This post will be a review of measurement units, specifically metric or “SI” (International System) units: how they are defined, how they are combined to make other measurements, and how they are used. ¹ Sounds pretty damn dry, yes? Not so! We will have some fun with this, toward the end. But first…

Some history

So, what do you want in a “unit”? Ideally, it should be convenient, i.e. applicable on the scales you need (for example, you don’t want to be measuring your property line in light-years), unchanging, and easily duplicated. Your meter should equal my meter, and today’s kilogram should equal next week’s.

The basic units are those of time, length (which can also mean distance or position, which amount to the same thing), and mass. The unit of time is the second (sec or s), the unit of distance is the meter (m), and the unit of mass is the kilogram (kg). ²

The second came about by dividing the day into 24 hours, each of which was comprised of 60 minutes, each of 60 seconds: thus the second was 1/86,400 of a day. ³ This was sufficient until the last few hundred years or so. This definition was supplanted by accurate clocks, which measured out seconds. Eventually, this led to atomic clocks (more on this later).

The meter was originally defined as 1/40,000,000 (one forty-millionth) of the Earth’s circumference, or 1/10,000,000 of the distance from the Equator to the North Pole. Needless to say, this was not easily duplicated by whomever needed a “standard” meter (scientists, surveyors). Eventually a Standard Meter was manufactured: the distance between two tiny marks on a platinum-iridium bar, kept at constant temperature in a vault in Paris. ⁴ You could visit the vault, and duplicate the scratches on your own bar, for your own use.

The kilogram was somewhat easier: once you had the meter, you could use water as your reference: a kilogram was the mass of a cube of water 10 cm on a side, at standard temperature. We now call this volume a liter. Eventually, the same tactic was used as for the meter, and the kilogram was defined as the mass of a cylinder of platinum-iridium. (Guess where?)

You may have noticed that we have skipped an important unit, that of temperature. The unit of temperature is the Kelvin (K) ⁵, which begins at absolute zero. I may come back to this in a subsequent post when we talk about entropy.

How they are defined now

The most basic defined unit is the second; all the others derive from this in interesting ways.

The definition of the second is: 9,192,631,770 cycles of the radiation that gets an atom of cesium to vibrate between two particular energy states. OK then. We don’t have to understand that, but note: it can be measured as accurately as your apparatus will allow, it can be duplicated in any sufficiently-equipped lab, and it will never (so far as we know) change. Good one!

The meter is now defined by the second and the speed of light, which is also constant and unchanging. Today, the meter is defined to be the distance light travels in 1 / 299,792,458 seconds. Again, can be duplicated, given the right equipment.

The kilogram is a bit more complicated. Only 3 years ago, a consensus was reached: the platinum kilogram in Paris was used to obtain a very precise result for Planck’s constant, which contains “mass” as one of its terms; now this accepted value of Planck’s constant is the basis for the definition of the kilogram. ⁶

Now the fun part starts

So these are the units for time, position, and mass. Now we will look at their “offshoots”.

Position, velocity, and acceleration

Position is measured in meters; velocity, which is the rate of change of position, is measured in meters per second, or $ \frac{m}{sec}$. Acceleration, the rate of change of velocity, is measured in meters per second per second, or $\frac{m}{sec^2}$. Simple enough.

Force, energy, and power

Force is pretty intuitive: it is just how hard you are pushing. Remember Newton’s first law? Force equals mass times acceleration, right? Therefore the unit of force is $\frac{kg\cdot m}{sec^2}$…acceleration unit, times kilograms (mass). The unit of force in the SI system is the Newton, abbreviated N. A force of one N, applied to a mass of one kg, will accelerate the mass at one meter per second per second.

Energy is a bit less intuitive, at first. It is defined as the force on an object, times the distance over which the force is applied. In other words, if you lift a 1 kg object by one meter, against Earth’s gravity (which has a force of 9.8 N, by the way), you will have added 9.8 energy units to the object. So the unit of energy will be force times distance, ie $\frac{kg\cdot m^2}{sec^2}$ (our force unit, multiplied by another distance unit). The unit of energy in the SI system is the Joule, or J. ⁷

Power (which is often used, erroneously, as a synonym for energy) is the rate that energy is added (or removed, or otherwise changed). ⁸ So power is measured in Joules per second. The unit of power in the SI system is the watt, equal to one J per second. We can rearrange this, and note that a J is a watt-second (that is, a power of one watt, for one second). Or: 3600 J=1 watt-hour. Or: 3,600,000 J=1 kilowatt-hour. Costs you about a dime to buy 3.6 million Joules! Such a deal!

One more unit

A commonly-used (unfortunately), very large, unit of energy, the megaton, is used to quantify the energy released in a nuclear explosion. It was first defined as the energy released by a million tons of TNT, exploded all at once (which is impossible). It is equal to about $4.18×10^{15}$ J.

Okay,NOW for some fun

Using Einstein’s formula, $e=m\cdot c^2$, let’s figure out how much $ worth of electricity you can get by converting a kg of mass (a liter of water, for example) into electricity.

c is the speed of light, $2.9979×10^8$ m/sec. We will round up to 3, OK? That number squared is $9×10^{16}$, then times the mass, which I conveniently called 1 kg, gives $9×10^{16}$…what? Joules, right? Yep! Well we already figured out how many J there are in a kilowatt-hour, $3.6×10^6$, so we can divide: we get $2.5×10^{10}$ kilowatt-hours from our 1 kg of mass, or (at 10 cents per kwh) 2.5 billion dollars’ worth of electricity!

A final observation

Going back to our number for J in a megaton, we can divide that, and find that our 1 kg of mass, converted to energy, yields about 6 megatons. The largest H-bomb ever exploded, the Tsar Bomba (Soviet), yielded 50 megatons: a bit more than 8 kg of mass were converted to energy in the explosion. ⁹ Kinda scary.

There will be very little here about the English system of units (pounds, feet, etc), which, ironically, is no longer used officially in England; in fact, it is the official system in only three countries: Liberia, Burma (Myanmar), and, of course, the good old USA.
You may wonder “why the kilogram, rather than the gram?” So do I. It just seems to be more convenient.
The second is the only unit in common between the SI and English systems.
An interesting side-story is how the French, advocating the metric system, ended up with this prestigious possession; the English, in return, kept the definition of “zero degrees longitude” in England, at the Royal Observatory in Greenwich, where it remains to this day.
Not “degrees Kelvin”, as you learned in school!
If that sounds like circular logic: well, it is.
Herein hangs a tale…the MCAT (Medical College Admission Test) covers inorganic and organic chemistry, biology, and physics. When I took the MCAT in 1997, I had recently taken courses in all of the above, except physics, which I had taken in 1971. Not exactly fresh on the physics questions! One question asked how much energy an electron gains in a certain circumstance. Well, I had no idea…but only one of the list of answers was even in the right units! Bingo!
Power is your hourly wage; energy is your bank balance.
Matter is nothing more than a very concentrated form of energy!

Units