# If $\mathbf{F}_{net} = m\mathbf{a}$ then how is $m\mathbf{a}$ not a force? [duplicate]

The question is in the title: If Newton's second law says that the sum of the forces acting on a body in a given direction is the same as the mass of the object times its acceleration in that direction, then how is $$m\mathbf{a}$$ not a force? Every book I have read on physics (all basic) says that $$m\mathbf{a}$$ is not a force. Are forces not "closed" under addition? Is this somehow a loose version of equality?  "University Physics" by Young and Freedman says...

Acceleration is a result of a nonzero net force; it is not a force itself.

I guess that's just not enough explanation for me. How can a force be equal to something that is not a force?

• Just want to add extra information, Newton's 2nd law doesn't state that "Force is mass times the acceleration" exactly... the 'purest' form of the law states that (cont.) Commented Dec 15, 2021 at 7:39
• (cont.) "The applied force is directly proportional to the rate of change of momentum of a body and takes place in the direction in which the force acts." Subtle difference which is important to note Commented Dec 15, 2021 at 7:41
• @AdilMohammed perhaps you can help…. What it says is “the vector that represents the net force is equal to the vector that represents mass times acceleration” ? Commented Dec 15, 2021 at 7:41
• @DavidHammen The OP can pick any answer they want to, and votes can change. Vote rankings are for the community, and they can also change over time. The accepted answer is for the OP specifically. This is why the recent change was made to not pin the accepted answer to the top. OP, do not worry; accept any answer you feel best answers your question. Commented Dec 15, 2021 at 12:18
• +1 not just because this is a common question and issue various people have, but also for OP continually asking questions about the answers to perfect his understanding. Even though it is technically a duplicate, I believe it's valuable to the site. Commented Dec 16, 2021 at 1:08

Mathematical equality is not the same thing as physical equality. If you have a mass $$m$$ undergoing an acceleration $$a$$, then we know the net force acting on the mass is mathematically equal to $$ma$$, but an accelerating mass isn't a force itself. You can't take the "$$ma$$" and use that to accelerate something else.

Most physics equations that are not definitions relate mathematical quantities that are not physically the same thing. This is what makes physics so useful. Saying "forces are forces" won't get you very far.

If this is true, can you perhaps give another example of two things that are mathematically equivalent but not physically equivalent?

The work-energy theorem that relates the net work done on an object to its change in kinetic energy: $$W=\Delta K$$. This is telling us that the net work done changes the kinetic energy, but work, which is the kind integral of force over a path, and kinetic energy, given by $$\frac12mv^2$$, are two different things physically, and they each have different definitions. They have the same mathematical value, but they are not physically the same thing.

• For real? That is going to be a lot for me to undo in my head. Commented Dec 15, 2021 at 6:24
• So it really is a loose definition of equality? Commented Dec 15, 2021 at 6:25
• @ChrisChristopherson They are mathematically equivalent. It is equality in the sense that if an object of mass $m$ experiences a net force $F$ then the acceleration $a$ is $F/m$. Commented Dec 15, 2021 at 6:33
• @MikeScott Hmm I have definitely read that their direction must be equal as well. But maybe I can logic my way as follows.... "the vectors that represent them are equal, but the things they are representing are not equal"? Commented Dec 15, 2021 at 7:17
• @ChrisChristopherson I have put in another example. Most physics equations that are not definitions relate mathematical quantities that are not physically the same thing. This is what makes physics so useful. Saying "forces are forces" won't get you very far. Commented Dec 15, 2021 at 12:28

I think you're confusing something "being" a force with some having force dimensions (If you don't know what I mean by dimensions you can mentally replace the word by "units"). More generally, you're confusing the entity being studied (the force), with the mathematical representation of such entity (a quantity with dimensions of a force).

In physics the term Force is used to describe an external influence on a certain object. In classical mechanics this influence is described as a vector with force dimensions. It is only this description that you can compare using the equality relation (the "=" sign). Thus the fact that two things are equal only means that they are mathematically described in the same way and that their "values" are the same (in the case of a vector by value I mean the three elements of the vector), but it says nothing about the physical meaning of the quantities.

So when you write: F = ma

You don't require that both F and ma are representing forces from a physics point of view, you only require that both quantities are equal, which entails that both quantity are of the same type (they are described by the same unit) and their values are the same (across all components of the vectors). You are making no implication about their physical meaning.

To use an analogy, the mathematics provide the grammar and the physics provide the syntax. This is how the equality relation is used in other contexts too. If you, for example, compare the number of siblings you have with the number of cities in your area, those are two different concepts, but you can compare their mathematical representations (not the ideas themselves) because both are expressed as numbers.

• I like what you’re saying and I believe it’s the same as my comment on the post itself: “F=ma” means that “the vector representing F is EQUAL to the vector representing ma.” While the actual objects they represent can be different objects. Commented Dec 15, 2021 at 8:55
• It took eight peoples help to get me there though so it’s much appreciated. Commented Dec 15, 2021 at 9:04
• @ChrisChristopherson To be fair, this answer just uses more words to say what others have been saying ;) Commented Dec 15, 2021 at 12:23
• +1 Very nice way to formulate the point. The idea behind a physical quantity or entity is often overlooked. The essense might be that being equal does not mean being the same. Commented Dec 15, 2021 at 13:10
• @ChrisChristopherson Yes, that's right but there's an additional layer on top of just comparing two vectors. In order for the the comparison to make sense both have to have the same dimensions (the quantities they represent must have the same units). You can't compare a vector with units of length with a vector with units of force, for example. On the other hand, you can compare two vectors with unit of force independently of their physical meaning. Commented Dec 16, 2021 at 9:55

$$F=ma$$ just gives a relationship between any force acting on a body and the acceleration experienced by it.

A force may result from electromagnetism, strong force, etc. - i.e., let's say a charged body with mass $$m$$ experiences electric force $$F$$. That force will cause it to accelerate with an acceleration $$F/m$$, as dictated by the $$F=ma$$ equality.

You can think of the equality in this way: you observe that objects accelerate when you push them, or two positively charged particles accelerate away from each other, or in other specific circumstances.

So you think to yourself that this acceleration must be due to something special, which you call a "force". This in turn gives you a prescription for measuring force. Since the very basis of identifying a "force" is the acceleration of a body, you come up with a physical law for measuring it, i.e. $$F=ma$$.

• I'm still definitely hitting my head on the desk for this one. Can you perhaps give another example of an equality where one side is not the same "thing" as the other? Commented Dec 15, 2021 at 6:42
• Consider Hooke's law: $F=-kx$. This is a physical law applicable to springs. It's an empirically derived expression for the force needed to extend or compress a spring by some distance. BUT this doesn't mean that we start calling $-kx$ a force that can affect any and everything else. That's why I mentioned in the answer - look at it as a physical law, a prescription. Hope that makes slightly more sense. Commented Dec 15, 2021 at 6:51
• I suppose. What I believe might be fruitful to me is reading things like "$\mathbf{F} =m\mathbf{a}$" as "The net forces on an object are proportional to it's mass times it's acceleration". Or like in hookes law, "the force is proportional to it's extension." Forgive my rigidness, I am very new to physics with my entire background in mathematics so I am trying to make sense of things. Commented Dec 15, 2021 at 6:56
• @ChrisChristopherson: The way you put it is, in fact, how those physical laws are properly stated in words. Commented Dec 15, 2021 at 6:59

It's of pedagogical and semantic reasons mainly. People tend to intuitively think of inertia as a force. When scouting for all present forces and drawing them on a sketch (on the free-body diagram), they might include the $$ma$$ "force" along with all the other actual forces.

But then the effect of the forces will be included twice when summed up with Newton's 2nd law. This is a typical and classical error. You shouldn't include the result of a summation in the summation. Insisting that $$ma$$ is not a force solves this issue.

It's mainly a semantic issue because we obviously all agree that $$ma$$ indeed equals the effect of the sum of all forces.

Also, note that while we wouldn't say that $$ma$$ is a force, we also wouldn't say that $$F_{net}$$ is a force. It is the sum of all forces. That is very important, at least pedagogically.

• Even more mind blowing! So vectors in physics are not closed under addition? Commented Dec 15, 2021 at 6:46
• There are sources of force vectors(stretched springs, gravity from nearby planets, wind from the north...) and results of those forces. F=ma simply tells about a result, without telling us a force vector that can be added with other force vectors. Commented Dec 15, 2021 at 7:39
• @ChrisChristophersen Essentially, you can't say that $F_{net}$ is the sum of all forces if you simultaneously say that $ma$ is a force. If $ma$ is a force then why isn't it added to the sum of all forces? There is a semantic conflict here. Commented Dec 15, 2021 at 7:44
• @Steeven quite a fair point. Commented Dec 15, 2021 at 7:56
• @ChrisChristophersen I see that you have a background in mathematics, so let's go at the problem like this: When calculating the sum, $N$, of all terms, $n$, $$N=\sum n,$$ within a descrete vector space, we shouldn't include the sum, $N$, itself as a term. $N$ is within the same space, since the space is stable and the map linear. Yet, it exists "after" the summation takes place, so to say, and thus shouldn't itself be included. Otherwise your sum becomes recursive and that's not what we see in nature in regards to force addition so that's not how the force-relation law should be formulated. Commented Dec 15, 2021 at 8:02

If I want to give n apples to each of m people, then the total number of apples I need, T, is given by T=nm. In the expression, m is understood not to represent people, but to mean the number of people. Likewise in Ohm's law, V=IR represents a relationship between the numerical qualities- it does not mean that a current is a voltage divided by a resistance. There are many other instances where an equality in an expression means an equality of the numerical quantities, but not of physical things. Take the formula for the volume of a sphere, for instance, or for the relationship between the object distance, image distance and focal length of a lens. In those expressions the equals sign does not mean 'is exactly the same thing as'- it means that the mathematical quantity on one side has the same value as that on the other.

Newton's second law is more fundamentally written as $$F_{\text{net}}=\frac{dp}{dt},$$ i.e. the net force is how much the momentum of an object is changing over time. Since mass is often constant this is then equivalent to $$F=\frac{d(mv)}{dt}=ma$$. We can make the following analogy: if $$p$$ is how much money is in your bank account (account balance) then a force is a transaction in- or out of your bank account. Newton's second law can then be written as $$\text{sum of all transactions}=\text{change in account balance}.$$ This sounds obvious when you write it like this but Newton's law defines forces as transactions. A change in account balance isn't the same as a transaction but in practice they always are. A hacker could set your balance to zero which would not count as a transaction. Similarly a change in momentum will always be caused by a force. Numerically they are the same, but fundamentally they are different objects.

• Newton's laws do not define forces as a transaction. Newton's Principia has a couple of corollaries that effectively say that individual forces on a body add vectorially. Newton's proofs of those corollaries are markedly circularly. I'm in the camp that says that those corollaries are essentially Newton's fourth law of motion. Commented Dec 15, 2021 at 12:51
• @DavidHammen I think it leads to the same conclusion, but instead of scalar numbers you exchange vector quantities. Commented Dec 15, 2021 at 15:30

Do not overthink it, physics is meant to be simple (the simplest explanation for the experiments we can find), and not too axiomatic. So what is meant here, you know that particles have trajectories x(t), from which you can get velocity and acceleration. A big problem of classical mechanics is to find the trajectory. What we find is we can find a simple law for the acceleration, F=ma. This has to be augmented with an expression of F for each mechanical setup.

But when you do overthink it, and you are actually on to something. In GR the gravitational force disappears and is described by a local acceleration.

In electromagnetism you only include forces from other things when computing your acceleration. So external forces on the left. The mass times acceleration is a quantity you use to model your own movement. I have not found any model for how you affect your own movement, they all seem to suffer from divergences or errors. So on the left you gather all your models of how other objects affect you, and on the right you get your acceleration. So if anything it is everyone else's force evaluated at me, and summed, equals my mass times my acceleration.

Then you replace "me" and "others" for each object in your scene, and you get a system of equations. I am not entirely sure what kind of "background forces" are allowed, but I guess since they are not related to any object they have to be treated with care.

So it's a bit like an equality and a deduction rule in one.

When you use the relationship distance = speed $$\times$$ time you are find a distance using two other quantities, speed and time.
Those two quantities when multiplied together enable you to evaluate the distance, and so it is with the relationship $$F=ma$$.

What you are missing is that forces appear to be vectors in Newtonian mechanics. Newton's Principia had a pair of corollaries to his three laws of motion that essentially said that forces are vectors, without calling them out as vectors (the concept of vectors postdates Newton's laws of motion by about 200 years). Newton's proofs of those corollaries is rather circular. His proof that the sum of two forces obeys the quadrilateral rule assumes that the sum of two forces obeys the quadrilateral rule.

Most physics education instructors and almost all basic physics texts simply teach that forces are vectors, without justification. A few physics education instructors teach that the justification is that forces appear to obey vector arithmetic is Newton's fourth law of motion.

The $$\mathbf F_\text{net}$$ in Newton's second law of motion reflects that forces appear to be three dimensional vectors. The $$\mathbf F_\text{net}$$ is not a force itself. It is instead the vectorial sum of all of the individual forces that act on a body.

"I'm so hungry, I'd be willing to pay \$100 for a hamburger right now. In other words, my willingness to pay is equal to \$100."

"But a dollar is not the same thing as a willingness! Dollars are green pieces of paper, or balances in checking accounts, but a willingness is a psychological state!"

"Maybe so. But you still understood me, didn't you?"

My answer is closely related to than given by Biophysicist, in fact inspired by it. This is based on my understanding, so there's a fair chance I may be wrong. The formula

$$F_{net}=ma$$

can be seen like this

$$\text {Cause=effect}$$

Cause and effect are not always equal. Usually, it's proportional and sometimes even inversely proportional, proportional to the square etc. In other words

\begin{align*} \text {Cause} \quad \alpha \quad \text {effect}\\ \end{align*}

Sometimes we put proportionality constant to equate them like in Newton's law for gravitational forces.

$$F=G \frac{Mm}{r^2}$$

Here mass is the cause and gravitational force is the result/effect if I am right. But just because Cause= effect doesn't mean they are the same thing. Consider the rotation of the Earth around the sun. Here the centrifugal force is provided by gravitational force. In other words

$$F_{centripetal}=F_{gravitational}$$ $$\frac{mv^2}{r}=G \frac{Mm}{r^2}$$ Where the gravitational force is the cause for the centripetal force. In other words, the centripetal force is provided by the gravitational force. But that does not mean centripetal is gravitational force. Physically, centripetal force is just a force that acts along the centre of the circle during a circular motion, while gravitational force is a force acting between 2 objects

The centripetal force can also be provided by electromagnetic forces, like in the case of an electron revolving a proton or it can be provided by frictional forces, like in the case of a car in a roundabout. (friction is also ultimately an electromagnetic force).

As Biophysicist said, Mathematical equality is not the same thing as physical equality and physically centripetal force and gravitational force are two different things. Similiarly Force and $$ma$$ are 2 different physical quantities, where the cause, Force $$F$$ is proportional to effect, acceleration $$a$$ with the proportionality constant being the mass or the inertia.

• Use the \text wrapper for things that are not variables Commented Dec 15, 2021 at 12:21
• @BioPhysicist better? Commented Dec 15, 2021 at 14:49