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I am currently studying advanced mechanics (meaning I do have an introduction in mechanics). I know the question is a bit weird (actually a lot). My question is: "Is it the acceleration which causes the force or the force which causes the acceleration"?
In Newton's Principia, he introduced his second law of motion as: $$ \vec{F} \propto \frac{d\vec{p}}{dt} $$ Now to remove the proportionality, he introduced a proportionality constant k $$ \vec{F}=k\frac{d\vec{p}}{dt} $$ Now we need to measure k, so we will measure the LHS and the RHS. but what is a force? It is not even defined! So newton just said $k=1$, so that the equation looks elegant. $$ \vec{F}=\frac{d\vec{p}}{dt} $$ and if the mass remains constant, $$ \vec{F}=m \vec{a} $$ Now here is my point: we say that a force causes an acceleration, but a force is fundamentally defined by acceleration, so why do we say that? I don't understand what force is now (I am heavily confused, thanks to my philosophical questions)? I also read somewhere that a famous physicist said that: "Saying that an acceleration causes a force is metaphysics". My question is: what is wrong in my approach? It will be really helpful.

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    $\begingroup$ There is an old philosophical text by Mach that reported that Hertz had written a mechanics textbook that did away with forces as a concept. Hertz's text is not actually good enough as a textbook, but the general idea is correct, you can redo mechanics without forces. However, your question about $k=1$ is really just that, at the time of initially doing mechanics, the units of force is yet to be chosen, and so it is convenient to choose $k=1$. $\endgroup$ Oct 23, 2023 at 17:24
  • $\begingroup$ "Now we need to measure k, so we will measure the LHS and the RHS. but what is a force? It is not even defined! So newton just said k=1 , So that the equation looks elegant.".Physics deals with abstract ideas or ideas which we dont know what they are.Take the wavefunction $\Psi$ of a particle.Do we know what the wavefunction is?No we dont.We only know that $\Psi \Psi^{*}$ gives the probability for a observable of the particle. $\endgroup$
    – Cerise
    Oct 23, 2023 at 18:22
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    $\begingroup$ Conceptually, acceleration is a property associated with the object itself (a time dependent property describing its motion, or a local property of its (classical) spacetime trajectory). A force is, again conceptually, an external influence that causes the motion of the object to change (and thus, it causes acceleration), in the way described by Newton's laws. It's just the way these notions are conceptualized, you don't need any deeper justification than that. $\endgroup$ Oct 24, 2023 at 15:55
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    $\begingroup$ F⃗ =ma⃗ is not the defining formula of force. This formula merely establishes a relation between two independently defined entities. $\endgroup$
    – Igor G
    Oct 25, 2023 at 11:06
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    $\begingroup$ “a force is fundamentally defined by acceleration” — why do you think that?  Newton's equation defines the relationship between force and acceleration (or, strictly, between force and change of motion); it doesn't define force itself (though, as Wikipedia notes, it's sometimes presented like that). $\endgroup$
    – gidds
    Oct 25, 2023 at 11:11

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Neither causes the other. They are essentially the same vector, differing only by a constant factor. To cause something implies that an event at one time results in an event at a later time. These vectors are always in lock step in time, never differing in their direction or the ratio of their magnitudes, $ \vec{F}(t) = m \vec{a}(t) $. If force "caused" acceleration, you would expect a relationship such as $ \vec{F}(t) = m\vec{a}(t+\epsilon) $ for some time of $ \epsilon $, but such a thing would be nonsense. Acceleration doesn't lag behind force.

They are the same vector, scaled slightly differently. One doesn't cause the other because they don't exist independently of each other; they aren't really separate entities. That's what it means to be equal to, philosophically. "Cause" is often a misleading and overused term in physics that should be avoided if possible.

EDIT: a generic, subscript-less $ \vec{F}(t) $ in the context of Newton's Second Law is universally assumed to be the total force vector, among physicists and those with any formal education in physics at all.

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    $\begingroup$ This doesn't appear to make sense to me, if we consider a case where there is a Force, but no movement. $\endgroup$
    – MikeB
    Oct 24, 2023 at 10:52
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    $\begingroup$ @Koschi When you stand still on a floor, is there acceleration? $\endgroup$
    – JimmyJames
    Oct 24, 2023 at 15:10
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    $\begingroup$ "To cause something implies that an event at one time results in an event at a later time" I'm not sure about this; for instance, I'm very willing to consider gravitational fields as causing gravitational force, despite each being a scale factor of the other (in Newtonian mechanics). $\endgroup$
    – Vaelus
    Oct 24, 2023 at 23:15
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    $\begingroup$ @JimmyJames I do not see how your comment helps for this discussion, tbh. Of course, on the molecular level there is always motion in general, and yes, if I push against a wall, there will be some motion due to my application of force of the atoms compressing. However, if the wall is connected to the ground, this compression will reach a maximum at which the molecular forces due to compression equal the force I apply to the wall. The wall will not move and I will also not compress it any further. Again, then there will be forces without acceleration. $\endgroup$
    – Koschi
    Oct 25, 2023 at 8:25
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    $\begingroup$ @Vaelus a gravitational field in classical (Newtonian) mechanics is merely a description that "if you place a mass m at a certain position in space, it will see the force which is described for that location by the field". There's no "cause". Not even the field is "caused" by anything. It's a mere mathematical description of what happens when 2 (or more) masses are at some finite distance from each other, albeit generalized for describing "all points in space" at the same time $\endgroup$
    – Apfelsaft
    Oct 25, 2023 at 9:32
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“but a force is fundamentally defined by acceleration”

Acceleration does not define a force. If it did then I wouldn’t be applying a force when pushing against a wall because the wall doesn’t accelerate.

It is the net force acting on an object that is defined as the change in momentum of the object.

Hope this helps

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  • $\begingroup$ so then the wall would experience no 'net force?' $\endgroup$
    – user45664
    Oct 23, 2023 at 17:38
  • $\begingroup$ @user45664 that’s correct $\endgroup$
    – Bob D
    Oct 23, 2023 at 17:40
  • $\begingroup$ The wall did accelerate. You just didn't see it, that all. It goes into the vibrational modes of the building. If you push hard enough, you'll see it. For instance, if you're coming along the curve on a 200 meter track, upwards of 20 MPH, loose traction and crash into the wall, besides fracturing your wrist, you'll see that dent in the doorway or wall that was made when that part of the wall was accelerated - as I can relate to you first-hand ... or first-wrist, as the case may be. And that dent in the wall in the alma maters' arena might still be there. $\endgroup$
    – NinjaDarth
    Oct 23, 2023 at 20:19
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    $\begingroup$ @NinjaDarth You're missing the point. The point is $F\ne ma$ where $F$ is the force I apply, $a$ is the acceleration of the wall, and $m$ is the mass of the wall. That acceleration, which for a rigid body wall plus the Earth would be miniscule to satisfy conservation of momentum, would certainly not define the force I apply. $\endgroup$
    – Bob D
    Oct 23, 2023 at 20:50
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    $\begingroup$ @NinjaDarth, Consider one more case instead, Where An ideal rigid solid cube is being pushed from two opposite sides by two different persons with equal forces, here acceleration of block is zero, that does not mean, forces didn't exist at all, that only indicates forces were balanced $\endgroup$ Oct 23, 2023 at 21:09
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You are right to be confused. The textbooks tend to teach this poorly. And yes, it is a bit of a philosophical question - physics is, after all, "natural philosophy."

  1. Force is not "defined by acceleration". If it were, Newton's 2nd law would have no content! Also, Newton's 2nd law does NOT say $\vec{F} = m\vec{a}$ or $\vec{F} = \dfrac{d\vec{p}}{dt}$ - it says $\Sigma \vec{F} = \dfrac{d\vec{p}}{dt}$. That sum is important, and means you have to add the forces on an object first before you can relate them to acceleration (or change of momentum).

  2. An important point that most textbooks neglect to emphasize is that a force, whatever else it might be, is always an interaction between two separate things, one exerting the force (the agent) and the other experiencing it (the object). (If this were made more clear from the beginning, lots of misunderstandings and mistakes about Newton's 3rd law would get cleared up.)

  3. We don't get to have a strict, general definition of force. Contrary to popular opinion, physics usually does not proceed from definitions, through syllogistic logic, to conclusions - when you're doing that, you're doing math, not physics. Of course, physics often gives provisional mathematical definitions of things so it can use mathematical reasoning on them, but those aren't physical definitions. The fact is, many of the most important quantities in physics resist definition - that includes force, momentum (I know you think it's $mv$, but wait till you learn about photons!), energy, quantum mechanical wave function, etc. That doesn't mean they aren't meaningful or we can't say meaningful things about them - on the contrary, as we work with them and develop theories, we gain an increasing understanding of what they are by connecting them with what goes on in the real world. Arguably the whole point of physics is to grope towards better and better understanding of these things that clearly have some physical reality but are hard to pin down. This is a profound fact about physics that most textbooks shy away from. Perhaps that's because the authors don't realize it, or perhaps it's because they are afraid it will lose credibility for physics since people are apt to think you don't know what you're talking about if you can't give a good definition of it, or perhaps it's because novices coming to the field are more comfortable with simple definitions they can grasp as a starting point.

So where does that leave us? Intuitively, objects push and pull on one another. When one object gives a push or pull to another object, let's refer to that interaction as a "force." That's obviously not a definition, but it's enough to specify what we're talking about. If an object has no forces on it at all, that is, nothing else is interacting with it, it maintains constant momentum (in an inertial frame) (Newton's first law), which tells us that forces have something to do with changing momentum. In fact, you can add up the forces acting on an object by all the different agents linearly (as vectors), and if you do, the sum tells you the rate of change of the object's momentum (Newton's second law). Meanwhile, for each agent-object interaction (i.e. force), the roles of agent and object are symmetrical: each agent experiences a force on it from the object (a "reaction"), of the same magnitude and in the opposite direction to the force it exerts on the object (the "action") (Newton's 3rd law).

The short answer to your question is, bodies exert forces on one another; an object with no forces on it does not accelerate; an object's acceleration is proportional to the net force on it. We suppose that the "agents" are exerting influences, and the "object" responds by accelerating. If you (as an additional agent) change what the other agents are doing the object's acceleration changes, but if you act directly on the object and change its acceleration, that doesn't necessarily do anything to the other agents. For those reasons we usually consider the forces as causes and the acceleration as an effect resulting from them, even though the "cause" is technically simultaneous with and identical to the "effect."

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  • $\begingroup$ I'm afraid I have to disagree with the statement that "physics usually does not proceed from definitions." We would not know what we are speaking about without an explicit definition. Of course, this doesn't exclude the possibility of poor definitions. $\endgroup$ Oct 25, 2023 at 21:15
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Your question seems to be about language used, That is ,"Cause" gives out a general meaning that "B happened as a result Of A only", Which Essentially means A "Caused" B, Here , if "Acceleration Caused Force to exist", It follows from previous analogy that "Force Existed because of acceleration produced only", Is it always true?, Verify this in case given by @BobD's Answer, Now think other way around, if "Force caused acceleration", It follows that "Acceleration happened/existed as a result of application of force only", Verify these statements yourself to find which seems to be the true case..

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I understand your confusion. The subject is already quite complicated from the force concept alone due to Newtonian mechanics' long history and evolution since its birth in the Philosophiae Naturalis Principia Mathematica. In turn, the concept of causal relation requires additional clarification when used in Physics.

Part of the difficulties has already been emphasized in other answers and some comments. Here, I do not pretend to write the ultimate solution, but I'll try to make it explicit and clarify some conceptual issues.

Causality

Let's start from the side of the concept of cause. I'll ignore the never-lasting and interesting philosophical debate around the idea of causality by confining the attention only to the more limited problem of the possible definition of a causal relation in Physics. Even with such a limitation, there are different facets to the causality concept. A non-exhausting list can be obtained by searching on this site using the keyword causality in Physics. Again, I'll further simplify the situation by focusing on two main concepts behind the idea of causal relation in Physics:

  1. the idea that the external conditions we can control and impose on a system of interest are responsible and then cause the system's status;
  2. the idea of special time correlations induced on an event at time $t$ by a previous event at the time $t'<t$ (provided the separation between the events is time-like). The less problematic example of such correlations is the case of a deterministic theory.

The two concepts are not exclusive and may coexist. The case of force and motion in classical mechanics is an example of such coexistence (see later) in the presence of a deterministic theory.

Force and acceleration

On the side of Classical Mechanics, there are many additional sources of confusion, mainly related to the concept of force and its relation with acceleration. In a way, most of the possible confusions originate in the long history of formulating the basic principles of the theory. It is not a mystery that Newton's three laws have undergone an evolution, and periodically, new interpretations or formulations have been proposed. The key problem is the consistent formulation of what are definitions and what are physical laws. Definitions are a matter of convention. They can be more or less useful but never wrong if they are not contradictory. Physical laws summarize experimental findings and can then be falsified by additional experiments.

Here, again, I'll simplify a much more complex history and focus on three main steps in the evolution of the concept of force.

  1. Newton defined force (definition IV) as follows (I am citing Motte's translation to English of the original in Latin): "An impressed force is an action exerted upon a body to change its state, either of rest or of moving uniformly forward in a right line". Here, previous ideas (1) and (2) about causality coexist. On the one side, force is seen as an external action to change the motion. On the other side, there is a deterministic relation between the net force acting on a body and the resulting motion, expressed by a second-order differential equation for the time evolution of the position. In such an equation (or system of equations), the force's main role is not to be a nickname for the product of mass and acceleration but to provide information about how acceleration depends on other bodies and the body's mass.
  2. In the nineteenth century, Mach's criticized many points of Newton's theory. Mach pushed forward the line of thoughts of Saint Venant, Kirchhoff, and continued after him by Hertz, aiming to eliminate the concept of force in favor of introducing only mass in addition to pure kinematical quantities. Mach's ideas still have a significant impact on the way Mechanics is reached.
  3. In the twentieth century, some extreme points of Mach's criticism were reconsidered, and the presence of Relativity forced a rethinking of the role of reference frames in defining the force.

The present day's situation is characterized by different teaching approaches where some hybridization between different formulations was introduced. In many cases, the assumed viewpoint is not explicitly stated, creating a situation of confusion about the status of the force. To make an example of possible confusion, in a pure Newtonian approach or modern post-Newtonian approaches, the force is not defined by the acceleration. Therefore, it is necessary an explicit proof of its vector character. In a naive Mach's approach, its vector character is granted by the vector character of the acceleration.

Summary

No meaningful answer is possible without stating explicitly i) what definition of causality and ii) which formulation of the principle of mechanics is used.

In an original Newtonian formulation, or in the modern Newton-like approaches, if we identify the cause-effect relation as the deterministic relation between the force (that we can control) and the motion, we could say that force is a cause of motion.

A causal relation is meaningless in a pure Mach-like approach where the product between mass and acceleration defines force.

In the case of hybrid approaches, it is necessary to examine the situation case by case.

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Loosely speaking a Force is a push or a pull. I don't think that works at all describing the weak nuclear force, but works well for classical mechanics. Does acceleration appply regarding WNF?

Newton's second law is: A force causes an acceleration inversely proportional to the object's mass. Apply the same force to a different mass, you get a different acceleration that still satisfies F=ma. Force is fundamentally different from acceleration. Consider a spring where $F=-kx$ and no reference is directly made to acceleration. Gravity, $F=\frac{-GM_1M_2}{r^2}$, likewise. No mention of acceleration. We do have $M_2a=\frac{-GM_1M_2}{r^2}\implies a=\frac{-GM_1}{r^2}$ so they are closely related.

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  • $\begingroup$ "Weak neutral currents" transfer energy and momentum via the weak nuclear force without changing any particle numbers. And the other (charged) weak interactions do also transfer energy and momentum as well as transforming particles, it's just that the "transforming particles" effect is considered more important. $\endgroup$
    – zwol
    Oct 25, 2023 at 15:22
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This boils down to a question of systems and how we choose to organize the universe in our heads to better understand it. The force on an object obeys an equality with its acceleration. Due to the equality, there is no real distinction between the two; we could choose to run philosophical causality either way. (There's no question of physical causality, because this is all at equal time; physical causality cares about different times.)

However, the force on an object is the sum over many forces from various sources, whereas the acceleration is a single vector that we observe in its momentum change. In order to perform the summation of forces, we have to understand where each of these comes from, e.g. an electromagnetic force, a gravitational force or a normal force. This decomposition allows us to introduce further explanatory power to our model of the system. Therefore, the most explanatory power can be had by choosing to say the force sources cause the forces that sum to the total force, and the total force causes the acceleration.

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The entire branch of physics called Statics are about systems on which forces exist but no acceleration. This only makes sense if force causes acceleration.

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According to Newton's second law , force is proportional to rate of change of momentum with respect to time as you have shown in your question so after simplifying it by substituting momentum as mass times velocity then the final equation includes the term which represent that force can be present even without change in velocity by just changing the mass, so it shows that neither define other one.
This is just another a tricky way to get to answer of your doubt.

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Phew! So many comments! The one I agree with most is @GiorgioP-DoomsdayClockIsAt-90. What can I add?

Let's be empirical since this is science we are talking about, not mathematics or philosophy (nothing wrong with mathematics or philosophy, but the meaning of "force" is fundamentally a physics question).

Empirically, we observe that in the absence of any interactions, an object moves with constant velocity according to any inertial observer. This is verging on metaphysics since finding objects which don't interact with anything is difficult in our every day lives. But if we look to deep space we see that it is commonplace there, at least to a very good approximation. So let's go with that. We can conclude that accelerations are caused by interactions between objects.

A force is how we quantify the strength of such an interaction. If an object only interacts with one other object then we can quantify the force as $\vec{F} = \frac{d\vec{p}}{dt}$. If it interacts with multiple objects then that should be a sum of forces, not a single force. This then gets more complicated because we might be able to measure $\frac{d \vec{p}}{dt}$ directly, but if there are multiple forces we can't determine one of them without knowing the rest.

The implicit choice of $k = 1$ so we don't have to write $\vec{F} = k \frac{d \vec{p}}{dt}$ is really just a choice of units. We could work in units where $k$ would have some other value (and it would have units), but this is an unnecessary complication. We build our system of units so that $k = 1$ because it is convenient. As you point out, in the case that the mass is constant we get to rewrite $\frac{d\vec{p}}{dt}$ as $m \vec{a}$.

So, now let's return to the question of whether force and acceleration are the same or different. To answer it let's again remember that we are doing science, so the answer is likely to involve thinking about what we measure, and how we measure it. Acceleration is a kinematic quantity. We can determine it entirely by looking at the motion of the object. From detailed position vs. time data we can at least get the average acceleration over short time intervals, and approximate this as the instantaneous acceleration, which is what is actually meant by the quantity, $\vec{a}$, in the equation.

Force is another matter entirely. It isn't a kinematic quantity. It is a "strength of an interaction" between two objects. Unfortunately, we can't measure it directly. All measurements of forces are really measurements of some proxy observable that we are able to calibrate to a force scale. The simplest example of an instrument that we use to (pretend to...) measure a force is a spring scale. It definitely doesn't really measure a force. It measures the length of a spring. But we calibrate the scale beside the spring by hanging known weights off of it. How do we know the weights? They are gravitational forces that cause a known, measurable acceleration (g) on known masses. So, acceleration and force are different things (a rate of change of velocity on one hand, and a strength of an interaction on the other hand), but the way we calibrate our force measurements generally trace back to knowing about accelerations of known masses. So, one could take the point of view that acceleration is a more "real" quantity than force is, since it is more directly measurable.

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  • $\begingroup$ More of like "general intelligence", We cannot measure it directly, But it is certainly proportional to IQ, And IQ is More directly measurable, But I have a question then, Is greater intelligence a cause of More IQ or More IQ is a cause of general intelligence? $\endgroup$ Oct 26, 2023 at 3:20
  • $\begingroup$ I think what's more important is the Giving Quantifiable means to Our "senses/observations", Just like we had a sense that "eating more Apples reduced hunger" , we quantified apples in terms of numbers like 1 or 2 apples etc. Similarly , we have a observation(more precisely our senses had an external stimuli) that more of "a certain interaction" will give more change in a particular quantity if subjected for same amount of time that we already measured as "Momentum", We quantified this observation as "Force", hence observations are primary, Our "quantification of that observation" is secondary $\endgroup$ Oct 26, 2023 at 3:52

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