# Is a causal relationship implied by Newton's 2nd Law?

Throughout my time learning physics I have been imbued with the notion that forces cause accelerations, period. Accelerations don't cause forces, and they aren't merely correlated phenomena. By causality, I am content with the following definition:

Connection between two events or states such that one produces or brings about the other; where one is the cause and the other its effect.

That is to say, an object experiences an acceleration because it is exposed to a net force; the force does not arise because of the acceleration. However, some philosophical thinking on Venturis has shaken my confidence in this idea. If the acceleration of the fluid through the constriction is caused by an unbalanced force, what causes the unbalanced force in the first place? Another way of asking the question is, how is the bounding geometry causally linked to the pressure distribution of the flow? My only answer as yet is that there's no other way to satisfy mass, momentum, and energy conservation simultaneously, but that seems decidedly unsatisfying. Is there any causality implied by Newton's 2nd Law?

• Causation is hard to define, and without defining it, there's no way to answer your question. See philsci-archive.pitt.edu/1214 – Ben Crowell Sep 19 '14 at 18:46
• @BenCrowell What I am after is probably closest to Aristotle's efficient or moving cause. – Bryson S. Sep 19 '14 at 19:00
• It would be pointless to discuss it in terms of concepts from 2000 years ago. See the link above for what a more modern formulation would look like. See also Earman, A primer on determinism; philsci-archive.pitt.edu/3003 ; and various other work that can be found with search terms such as "norton's dome" and "staccato run." – Ben Crowell Sep 19 '14 at 19:26
• @BenCrowell Are you saying $F=ma$ will no longer be useful in two-thousand years? – Bryson S. Sep 19 '14 at 19:28
• Cause and effect are not, to my mind, physical terms. They're squishy terms, difficult to cast into precise notions, and so I would maintain that it is unclear what you are asking - not because I do not understand what you write, but because these things are ill-defined. – ACuriousMind Sep 21 '14 at 1:25

## 4 Answers

One of my favorite quotes, and I think this complements Ján Lalinský's answer:

"Does the engineer ever predict the acceleration of a given body from a knowledge of its mass and of the forces acting upon it? Of course. Does the chemist ever measure the mass of an atom by measuring its acceleration in a given field of force? Yes. Does the physicist ever determine the strength of a field by measuring the acceleration of a known mass in that field? Certainly. Why then, should any one of these roles be singled out as the role of Newton's second law of motion? The fact is that it has a variety of roles." - Brian Ellis, The Origin and Nature of Newton's Laws of Motion (1961), as cited by A P French, Newtonian Mechanics. (a fantastic book)

$F=m\ddot{x}$ isn't a definition of force or a definition of mass, it's a relationship.

As for your specific example, I can't help on the dynamics, but from the fact that the water accelerates, it must be pushed from behind (or pulled from the front, but you know with pressure the two are equivalent). This image on wikipedia:

http://en.wikipedia.org/wiki/File:Venturi.gif

in which high pressure is indicated by a dark blue color, gives you a pressure gradient. Clearly the change in pressure is enough to explain the acceleration/deceleration. Why is there a pressure gradient? That is deserving of its own question*.

*I'm answering the question "Is a causal relationship implied by Newton's 2nd Law?"

• Great quote, and as regards the explanation for the pressure gradient, therein lies the rub. I am looking for an intuitive and satisfying explanation as to why the pressure should decrease. And by the way, the pressure increases through a constriction for a supersonic flow, so we have to explain that as well... – Bryson S. Sep 19 '14 at 19:51

Is there any causality implied by Newton's 2nd Law?

It depends on what you mean by "causality" and "Newton's 2nd law".

I would rather not go into various meanings. Newton's original formulation of the 1st law seems causal:

Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

But his formulation of the 2nd law does not:

The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.*

Also, it is not necessary to formulate mechanics this way. The equation

$$\mathbf F = m\mathbf a$$

can be used without the verbal or causal formulation. In this equation, there is nothing referring to causality; it just puts two quantities into mathematical relation; whenever there is acceleration there is also the corresponding force. There is no delay and no subordinate relation between the two.

Jakov Frenkel made a very good observation that we think of force as cause because we see its accelerating effect only after the body acquires substantial speed, which happens after some time. But in fact, we believe that the acceleration is present exactly at the same moment the force is, so there is no delay between the two and no cause-effect relationship.

• Why the upvotes? The correct form is $\mathbf F = m \mathbf a$. There is no derivative of acceleration in Newton's second law. – David Hammen Sep 19 '14 at 20:25
• @DavidHammen The correct form is more like ${\bf F}=d{\bf p}/dt$... – Kyle Oman Sep 19 '14 at 20:52
• @Kyle: That's only for constant mass systems, and then arguing whether it's F=dp/dt or F=ma is a "great taste ... less filling" kind of debate. With variable mass systems, it's either F=ma or nothing (and there are some who say Newton's 2nd applies only to constant mass systems). Those who do think Newtonian mechanics applies to variable mass systems and use Newtonian mechanics to address said systems (i.e., "rocket scientists") invariably use F=ma because that leaves force as an invariant. – David Hammen Sep 19 '14 at 21:59
• Oops I made an error, thanks for pointing that out. I've corrected it. – Ján Lalinský Sep 20 '14 at 20:01
• @Hjan, in this paper: J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534. dx.doi.org/10.1007/BF01331692 – Ján Lalinský Jan 1 at 11:31

What makes you believe that there us an unbalanced net force in a Venturi tube? The force required to change the momentum of the liquid comes from the walls of the tube. This can be easily observed with an open nozzle (a garden hose will do!) and is used in rocket motors. In closed form a U-shaped tube is used as one of the most precise flow sensors, which measures the flow of dense liquids by measuring the forces on the tube, see the Coriolis flow meter: http://en.wikipedia.org/wiki/Mass_flow_meter

• I am interested in the fluid, not it's container. The fluid particles accelerate into the throat, then decelerate as they exit. If you know how to accelerate an object without an unbalanced net force then you may be the next Einstein. Also, the force vector doesn't make sense for your explanation. The force from the walls points backward at the entrance to the tube, and forward at the exit; this is the opposite of what's needed to produce to observed accelerations. Not to mention that the opposite effect occurs for supersonic flows. – Bryson S. Sep 19 '14 at 19:22
• @BrysonS.: Without its container the fluid would not undergo this particular type of motion to begin with. Your problem seems more concerned about separability than causality. You want to separate parts of the system in a way that changes the system, which, of course, is not a proper physical analysis. You can't have your cake and it it, too. – CuriousOne Sep 19 '14 at 19:25
• Again, the explanation that axial force from the walls is alone responsible for the accelerations cannot be correct because the trend reverses for supersonic flow in the same geometry. – Bryson S. Sep 19 '14 at 19:27
• @BrysonS.: Why would the difference in flow pattern between two velocity regimes change the physics of the flow at ANY velocity???? – CuriousOne Sep 19 '14 at 19:29
• It doesn't. What I am saying is that the force from the walls on the flow points in the same direction for both subsonic and supersonic flows. However, the acceleration of the flow in the vicinity of the throat is completely reversed for subsonic and supersonic flow. Thus, the axial force from the walls cannot be the cause of the acceleration, because the direction of the acceleration changes even when the force direction doesn't. Q.E.D. – Bryson S. Sep 19 '14 at 19:32

"what causes the unbalanced net force in the first place?" - potential. The energy of molecular motion upstream of the flow is greater than downstream. And we see the energy as pressure or potential (energy).

But regarding the main question - Is there causality wrapped up in Newton's 2nd law? Maybe.

Consider F = d/dt(momentum). The fact that I had to write this expression using a derivative implies that there is prediction going on to figure out what force I have from the momentum of a system. On the other hand if I write a = integral(F/m), I'm only relying on the present and past states to determine motion.

I apologize - I'm a beginner and not skilled yet in properly formatting these posts.

• Yes but why is the energy different? If anything I see that as more effect than cause. Can you expound on exactly how the geometry causes this difference? – Bryson S. Sep 19 '14 at 17:37
• Not geometry - the operations of integration and differentiation. Differentiation implies prediction, forecasting which is a non-causal activity. Integration sums the present and past which is completely causal. So by this interpretation one could say Newton's 2nd law is only causal if solving for accelerations. For practical operation of the Venturi nozzle, you don't start with flow but rather the pressure. Pressure has to come first. You have to have energy to start flow. Flow (acceleration of atoms) arises from the pressure (energy) differential. – docscience Sep 19 '14 at 18:20
• This doesn't seem to be helping. So let's take a thought experiment. Imagine a windtunnel with a variable geometry throat. I start up the flow and keep the upstream conditions constant. I then start to contract the throat dimension. How is this geometrical change changing the pressure, which then changes the velocity distribution? – Bryson S. Sep 19 '14 at 19:05
• @Bryson, In your example by keeping the upstream pressure constant that sets the energy (per unit mass) constant. Assume no energy losses at the walls of your tunnel. As you reduce the neck of the Venturi the energy needs to go somewhere, but the repulsive (electromagnetic) force between molecules resists, and so energy conservation is maintained by speeding up the molecules. The pressure (potential) energy is reduced and the velocity (kinetic) energy is increased. – docscience Sep 20 '14 at 14:53
• Downstream of the neck energy is often said to 'recover' but actually the kinetic energy becomes potential and pressure increases. By not having lost any energy through the walls of the tunnel, the process is adiabatic within the space (no heat loss). But according to the state equation temperature was changing during the process. The temperature is a measure of the kinetic energy. – docscience Sep 20 '14 at 14:53