The measure of life

Every now and again while, when running though a long physics calculation, I stop for a minute to bask in the beauty of the SI system. I recommend you do the same.

I appreciate that as a reasonable percentage of my audience is American, you view yourselves as ‘not using the metric system’. That’s a bit wrong. Americans, what is the smallest measure of time you use? What do you use as a base measure for electric current? What measure do you use for working out relative mass? The answer to those three (in order of decreasing popularity) is the second, the amp and the mole. All of which are SI units of measurement, derived from the metric system. You lose, America.

At this point I should probably explain the distinction between the metric system and the Système International (as SI is properly called) of units. The metric system was created in Paris immediately after the French Revolution, in response to the ‘Measures Crisis’ and grain shortage. In Paris around this time, there were over 250,000 different types of measurement in regular use – a system that is fantastically, extraordinarily bad for trade and very easy to exploit (for more detail, see this episode of Radiolab). Just before the revolution, a grain shortage caused a massive rise in flour prices – but bakers couldn’t raise shop prices, or a mob would literally kill them. So you switch to a slightly lighter unit of weight and keep the price the same – sell less bread for the same price.

So when the Revolution came, one of the things that the General Assembly was crying out for was standardized measurements. But (this is almost a direct Radiolab quote) not a measurement based on some arbitrary human thing like a king. Let’s use something natural, unchanging, beautiful. Like the Earth.

So they measured the distance from the North Pole to the Equator by way of the Paris Meridian, divided that by 10 million and got the metre. They divided that by 10, made a cube of that side and got the litre. Then they filled that with water, weighed the water and got the kilogram. And the metric system is born.

And from that, they then got lots more units. They had lots of other things that needed measuring (like electric current, power, pressure, acceleration and so on), so they took those three units and extrapolated. From those, you get units like the gal, dyne, erg or barye. Or m/s² (acceleration), Newton (force), Joule (energy) and Pascal (pressure).

Which is all fine and well. But then, around 1870, they remeasured the diameter of the Earth. And it was wrong. 1975 metres out. So no big deal, right? It only affects the metre, right? So we reconfigure the metre, and everything is cool. Right?

Wrong. Remember the definitions. A litre is a tenth of a metre cubed – the metre is wrong, the litre is wrong. The kilogram is the weight of a litre of water – the metre is wrong, the litre is wrong, the kilogram is wrong. The dyne is the force needed to accelerate 1 g of water at a rate of one centimetre per second squared – the gram and the centimetre are both wrong, the dyne is wrong.

When you redefine the metre, you redefine everything. Which would be chaos – imagine what would happen if the government suddenly declared that the second would now last 1.3 seconds. Now do that with every single measurement in the world. Anarchy would ensue.

So the General Conference on Weights and Measures shook things up. They created a new system, with simple (non French-sounding) names. They created the Système International. So much for dropping the French.

The SI system has evolved slightly over time in definition and stuff, so rather than go back to the 1940s and 50s and work forwards from there, I’m going to start now.

There are seven SI base units, each of which is measured by the fundamental forces of the universe (apart from the kilogram, which done fucked up). They are as follows (fundamental force in brackets). The metre (speed of light in a vacuum), second (vibration of caesium atoms), ampere/amp (electrical things what I don’t understand), kelvin (boiling/melting point of water), mole (number of molecules in 12 grams of carbon-12) and candela (a certain type of brightness). Plus the kilogram, which will soon be measured by either magnetic repulsion or a literal atom count. But that is a whole other story which will almost certainly be covered at a later date.

So seven measurements doesn’t sound like a lot, right? That’s true – it would take more than that to measure anything you wanted measuring. And that is where half of the incredible beauty of the Système kicks in.

As well as those seven base units, there are also countless derived units – units which you get by multiplying out those seven bases plus mathematical constants. Any constant is just a factor that is the same at all times. So a mathematical constant just means a number, like 2 or -1 or √109. A natural or physical constant is a force of nature, that you can’t predict or calculate, but have to measure. The distinction of the 7 base units is that they are derived from physical rather than mathematical constants. In case you were wondering.

The watt is a unit of power. It is calculated by energy over time.

P(watts) = E(joules) / t(seconds)

So watt is a derived unit, taken from a base unite (seconds) and another derived unit (joules). What about joules?

E(joules) = (m(kilograms) * 10) * d(metres)

There you go. 1 joule is described as ‘the amount of energy required to lift a weight of 1 Newton (100 g) a distance of one metre’. So we can actually express the watt equation as:

P(watts) = (m(kilograms) * 10) * d(metres) / t(seconds)

But obviously it’s easier to substitute. And thus, using three SI base units and a mathematical constant, we can describe the watt. Which you see written on the side of lightbulb boxes. So the real meaning of a 60 W lightbulb is:

‘This lightbulb uses as much energy that if it were transferred entirely into kinetic energy, could lift a mass of 6 kilograms one metre vertically in one second, if such an event were to occur at sea level on the Earth’s surface’.

And now you will never be unable to un-know that fact. You’re welcome.

Side note: The ’10’ isn’t actually a mathematical constant, but it might as well be. In physics all weight is described in newtons, which factors in the strength of the gravitational field. So on the moon, your 60 W lightbulb could lift about 36 kg rather than 6.

So that’s half the beauty of the SI system. The second one is much more simple.

When you are doing a calculation in the imperial system and you need to change the magnitude of the calculation(changing a number from inches to miles, for example), you need to check what numbers to use. Because the difference between these measurements changes. To go from inches to yards, you multiply by 36. But to go from yards to miles, you multiply by 1760. Which is a right old pain in the arse.

But in the SI, everything is in powers of tens. Which means you can switch between orders of magnitude with fantastic fluency. Take my piece on the size of subatomic particles. In that, I was switching between measurements in kilometres (10^3 metres) to picometres (10^-12 metres), often in the same calculation. In imperial, this would mean multiplying by 7 different numbers. In metric, to get from picometres to kilometres, you multiply by 10^15. Or 1,000,000,000,000,000.

Here’s an example. Last week, someone asked me if it would be plausible to move the UK to Mars. So I estimated the volume of the UK and underlying mantle in kilometres (1000*500*5000) to give a number in cubic kilometres. I then divided that by 1 billion to get litres and multiplied by the volume of rock (roughly 3 kg per litre) to get weight in kilograms. Then divide by a thousand to get tons. Then I happen to remember that to get to orbital speed you need around 15 times your own weight in fuel. I then divided that total by the SI prefix to get a round number. The answer?

120 petatonnes of weight to orbit, or 120,000,000,000,000,000 tonnes.

I did that in my head. You can go and eat your imperial system, America.