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Why are we using Newton’s second law of motion to re-define the units of force when we must already have units to measure the force?

According to Newton’s second law:

$$a\propto F \tag{1} $$ $$a\propto \frac{1}{m} \tag{2} $$ $$a\propto \frac{F}{m} \tag{3} $$ $$F = k\cdot ma \tag{4}$$

From here onward the textbooks re-define the unit of force (using Equation 4) as that force that produces the acceleration of 1 $\frac{\textrm{m}}{\textrm{s}^2}$ of a body of mass 1 kg (thus making $k=1$). But why are we re-defining the unit of force when we must already have a unit to measure force?

Relation 3 (which comes from 1 and 2) can only be established through experiments. Now since we are doing experiments we must be measuring mass, acceleration, and force. Since we are measuring force, we must be having some units for force to work with. Now my question is when we are already having the units to measure the force then how can we use Newton’s law to redefine the unit of force?

Wrzlprmft
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    You are talking about defining the unit of force called the Newton. You aren't talking about the actual definition of force. If $m$ is measured in kilograms and $a$ in meters per second per second, if $k=1$ then you can say that the result of the calculation will have units of Newtons. Can you please clarify your question. Do you mean to talk about the unit, or the actual definition of a force? – BioPhysicist Apr 07 '20 at 18:53
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    I don't understand your question. Do you ask, why do we use $N$ instead of $kg m/s^2$? – Semoi Apr 07 '20 at 18:54
  • there was a different Definition of force , the pond or kilopond, the weight of a mass of 1kg, but this is not constant over different locations, but still one one specific location can compare forces, so you can find the proportions you mentioned, and then you define the unit by the other units kg, m,s, one tries to have as few units as possible, and to have definitions that you can use in any location like in space or on the moon. – trula Apr 07 '20 at 20:12

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Imagine that you have a relatively stiff spring in your hands. Hold one end of the spring in each hand, and hold your left hand stationary while you extend your right hand away from you. You will observe that the spring gets longer, and that you feel it pull on your left hand. Furthermore, the longer the spring is, the harder it pulls on your hand.

Terminology: We say that the stretched spring exerts a force on your left hand.

At this point, you would be justified in claiming that there is a one-to-one relationship between the length of the spring $\ell$ and the force exerted on your left hand $F$. This spring is now your force gauge - you can now subject various objects to the same force by attaching the spring and ensuring that it maintains a constant length.

Now that you have an instrument, consider an experiment. In your laboratory, you have a long, frictionless track (like an air track) and a cart whose mass you can change by stacking blocks on it. You notice that when you push on the cart, it accelerates, and you would like to quantify this relationship.

You attach your spring to the end of your cart and pull in such a way that your spring maintains a constant length of, say, $\ell=5$ cm, and therefore exerts a consistent force $F$. You measure the resulting acceleration, and then repeat the experiment by stacking different combinations of blocks on it. At the end of the experiment you find that the acceleration of the cart is inversely proportional to its mass when the force exerted upon it is held fixed.

For the next experiment, you take several identical springs and attach them all to the block "in parallel." You reason that if one spring exerts a force $F$ on the block, then $N$ identical springs (all stretched to the same length as the original) will exert $N$ times the original force. Under this fairly mild assumption, you hold the mass of your cart fixed and measure its acceleration when subjected to various different forces. At the end of your experiment, you find that the acceleration of the cart is proportional to the force exerted upon it when its mass is held fixed.

At the end of your long day of experimentation, you are therefore led to postulate that

$$F \propto m a$$ $$\implies F = k ma$$ for some constant $k$. If you choose a value for $k$, then this relationship allows you to assign a numerical value to $F$, which would be quite useful. It is, after all, much easier to report a numerical value than to say "the force exerted by my favorite laboratory spring when it is stretched to a length of $5$ cm."

All that remains is to choose a value for $k$. Because you love to measure mass in kilograms, distance in meters, and time in seconds, you say

This quantity force is to be measured in units of Newtons, such that a total force of $1$ N will accelerate a $1$ kg object at $1$ m/s$^2$.

This amounts to choosing $k=1 \frac{\text{N}}{\text{kg}\cdot\text{m}/\text{s}^2}$.

J. Murray
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  • Until you have a definition of units, you do not have a quantitative definition of force which is required in any scientific discussion. – Charles Francis Apr 07 '20 at 20:28
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    @CharlesFrancis I don't need a system of units to discuss the fact that I have to push harder to lift two books than I do to lift one book. The concepts of force, mass, and acceleration are logically distinct from one another, and Newton's 2nd law is a physical assertion which relates them. You are claiming that Newton's 2nd law is a tautology. – J. Murray Apr 07 '20 at 20:36
  • But if you are using only a qualitative notion of force, your argument is not appropriate to a physics site. You may as well claim you are doing work when you hold a weight at constant height. The same word can have many definitions, but we choose unique scientific definitions in so far as is possible. And I have made no tautological claim. I said force was defined by the law, not by the units. The units are defined by the same law. – Charles Francis Apr 07 '20 at 20:48
  • @CharlesFrancis I am not being at all qualitative in my answer, in which I give an operational definition of force as the influence exerted by e.g. a spring. But I'm not sure what you mean when you say you are not making a tautological claim - if the concept of force is defined by Newton's 2nd law, then Newton's 2nd law is true by definition. – J. Murray Apr 07 '20 at 20:55
  • A definition must be consistent to be a valid definition. This is an empirical definition which must be consistent with observation. You describe experiments to confirm that the definition is valid. But springs, being somewhat variable in reality, do not give us a useful definition of units. They are calibrated by Hook's law to the definition we actually use. – Charles Francis Apr 07 '20 at 21:10
  • @CharlesFrancis This discussion has gotten off track. The key point of contention is that I claim that force, mass, and acceleration are conceptually distinct, but are related by the (non-definitional) claim that $F=ma$. You are claiming that $F=ma$ is the very definition of force, and that the notion of force cannot be separated from those of mass and acceleration. I think we've heard each other out, but we seem to have settled on a fundamental disagreement. – J. Murray Apr 07 '20 at 21:27
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Force is defined by Newton's second law, and has no other (independent) definition. Since force is defined by Newton's second law, its units are also defined by Newton's second law. Whatever instrument you might be using to measure force has previously been calibrated to that definition.

Charles Francis
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    I am pretty confident that the concept of physical force has been around much longer than Newton. Newton’s laws may allow us to arrive at modern, less tedious definition and so on, but that’s not the only way to do it. I could also define force via pressure, springs, etc. – Wrzlprmft Apr 07 '20 at 19:58
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    Yes, for example there was the Aristotelian idea that a force was needed to keep a body in motion. And there were discussions by thinkers like Galileo and Leonardo da Vinci. The quantitative definition, however, in modern science is Newton's second law. You refer only to things which require a prior definition of force. – Charles Francis Apr 07 '20 at 20:09
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    @CharlesFrancis I would argue that Newton's 2nd law is not a definition, in the sense that it is in principle possible that $F = ma + \gamma a^3$ for some $\gamma$. The LHS and RHS of Newton's 2nd law can be measured independently, and experiment suggests rather strongly that they are proportional to each other (with the choice of proportionality constant fixed by a definition of the unit of force). – J. Murray Apr 07 '20 at 20:19
  • @J.Murray, of course a definition could be replaced by a different definition. That is precisely why it is a definition. Naturally we choose a useful definition. I refer you to the definition of the Newton "One newton is the force needed to accelerate one kilogram of mass at the rate of one metre per second squared in the direction of the applied force". – Charles Francis Apr 07 '20 at 20:24
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    @CharlesFrancis That is a fine definition of the Newton, but my point is that it is not trivial that if 1 N causes a 1 kg mass to accelerate at 1 m/s$^2$, then a 2 N force will cause that same mass to accelerate at 2 m/s$^2$. That statement has physical content, which cannot be ascribed to a matter of definition (which would make it a tautology). – J. Murray Apr 07 '20 at 20:30
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    Yes, the definition of force by Newton's second law does have additional content to the definition of units. It is worth noting that this definiton does break down in relativity and in quantum theory, but conservation of energy and momentum remain and is equivalent to Newton's second and third laws together. – Charles Francis Apr 07 '20 at 20:43
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It’s all down to historical convention and how relevant a certain relation is.

In the case of force, if we did not set $k=1$, we would always have to work with this constant when translating acceleration to force (in a classical setting). We also would have to have an eight SI unit for force. This would be possible, but impractical.

You might as well ask why we not have a separate unit for velocity, though the relationship $v ∝ \frac{d}{t}$ (with $d$ being the distance) needs units for velocity, distance, and time to be empirically established. The only difference here is that the relationship is so obvious that nobody managed to get their name eternalised for discovering it.

Contrast this with the relationship $m = \frac{E}{c^2}$: The mass of an object is proportional to the energy required to create it out of nothing. Like Newton’s second law, we could use this relationship to equate energy and mass in terms of units and not worry about this annoying constant $c$ anymore. This would also allow us to equate distance and travel times of light. So, why do we not do it? This relationship was established much later historically when the physical unit system was much more settled. Also, the respective phenomena do not pervade our everyday lives, and thus using the same unit for time and distances would cause more confusion than good. However, natural unit systems (which are made for use by physicists and not for everyday life), actually do directly equate mass and energy and set $c=1$.

The question Why is the meter considered a basic SI unit if its definition depends on the second? may also help you.

Wrzlprmft
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