The Newton unit

To talk about ‘how much force’ there is, we of course need a unit for force. Should that be a fundamental unit?

Nope. Force is not a fundamental property because it can be created from mass and acceleration via Newton’s 2^nd law. We can thus also derive its unit directly from that.

Newton’s 2^nd law, $\sum \vec F=m\vec a$, tells us that some force is some ‘mass times acceleration’, so ‘some grams times some metres-per-second-per-second’. If we stick to the usual SI base unit for mass, $\mathrm{kg}$ instead of the $\mathrm g$,¹ then it is ‘some kilograms times some metres-per-second-per-second’:²

$$\mathrm{[kg]\cdot \left[\frac{m}{s^2}\right]}= \underbrace{\mathrm{ \left[ kg\cdot \frac{m}{s^2}\right]}}_\mathrm{[N]}$$

In honour of the discoverer, let’s give this chunk of units its own name: the Newton, symbolised $\mathrm N$:³

$$\mathrm{[N]}= \mathrm{ \left[ kg\cdot \frac{m}{s^2}\right]} $$

Now we can say that ‘it takes 1 Newton to hold up an apple’, or ‘he hit me with 2500 Newton’ and write it as $F=1\,\mathrm N$ and $F=2500\,\mathrm N$.

For a feeling of the Newton unit, here is an overview of the force magnitude scale: