Why are we using Newton’s second law of motion to re-define the units of force?

Question

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?

Answer 1

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.

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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.

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

Answer 2

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.

Answer 3

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.