Formulation of Newton's second law: $F=ma$ or $a=\frac{F}{m}$?

Question

Assuming a constant mass, which would be best way of expressing Newton's second law? $F=ma$ or $a=\frac{F}{m}$? The most common way is the first one, but I feel more comfortable explaining it the second way. It seems more natural to talk about the consequence (acceleration $a$) of applying the cause (force $F$) on an object (of mass $m$). How is the second law stated in each case?

Edit: What I mean is that a scientific law is more than the formula. I want to know of the law itself would be closer to Newton's interpretation if phrased as "the acceleration of an object is proportional to the net force applied on it and inversely proportional to the mass" or as "the net force needed to accelerate an object is proportional to the acceleration and the mass". I usually I see it explained using the first, but accompanied by $F=ma$, which I find weird (and confusing for students). I am aware of the relationship between force and change of momentum, but I am working with 10th graders.

Answer 1

There are a number equivalent mathematical forms of Newton's second law. The choice is a matter of convenience for solving different problems or understanding different concepts.

There is, of course

$$F=ma$$

Which is useful in understanding Newton's first law relationship to the inertial property of mass, i.e., the larger the mass the larger the force needed to effect a change in velocity (acceleration).

And what you prefer

$$a=\frac{F}{m}$$

When one is interested in the acceleration of a mass due to a net force.

Then there is

$$F=m\frac{dv}{dt}$$

which can be re-written as

$$F=\frac{d(mv)}{dt}=\frac{dp}{dt}$$

which demonstrates that a net force results in the change in momentum of an object.

I want to know if it would be closer to Newton's interpretation if phrased as "the acceleration of an object is proportional to the force applied on it and inversely proportional to the mass" or as "the force needed to accelerate an object is proportional to the acceleration and the mass". I usually I see it explained using the first, but accompanied by =, which I find confusing for students.

I usually see it phrased the first way (except it's missing a key word, as noted below), together with the formula $F=ma$. I agree with you the equation expressed in terms of $a$ better fits the phrasing. And I also agree it can be confusing to students. The problem is, that is the way they will see it if they continue their education in physics. But if I were teaching it, I would present it both ways like you.

Equally, if not more, important is your phrasing is missing the key adjective "net" in front of force. You can have a force without acceleration, like when you push against a wall. I would really stress the phrase "net force" with students. If that were more often done, perhaps students would not get so confused when the learn the third law and wonder why forces don't always cancel each other and no acceleration occur.

Hope this helps.

Answer 2

The two expressions are mathematically equivalent, so you can use either depending on the context. Your preference might be influenced depending on the factor your are trying to calculate. For example, if you are given F and m it would be more natural to think of the relationship as a = F/m.

Answer 3

In my opinion $F = ma$ is not a good way to describe Newton's 2nd Law as one will see in rocket equations, Impulses, Variable mass systems, etc.

According to me the best way to describe 2nd law is $\vec F = \cfrac {d \vec p}{dt}$ As it shows how Force induces a change in momentum of the particle.

Answer 4

It seems from your comments that you are interested in the relation between our usual formula $F=ma$ and with the actual original phrasing by sir Isaac Newton himself.

Newton's own phrasing was actually entirely different. He didn't mention acceleration and arguably also not even force. Instead he described the proportionality between the total impulse $J$ (which he called "impressed force" or "impressed motive force") and motion change $\Delta v$: $$J\propto \Delta v$$

Newton didn't really write out his law as an expression like this. In his original Principia publication in the 1680s he wrote it in words like this (an exact quote):^*

"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" - Sir Isaac Newton

To make use of his law, we can rewrite this proportionality into an equation with a proportionality constant that we call mass $m$:

$$J=m\Delta v$$

Often his words are interpreted as not "change in motion" but "change in amount of motion", which might mean "change in momentum". So, you will often see the interpretation of his words instead written as:

$$J\propto \Delta p$$

which translates to $J=\Delta p$ with a proportionality constant of 1, and is equivalent to the motion-interpretation since $m\Delta v=\Delta (mv)=\Delta p$. This latter version turns out to be more accurate, since it doesn't have to assume constant mass.

The background of and Newton's own thoughts on and doubts about his three laws of motion is an interesting read that you can dig deeper into here.

To finalise, impulse relates to force: $$J=F\Delta t$$ and after a rearrangement, momentum change over time relates to acceleration: $$F\Delta t=\Delta p\quad\Leftrightarrow \\F=\frac{\Delta p}{\Delta t}=\frac{\Delta (mv)}{\Delta t}=m\frac{\Delta v}{\Delta t}\quad \rightarrow \quad F=m\frac{d v}{d t}=ma$$

So, the different formulations are equivalent (assuming constant mass $m$).

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^{* $\quad$‘Philosophiæ Naturalis Principia Mathematica’, Isaac Newton, 1st ed., vol. 1, 1687, (English translation published 1728)}

Answer 5

I favour your choice, $a = F/m$ and I would explain it to pupils as follows. That the acceleration of a body is directly proportional to the net force on it and is inversely proportional to its mass. Thus, $a \propto F/m$. We choose units of physical quantities in such a way that the constant of proportionality is zero. I would also emphasize that although $F = ma$ and $a = F/m$ are mathematically identical, the physics of the situation comes out clearer in the latter.

$F = ma$ has led several pupils astray into believing that force is equal to mass into acceleration and therefore concluding that 'there is force because the body is accelerating'.

Answer 6

I think all the answers are ignoring the fact that we are talking about a law . What are laws in physics? They are the corresponding axioms that define which solutions of the general mathematical set up for a physics theory fit the data, and can make the theory predictive.

Axioms mean they cannot be proven. As in mathematics, a theorem can replace an axiom and then the axiom becomes a theorem, the same with laws of physics, so it is OK to use different forms .In physics though, Occam's razor is used: trying to give to the laws the simplest form. $F=ma$ to me seems clearest, using the known measurable in the lab quantities of mass and acceleration, to define axiomatically what a force is.

Answer 7

There is no one form suitable for all. Concrete law should be expressed in a way most suitable to context needed.

We can even express Newton second law as: $$ \large{ F = m \cdot \begin{bmatrix} \partial ^{2}x \over \partial t^{2} \\ \partial ^{2}y \over \partial t^{2} \\ \partial ^{2}z \over \partial t^{2} \end{bmatrix} } $$

Or

$$\vec{F} = m \left(x''(t)\space\hat{\textbf{i}} + y''(t)\space\hat{\textbf{j}} + z''(t)\space\hat{\textbf{k}} \right) $$

So really, everything depends on context & notation.

Answer 8

I would say it actually depends on what you want to do with the formula. If you want an interpretation on relations between mass, force and acceleration, I would agree with you. I think it is good to see the consequence in terms of the cause (and a property of the object).

I think the second version is more useful for problem solving. If you solve a force or torque-related problem in Newtonian mechanics, you use force diagrams and calculate the net force in each direction, then apply Newton's Second Law. It is useful to have the geometric-dynamic part on one side and the rest on the other one.