the Lagrangian. Of course, from a practical point of view, engineers need to know about forces. But this is a physics forum, not an engineering forum. From a physicist point of view, forces don't add much to our physical understanding of the world.

Are we going to throw away Newton?

What a really mean to say is that the idea of a force doesn't really provide any new physical insight. All the physics is contained in the Lagrangian. Of course, from a practical point of view, engineers need to know about forces. But this is a physics forum, not an engineering forum. From a physicist point of view, forces don't add much to our physical understanding of the world.
Baloney. There are places where you cannot construct a Lagrangian. The real world is messy, and physicists are called upon to help analyze problems in the real world. A physicist who is asked to consult with NASA had better be able to deal with the world of forces and torques because that is the world in which NASA works for the most part. A physicist who is asked to help formulate a high-precision model of the solar system had better be able to deal with forces (with general relativity modeled as perturbative forces) because a full-blown GR solution to the N-body problem is downright intractable.

This cartoon says it all:

A dirty little secret of physics education:
• The problems presented in high school physics classes are specially constructed so that high school students can solve the problems with the limited math skills available to them.
In other words, they are given problems can be solved in a relatively short amount of time using simple algebra.

• The problems presented in introductory college physics classes are specially constructed so that freshmen and sophomores can solve the problems with the limited math skills available to them.
In other words, they are given problems can be solved in a relatively short amount of time using simple freshman level calculus.

• The problems presented in upper level undergraduate college physics classes are specially constructed so that juniors and seniors can solve the problems with the limited math skills available to them.
In other words, they are given problems can be solved in a relatively short amount of time using simple calculus of variations techniques, simple Green's functions, simple orthogonal functions.

• The problems presented in lower level graduate college physics classes are specially constructed so that first year grad students can solve the problems with the limited math skills available to them.
In other words, they are given problems can be solved in a relatively short amount of time using all of the techniques they have learned from high school on.

Once PhD candidates start working on their theses or go out into the real world they find just how simple those homework problems really were. A good physicist needs to be very adept and be able to use every technique they have learned along their educational journey. Sometimes that means going all the way back to the language of forces and torques.

I think you misunderstand me. I don't think the idea of a force is unimportant, and I know that some problems are intractable without the use of forces. But I have to stick to my guns, ALL of the physics is encoded with the Lagrangian (besides specification of the initial state). Whether you can solve the particular problem or not using the Lagrangian formalism or Newtonian mechanics is a different question, and that is not what I am arguing against. And just because you can't construct a Lagrangian, doesn't mean it does not exist! I am not arguing about solvability of complex problems, but more saying that the Lagrangian (or equivalently the group of symmetries governing the dynamics) is the most fundamental object, in which in principal, all the physics (equations of motion) is derived from, regardless whether there is a much better, faster, more tractable or more convenient way to arrive at a particular solution. And I do find your tone, DH, a bit condescending. Although you did not explicitly call me a bad physicist, you did imply it with your cartoon and your reference about PhD candidates, as if I am some amateur.

http://dictionary.reference.com/browse/define" [Broken]: to explain or identify the nature or essential qualities of;

Do humans define the speed of light and make it constant for all inertial observers? Do humans define Galilean relativity? Or does nature do this and leave humans to just work out the units? Perhaps it is just a different way of looking at it.

I've often heard that all our theories are human inventions. They do not describe Nature, since all of them are provably wrong.

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I've often heard that all our theories are human inventions. They do not describe Nature, since all of them are provably wrong.
That is incorrect. Quantum mechanics, for example, says the photon has zero intrinsic mass. In every experimental attempt to measure the mass of the photon, the measured value has been extremely small and has been consistent with no intrinsic mass. All that improving experimental techniques has done is to make the measured range of the photon's rest mass ever smaller -- and still consistent with a null intrinsic mass.

You are perhaps confusing falsifiability with falsification. Any scientific theory should be falsifiable. A viable scientific theory must not have been falsified, at least not within its domain of applicability. Were that to happen, the experiment would be examined critically (e.g., OPERA results). If the results held, there are only three possible outcomes: Toss the theory out (example: caloric theory), reduce the domain of applicability of the theory (example: Newtonian mechanics), or modify the theory so that it accommodates the new results (example: neutrino oscillations and the standard model).

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That is incorrect. Quantum mechanics, for example, says the photon has zero intrinsic mass. In every experimental attempt to measure the mass of the photon, the measured value has been extremely small and has been consistent with no intrinsic mass. All that improving experimental techniques has done is to make the measured range of the photon's rest mass ever smaller -- and still consistent with a null intrinsic mass.

You are perhaps confusing falsifiability with falsification. Any scientific theory should be falsifiable. A viable scientific theory must not have been falsified, at least not within its domain of applicability. Were that to happen, the experiment would be examined critically (e.g., OPERA results). If the results held, there are only three possible outcomes: Toss the theory out (example: caloric theory), reduce the domain of applicability of the theory (example: Newtonian mechanics), or modify the theory so that it accommodates the new results (example: neutrino oscillations and the standard model).

But the standard model is internally inconsistent (unless it is asymptotically safe). So it cannot be correct.

Newton already had a somewhat vague definition of force, and he ways to measure forces independent of acceleration.

Newton attributed his first two laws to Galileo. Galileo's experiments showed that force is proportional to (modern view) the product of mass and acceleration. Newton did even more experiments in this regard.

Sorry if this is considered necroposting! :uhh:

I just wanted to see if you could direct me to resources that show how Newton measured force and of Galileo and Newton's experiments that showed force being proportional to the product of mass and acceleration.

Sorry if this is considered necroposting! :uhh:

I just wanted to see if you could direct me to resources that show how Newton measured force and of Galileo and Newton's experiments that showed force being proportional to the product of mass and acceleration.

When two falling unequal masses accelerate at the same rate, in the absense of air resistance, this is because the greater mass experiences a greater gravitational force being resisted by a greater inertia, and the smaller mass experiences a smaller gravitational force being resisted by a smaller inertia.
Their common and constant acceleration a = the ratio F1/m1 = the ratio F2/m2.
This is experimental confirmation that a=F/m, which is the same thing as F=ma.

When two falling unequal masses accelerate at the same rate, in the absense of air resistance, this is because the greater mass experiences a greater gravitational force being resisted by a greater inertia, and the smaller mass experiences a smaller gravitational force being resisted by a smaller inertia.
Their common and constant acceleration a = the ratio F1/m1 = the ratio F2/m2.
This is experimental confirmation that a=F/m, which is the same thing as F=ma.

How did Galileo measure the forces? Did he equate the weight of the masses with the gravitational force?

For some reason I always assumed that, in Galileo's time, weight wasn't associated with the gravitational force; I just thought they had a very "primitive" understanding of what weight was, only seeing it as the "heaviness" of an object due to its size.

If the claim was that "falling unequal masses accelerate at the same rate", you don't need to measure the force to prove it, you just need to simultaneously drop several objects of different weight/mass and see if they hit the ground at the same time. Which he is believed to have done, though the specifics of the experiments are in doubt: http://en.wikipedia.org/wiki/Galileo_Galilei#Falling_bodies
http://en.wikipedia.org/wiki/Inertia#Classical_inertia

How far he took the logic I'm not sure, but if we know acceleration due to gravity is constant, then a=g=f/m

If the claim was that "falling unequal masses accelerate at the same rate", you don't need to measure the force to prove it, you just need to simultaneously drop several objects of different weight/mass and see if they hit the ground at the same time. Which he is believed to have done, though the specifics of the experiments are in doubt: http://en.wikipedia.org/wiki/Galileo_Galilei#Falling_bodies
http://en.wikipedia.org/wiki/Inertia#Classical_inertia

How far he took the logic I'm not sure, but if we know acceleration due to gravity is constant, then a=g=f/m

But in order to make the proportion $\frac{F_1}{m_1}\textit{=}\frac{F_2}{m_2}$, as mikelepore stated, wouldn't force need to be quantified in some way? If you don't need to measure the forces, how did Galileo know to attribute the accelerations to the forces?

I'm not sure about Galileo's time, but after Newton people were able to say:

a1=a2 empirically from Galileo dropping things.
tentatively suppose a=F/m
F1/m1 = F2/m2
substitute the law of universal gravitation
(G m1 m_earth/r^2) / m1 = (G m2 m_earth/r^2) / m2
cancel out values
arrive at the true statement 1=1,
therefore the supposition that a=F/m must have been true.
This doesn't involve measuring the weights experimentally.

If you don't need to measure the forces, how did Galileo know to attribute the accelerations to the forces?
He figured out that friction was tripping-up Aristotle. Once you eliminate friction, it is easy to see that force causes acceleration and lack of force means no acceleration.

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But in order to make the proportion $\frac{F_1}{m_1}\textit{=}\frac{F_2}{m_2}$, as mikelepore stated, wouldn't force need to be quantified in some way? If you don't need to measure the forces, how did Galileo know to attribute the accelerations to the forces?
Galileo attributed changes in motion to forces in a general way. But, at least in my understanding, it was not Galileo but Newton who attributed gravitational accelerations to gravitational forces. Galileo determined that in the absence of friction or resistance, all objects fall at the same rate. He determined that the relationship between time, t, of fall and height, h, of fall of an object was $h = at^2/2$. But Galileo did not conclude that they fall at the same rate due to gravitational force being proportional to mass.

AM

Galileo attributed changes in motion to forces in a general way. But, at least in my understanding, it was not Galileo but Newton who attributed gravitational accelerations to gravitational forces. Galileo determined that in the absence of friction or resistance, all objects fall at the same rate. He determined that the relationship between time, t, of fall and height, h, of fall of an object was $h = at^2/2$. But Galileo did not conclude that they fall at the same rate due to gravitational force being proportional to mass.

AM

Ahh so it was Newton who made the connection. But I still don't see how he'd figured a way to have a relative scale of forces. Did Newton use the weights of the bodies or something like spring scales to do this?

Also, based on DH's post Newton seemed to have related the change in momentum to the force, without reference to it's rate change with respect to time. If this is the case, how did we come to interpret it as $\textit{F}\propto{\frac{Δp}{Δt}}$?

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Also, was weight seen as the force due to gravity by Galileo's time? I've read that there was confusion among physicists at the time about the nature of weight; I wasn't sure if it was seen as synonymous with the gravitational force though.

Why did Newton define force as F=ma instead of stuff like F=m*(a^2) or F=(m^2)*a?

Newton needed to name the quantity ma because it entered into the physical description of things. Since ma corresponds to our intuitive notion of force, he named it force.