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We know that Force equals Mass times Acceleration, $F = ma$. But what if we had a law like $$F=m^2a$$ or $$F = 2ma~?$$

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    $\begingroup$ As we all know we live in a classical world where everything works on the basis on this law Ever heard of quantum mechanics? $\endgroup$ – Gert Apr 24 at 14:13
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    $\begingroup$ Related, if not dupe, physics.stackexchange.com/q/104101/25301 $\endgroup$ – Kyle Kanos Apr 24 at 14:15
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    $\begingroup$ The proposed duplicate asks why there's no numerical constant out front (see also), but the question of why the constant of proportionality is "mass" rather than "mass squared" (or the like) is an interesting twist. $\endgroup$ – rob Apr 24 at 14:22
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    $\begingroup$ Ultimately, this depends on what your particular definition of mass is. If by "mass" you mean "inertial mass," then the question is trivial - inertial mass is defined to be the proportionality constant between force and acceleration. So which definition of mass are we intended to use? $\endgroup$ – probably_someone Apr 24 at 14:28
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    $\begingroup$ Without a clear definition of the terms mass and force the question is too broad for a reasonably confined answer. $\endgroup$ – GiorgioP Apr 24 at 15:53
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It wouldn't change anything. We would just have defined mass differently. What is relevant about Newton's 2nd law is not $m$ but rather the fact that $F$ and $a$ are proportional, $F\propto a$.

The proportionality constant can then be called $m$ or $2m$ or $m^2$. That doesn't matter for the relationship that this law describes.

Newton's 2nd law defines mass (inertial mass) as the proportionality constant in $F=ma$. Had we decided upon $F=2ma$, then we would just have defined mass as half of the proportionality constant. It would change our language regarding mass (you would only weigh 40 kg rather than 80 kg i.e.), but it wouldn't change the way the world works.

If you on the other hand had asked about $F=ma^2$ or so, then you are messing with the world entirely with disastrous consequences that we can only try to guess as a result!

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    $\begingroup$ It would change something though as we wouldn’t have eg universal acceleration. Galileo would not have found the balls hit the ground at the same time when dropped from the tower ;) $\endgroup$ – innisfree Apr 24 at 14:42
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    $\begingroup$ I disagree that it would be so simple. It would also have the consequence that when putting two objects together, we would have to add their masses squared instead of their masses, which seems pretty strange. Clearly what we now call mass is a better option than the others. $\endgroup$ – Javier Apr 24 at 14:52
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    $\begingroup$ We’d also have to use some different units or fix the units somehow, eg introduce another dimensionful constant of nature or maybe just change the units of Newton’s constant $G$ $\endgroup$ – innisfree Apr 24 at 14:53
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    $\begingroup$ It would be nice if this answer (which is the accepted one) made mention of gravitational mass (and other non-inertial properties we expect mass to have, like additivity) and how they change. $\endgroup$ – jacob1729 Apr 24 at 15:23
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    $\begingroup$ If it was $m^2$ then it would definitely change something if the $m$ also refers to the mass associated with the force of gravity $\endgroup$ – Aaron Stevens Apr 24 at 17:59
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In contrast to the other answers, I am going to go a different direction. This is not a question of mathematics, where we can just say "oh yeah just redefine your variables." This is a physics question.

Going further, The OP is assuming $m$ stands for the mass of an object, not some arbitrary proportionality constant that ends up being the mass. Therefore, having a law like $F=km^2a$ ($k$ as just some constant for unit's sake) would not leave the universe unchanged. This is saying, for example, that if we double the "amount of stuff" associated with an object, that we would need $4$ times as much force to achieve the acceleration. This is certainly not true in our universe today, so we cannot say this law would be found in a similar universe.

There is also the question of the relation between inertial mass and gravitational mass, which we take to be equal. i.e. we know the force of gravity to be proportional to the mass of the object that gravity is acting on. Therefore, if forces behaved like $F=km^2a$, then we would find that, near the surface of the earth, that the acceleration due to gravity would actually depend on the mass of the object! Certainly this is not the same as our universe.

I understand the other answers' attempts to try and teach a lesson about definitions in physics, and how there are some things that are around purely by convention. But this is not one of those cases I believe. The OP is starting from $m$ means mass. So we should start there too and then see what that means for the proposed new "force laws".

This would be like me saying that it would be fine to take what we understand kinetic energy to be and rewrite it as $mv'$. Sure, I could say that I have "redefined" the velocity so that $v=\sqrt{2v'}$. And this is valid mathematically if we stay consistent with this in the rest of our equations. But if I start off by saying "let $v$ be the velocity of the object, i.e. the rate of change of displacement", then I cannot "redefine" my equations so that kinetic energy is $mv$.

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  • $\begingroup$ How would $g$ depend on $m$ here (for $F=km^2a$)? $\endgroup$ – KV18 Apr 24 at 18:41
  • $\begingroup$ @KV18 If you assumed that near the surface of the earth that the force of gravity is $mg$, then you just set up the equation: $km^2a=mg$ and solve: $a=g/km$ i.e. a more massive object would experience a smaller acceleration, which would make sense for this model of newtons second law $\endgroup$ – Aaron Stevens Apr 24 at 18:56
  • $\begingroup$ But will not the force of gravity itself be $F=km^2g$? $\endgroup$ – KV18 Apr 24 at 19:04
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    $\begingroup$ @KV18 Maybe, maybe not. Under relativity, gravity is a 'fictitious' force. GR gives us the equivalence principle, relating gravity (locally) to acceleration. In the constructed universe, that doesn't need to hold; gravity might still be proportional to m, not $m^2$. To keep it the same, other parts of physics need to change too. $\endgroup$ – jaxad0127 Apr 25 at 1:22
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    $\begingroup$ @KV18 What you are proposing is then what the other answers suggest. That we just do a mathematical operation and denote mass as $km^2$. As it stands, $F=ma$ doesn't define the gravitational force. It is a statement that is true for all forces independent of what $F$ actually is. Therefore, in this hypothetical scenario we wouldn't change what the force of gravity actually is, only how it relates to the acceleration it causes. $\endgroup$ – Aaron Stevens Apr 25 at 2:17
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One of the things that allowed us to invent and understand Newton's laws, which govern the motion of classical objects, is that they have the mathematical property of linearity. Linearity is a class of mathematical behaviors that people tend to be good at. If there are two or more forces acting on an object, the net force is the simple vector sum of the forces. If there is a system made of several masses, the total mass is the simple arithmetic sum of the individual masses.

For example, an Aristotelian misconception that was not properly refuted until the Renaissance is that "heavier objects fall faster." An argument against this, which I believe was due to Galileo, is a thought experiment, as follows. Imagine three identical cannonballs dropped from the same height: they should fall at the same rate because they all have the same mass. But tie two of them together with a low-mass connector, like glue or silk, and you have two objects whose masses are different by a factor of two. Can connecting two heavy masses with a silk thread change their fall time by a factor of two? It's a clever thought experiment.

In your formulation, two identical masses $m$ accelerating in tandem with the same acceleration $a$ would have the same amount of force,

$$ F = m^2 a, $$

acting on them. But if you tied them together so that you had a single object with mass $2m$ being acted upon by a force $2F$, then you would have

$$ 2F = (2m)^2 a $$

which predicts a different, smaller acceleration. Either you'd have to come up with an additional non-linear rule for adding forces, or you'd be describing a world that is different from the one where we live.

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"Force" is just a word. You don't need it at all to do Newtonian mechanics, if you start from conservation of energy and momentum and the idea of gravitational potential energy. "Force" is then just a mathematical invention to keep track of how the momentum is transferred between different bits of stuff, when things move around.

The world doesn't change just because humans invent random new words.

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  • $\begingroup$ I do not understand the down votes to this answer. They only make clear that who gave them does not know anything about the point of view leaded by Kirchhoff, Mach, Hertz in the nineteenth century and nowadays present, anonimously, in several Mechanics textbooks (basically all of them which write that F=ma is a definition of F). $\endgroup$ – GiorgioP Apr 24 at 15:35
  • $\begingroup$ @GiorgioP not only this one. It's strange that three answers had +2/-2 when I opened this page, and others 0/0. $\endgroup$ – Ruslan Apr 24 at 17:24
  • $\begingroup$ @GiorgioP Down votes aren't only for answers that are incorrect or that people do not agree with $\endgroup$ – Aaron Stevens Apr 24 at 18:07
  • $\begingroup$ -1: Force is not a random new word. It is an important concept and an aspect of reality that nature forces us to discover. $\endgroup$ – Dvij Mankad Apr 27 at 22:45
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Then we would recall mass as $m' = m^2$ or $m' = 2m $. The whole idea of Newton's second law is that force and acceleration are proportional. We can then define inertial mass as the constant factor between them. So the world would look exactly the same.

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Writing $F=kma$ and putting $k=1$ or $k=2,3...$ will not change the consequences of how things would actually behave.

All it does is put up a scaling factor to the value. But due to convenience, we agree on $k=1$ implying $ F=ma$.

Writing $a \to 2a$ instead does not change anything but the convention you have to abide by, for it is the same argument as writing $kma=F$.

If $k=m$ as you have included ($F=m^2a$), things would be the same if you assume $m^{'}=m^2$ and assume that $m^{'}$ is the actual mass.

Besides, $m^{'}$ is meant to be a constant of proportionality, and so it makes no difference to the consequences of physics. $F=ma$ is based on the understanding that $F$ is proportional to $a$.

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    $\begingroup$ Writing $k = m$ in $F = kma$ would change consequences of how things behave, at least in our current system, no? $\endgroup$ – JMac Apr 24 at 14:35
  • $\begingroup$ No. If $a$ becomes $2a$, then we are basically using a "different" number system. I meant that the physics does not change, only a modification to the math - like using a shifted coordinate system where $X=x+2$ - kind of like that. $\endgroup$ – KV18 Apr 24 at 14:41
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    $\begingroup$ $k = m$ doesn't make $a$ become $2a$ though, it makes $F = m^2 a$. $\endgroup$ – JMac Apr 24 at 14:43
  • $\begingroup$ I only meant $k$ for integers. $\endgroup$ – KV18 Apr 24 at 14:45
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    $\begingroup$ The question uses a non-integer $k$ as an example. $\endgroup$ – JMac Apr 24 at 14:46
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I have not read all the answers but most of them are conceptually wrong. Newton's second law does not state that force is proportional to acceleration. Mass is not just a proportionality constant.

Newton's second law of motion states that

The rate of change of motion (momentum) is directly proportional to the motive force and the direction of the change is parallel to the force.

Hence instead of $F\propto a$ we have $\vec{F}\propto \frac{d\vec{p}}{dt}$. Changing the proportionality into equality $$\vec{F} = k\frac{d(m\vec{v})}{dt}$$ Solving through $$\vec{F} = km\frac{d\vec{v}}{dt} + k\vec{v}\frac{dm}{dt}$$ (This equation is actually wrong, but that is not the point here). In most cases mass of system is constant and hence the second term becomes zero, but the most general form of second law is as given above.

Now returning to the question. What if $k$ is something other than $1$. That will not change the physics. The only difference will be that accelerating objects will be harder (if $k>1$).

The other question is invalid because $m²$ is not possible since $\vec{p}=m\vec{v}$. One can again ask why is momentum not $m²\vec{v}$? You can consider momentum as $m²\vec{v}$ just don't call $m$ as the mass, since momentum is defined that way.

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No matter the transformations you make into Newton's second law, as long as it's reverse transformation admits $\rm a$ to be isolated.

$F(r,v,t)$ in Newton's second law as expressed by equation $$ F(r,v,t)=ma \tag{1}$$

Is a very abstract concept that must be adapted to solve the observed motions in the universe. More importantly, $F(r,v,t)$ is an abstraction of some kind of position, time, and velocity function that satisfy the experimental observations. Since equation (1) can be written

$$F'(r,v,t)=a \tag{2}$$

and $F'(r,v,t)$ has no form a priori, any transformation in (1) that can be reversed into (2), will give the same results as Newton's second law. However, What one would call force is not necessarily what we currently call it but they should be easily converted into one another.

For example, lets analyze the definition of gravitational force in the advent that Newton's second law were

$$F(x,v,t)=2ma^2 \tag{3}$$

In this particular case, the definition of gravitational force $F_G$ here must satisfy (at least) the following observed facts:

  1. (By Galilee): “The acceleration of a falling body doesn’t depend on its mass”;
  2. (By Kepler): “The motion of planets around the sun are ellipses…”

The simplest definition of $F_G$ that satisfies both conditions is

$$F_G=\frac{4m^2}{r^4} \tag{4}$$

However, with time, people would realize that $\sqrt{F_G}$ was actually a better physical quantity, since more intuitive generalization of it, based on geometrical insights like Gauss law would arise. The same goes for the use of $m$ instead of $2m$.

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