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This question follows off the other question posted here: How do we know that $F = ma$, not $F = k \cdot ma$

I hear that when writing $F = kma$ we can always set $k$ to be $1$. How is that possible?

I think the confusing part is where the newton, along with the kilogram and $\mathrm{m/s^2}$, which are the units which are widely used in the scientific world, just so happen to be the exact units that allow for $k$ to be $1$, and not $2.1$, $3.6$, $9992$, etc. Why is this so conveniently possible?

Please note, I am not asking why it is $F=ma$ and not $F=kma$ assuming that $k=1$.

David Z
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user352935
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    Possible duplicate of How do we know that $F = ma$, not $F = k \cdot ma$ – AccidentalFourierTransform Feb 26 '17 at 22:22
  • I did link to the original question, but mine is based off of that question. It's not the same question. – user352935 Feb 26 '17 at 22:28
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    I fail to see the difference. You should make it clear what you want to know, and why this post is different from the linked one. – AccidentalFourierTransform Feb 26 '17 at 22:29
  • Done. Please explain why you don't see a difference and I will try to edit it further – user352935 Feb 26 '17 at 22:32
  • I still fail to see the difference. You are asking why, given the formula $F=kma$, $k=1$, and the other question is also asking why $k=1$. ("...how do we know that this is the case? How do we know that the constant isn't 2...?") Though thanks for editing the question to try to clarify why it's not a duplicate. That is the proper way to handle it when someone says your question is a duplicate and you don't think it is. – David Z Feb 26 '17 at 23:43
  • @DavidZ I think the user is asking about why the Newton specifically is the unit that makes $k=1$. It does seem like a pretty crazy coincidence, until you realize that the Newton was defined after $F=ma$. For example, we frequently use units where $k\neq 1$ in E&M, since the units were defined before a complete theory of E&M was. – Jahan Claes Feb 27 '17 at 02:02
  • @JahanClaes Ah, if that's the question, then I'm no longer so convinced that it's a duplicate as to force the question closed. (Even if personally I still don't see how this question isn't addressed by the other one.) I'll edit the latest revision to clarify that and then reopen it. user352935, please take a look at my edit and make sure it still reflects what you wanted to ask. – David Z Feb 27 '17 at 02:33
  • Here's another answer I've posted which is probably relevant. – David Z Feb 27 '17 at 02:38
  • This has a neat duality with the other question. On the dimensional side, force is defined by Newton's second law to satisfy $F = ma$. On the units side, the unit of $F$ is defined to make $[F] = [m][a]$ where $[x]$ means the number associated with the dimensionful quantity $x$ in SI units. – knzhou Feb 27 '17 at 02:53
  • It does, the other answer though, although extremely detailed, is too complex for me to understand the reasoning behind having k=1. I think the simplicity from Jahan Claes response (in comments) was probably better in addressing what I really wanted to know. – user352935 Feb 27 '17 at 03:26
  • @user352935 -- Essentially you are asking whether force is a fundamental or a derived quantity. If it is fundamental, then it would be more correct to write $F=kma$, with units chosen so that $k=1$ becomes a convenience. If it is derived, then using $F=kma$ with $k \ne 1$ means you have an inconsistent set of units. Related: http://physics.stackexchange.com/q/11648/. – David Hammen Feb 27 '17 at 05:13

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Let's say you've chosen some unit of mass, the jilogram, and a unit of force, the Mewton. Let's denote by $F_M$ the force measured in Mewtons, and by $m_j$ the mass measured in jilograms. Let's say that in these units, you find

$$ F_M=4m_ja $$

This is not Newton's law! But maybe we can make it work. For example, if we define a new unit of force, the Newton, such that (1 Newton)=(4 Mewtons), then you'll agree that if we measure force in Newtons, we'll have $F_N=F_M/4$. Thus, $F_M=4F_N$. If we substitute this in to our first equation, we find

$$ 4F_N=4m_ja $$ or $$ F_N=m_ja $$ So we've recovered Newton's law, by choosing a different unit of force!

Now, let's say we don't want to redefine force. Perhaps we think the Mewton is a really good unit of force, for some reason. Then we can still choose a new unit of mass. Let's define a kilogram by (4 kilograms)=(1 jilogram). Then if we measure things in kilograms, we'll have that $m_k=4m_j$. Substituting this in, we get

$$ F_M=m_ka $$ which is again Newton's law.

Of course, if you randomly chose to measure mass in jilograms and force in Mewtons, and didn't want to redefine either force or mass (or distance or time), you'd be stuck with a $k\neq 1$. But most systems of units define three of the four quantities (length, time, force, mass), and then choose to define the last one to make Newton's law have $k=1$, just like we did above.

Jahan Claes
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  • Thanks for the response. I think the part where 1 Newton, along with 1 kg and 1m/s1^2 which are the units which are widely used in the scientific world, just so happen to be the EXACT units to allow for k to be 1, and not 2.1, 3.6, 9992, etc. Why is this so conveniently possible? – user352935 Feb 26 '17 at 23:58
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    @user352935 Because Newton got to define the Newton! We didn't have that unit for force before Newton's second law. We defined the Newton precisely so that k=1. It wasn't a coincidence that the units used by the scientific world happen to make Newton's law nice. It's that we literally chose our unit for force for the sole purpose of making Newton's law nice. In face, now $F=ma$ is a definition for what we mean by the Newton. The Newton is defined to be the unit that makes $k=1$ when you measure mass in kg and a in $m/s^2$. – Jahan Claes Feb 27 '17 at 01:33
  • @user352935 The fact that we defined a unit specifically to make Newton's law look nice should tell you something you hopefully already knew: namely, Newton's laws are very, very important. – Jahan Claes Feb 27 '17 at 01:34
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    Wait, you're saying that the units that we use TODAY, have come from someone trying to make that specific Newton's law convenient for them? I think I just had a Eureka moment. – user352935 Feb 27 '17 at 01:54
  • @user352935 Yes, that is exactly correct. See https://en.wikipedia.org/wiki/Newton_(unit)#Definition

    Note people have also tried to do this with electricity and magnetism, which is why you'll see some unit systems where the electric force is given by $F=\frac{ke_1e_2}{r^2}$, and other unit systems with no $k$.

     – Jahan Claes Feb 27 '17 at 01:58
  • @JahanClaes -- Newton did not define the Newton. The Newton was defined about 200 years after Newton wrote his Principia. Newton wrote in terms of proportionalities. You will find nothing like $F=ma$, let alone $F=kma$ in the Principia. – David Hammen Feb 27 '17 at 05:04