Newton's Second Law assumptions

Hi people
Newton's Second Law a=F/m assumes that either a body is pointlike or that force is transmitted instantaneously across it. Special relativity which applies to Newton's First Law and which was developed from the assumption of a finite velocity of energy transmission, has enjoyed tremendous success- it is absurd that we have not applied the 'finite velocity of energy transmission' principle to the Second Law as well.
I ended up with F=ma+P/v where P is power and v is velocity of energy transmission which seems too simple and even worse difficulties when extended to energy eqns.
Have I got it wrong in thinking that Newton assumed instantaneous transmission of force across a body?

Here's the actual statement of Newton II by Newton:

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

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to, or subtracted from the former motion, according as they directly conspire with, or are directly contrary to each other, or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both."

The word "motion" as used by Newton is short for "quantity of motion", what we would call "momentum".

Therefore, all the above by Newton is probably best rendered by the formula:

Δρ=FΔt

Euler is, apparently, the one who thought the Second Law of Motion ought to be F=ma. Newton doesn't ever seem to have expressed it that way.

Interesting. Reminds me of the Heaviside eqns being labelled Maxwell's eqns. Euler's form does make the assumption of instantaneous transmission of force more obvious, and more to the point it is the form most commonly taught/used.
I take it that you are not convinced that there is an assumption of instantaneous transmission of force?

CWatters
Homework Helper
Gold Member
Hi people
Newton's Second Law a=F/m assumes that either a body is pointlike or that force is transmitted instantaneously across it.

It's hard to believe he had never played with something compressible like a ball or peach and didn't realise this would affect the way forces were transmitted. But I can see he might have assumed instantaneous transmission for rigid bodies.

Interesting. Reminds me of the Heaviside eqns being labelled Maxwell's eqns. Euler's form does make the assumption of instantaneous transmission of force more obvious, and more to the point it is the form most commonly taught/used.
I take it that you are not convinced that there is an assumption of instantaneous transmission of force?
I don't know enough to follow what Euler was up to. If you can understand this:

maybe you can determine what assumptions he made.

It seems clear that Newton, at any rate, did not assume momentum was necessarily altered instantaneously: "...whether that force be impressed altogether and at once, or gradually and successively..."

Chi Meson
Homework Helper
In the form F=∆p/∆t, "F" is assumed to mean "average net force" on the object in question, and that would be the force averaged over the entire ∆t duration.

As I recall from 30 years ago (omfg) the formulas for the changes in motion of non-rigid bodies, and especially for objects like golf balls through tiny increments of time, quickly became absurdly complicated series functions. But if you waited for like, half a second, for the object to settle out, the functions fell directly into the simple format we learned first.

The simple law is still true on a particle-by-particle basis. The problem gets tricky because every "object" is actually a very complicated system of particles and the interaction between these particles is complicated in solids, and utterly chaotic for fluids.

For solid objects, we mostly need to analyze the outcome of an interaction after the force has been applied. But if the close-up or slow-motion analysis is required while the force is being applied, there is no simple "tack-on" fix to the 2nd law. It would be dependent on the material, distribution of density, geometry, and orientation of the object. I'm pretty sure Newton understood this, because he had to go and invent Calculus just to complete his analysis [/oversimplification]

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sophiecentaur
Gold Member
Have I got it wrong in thinking that Newton assumed instantaneous transmission of force across a body?

Δρ=FΔt

Stating it this way, the time taken for the force to 'make itself felt' throughout a resilient object is taken care of.

AlephZero
Homework Helper
it is absurd that we have not applied the 'finite velocity of energy transmission' principle to the Second Law as well.
...
Have I got it wrong in thinking that Newton assumed instantaneous transmission of force across a body?

That would be absurd if it was true, but it's not.

Newton didn't have any clear concept of "energy", so he couldn't think in terms of "velocity of energy transmission", but he certainly attempted to calculate the finite speed of sound in a medium. There was nothing wrong with the basic idea of his solution, though he got some details wrong.

Hi Sophie, the form that you and Chi Meson are describing waits for the transient to settle into a steady state. Chi Meson then identifies the crux of the issue-the transient.
Say a bullet is travelling faster than the speed of sound in lead. When it impacts, the back half doesn't know that the front half has stopped and keeps plowing forward.
This suggests an additional force which is temporary- the problem is reconciling that with the 'before and after' steady state. The excellent answer by Chi Meson points out most of the difficulties involved with a golf ball- but now add in the complication that the golf ball is struck with a velocity greater than the speed of sound in golf balls?
Do shockwave effects alter the force law? Consider a steel rod that's 10 km long and we apply a force- until the sound wave returns, the rod cannot obey a=F/m no matter how large the force applied.

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sophiecentaur
Gold Member
You seem to be making some very unreasonable demands on poor old Sir Isaac.
The "force law" that you want to use would apply perfectly well to elemental pieces of the system and that would lead you to a wave equation which would be soluble for a simple case. It actually sounds like it could be a piece of book work which someone would probably be able to remember (not me though).
So, to answer your original points - the force law applies to separate bits of an object but, of course, it doesn't allow for resilience, as it stands - because that's not its purpose. After all, it's not called Newtons Law of Collisions is it?

You seem to be making some very unreasonable demands on poor old Sir Isaac.
The "force law" that you want to use would apply perfectly well to elemental pieces of the system and that would lead you to a wave equation which would be soluble for a simple case. It actually sounds like it could be a piece of book work which someone would probably be able to remember (not me though).
So, to answer your original points - the force law applies to separate bits of an object but, of course, it doesn't allow for resilience, as it stands - because that's not its purpose. After all, it's not called Newtons Law of Collisions is it?

First of all, many thanks to you and Chi Meson for confirming the identified assumptions in the 2nd Law. I agree that wave theory is needed to model the 'steel rod' scenario. To calculate the extra force applied, I assumed that the velocity of sound stayed constant (despite the changes in density), and ended up with an expression for radiation pressure with the area terms dispensed with. Hence 1/v describes in newtons per watt the extra force applied. The reason for attempting this is to simplify 'systems of particles' calculations which, as Chi Meson points out, are amazingly complex, and to allow for 'resilience'. However I am not sure that my add-on is correct in light of the fact that the velocity of sound isn't constant in a material. ie a shockwave usually travels at a higher velocity than the normal speed of sound.

That would be absurd if it was true, but it's not.

Newton didn't have any clear concept of "energy", so he couldn't think in terms of "velocity of energy transmission", but he certainly attempted to calculate the finite speed of sound in a medium. There was nothing wrong with the basic idea of his solution, though he got some details wrong.

Newton, in relation to his theory of gravitation, commented on the 'absurdity of instantaneous action at a distance', so even though he couldn't think in terms of velocity of energy transmission, there is evidence that he thought in terms of velocity of force transmission. So Newton proposed it, Einstein seconded the proposal (for finite velocity of energy/force transmission)- sounds like an interesting avenue to explore!