I’d like to share a joke with you. it’s a joke about systems of measurement, which
I think is pretty funny, but before I tell it I’m going to have to go over a few basic
concepts. I’ll try to make this quick.
so there’s this concept in metrology called “coherence”, where you can do math to
stuff that you’re measuring without needing to worry about converting the units. it’s
a pretty good property for a system of measurement to have.
like, okay, speed is just how far something goes divided by how long it takes, right?
but like, if you want to know how fast something went in miles per hour and you know how far
it went in feet and how long it took in minutes, just doing the division gives you the answer
in feet per minute, and you have to do extra math to get it in the actual units you want.
so, the set of units you’re using, the foot, minute, and mile per hour, don’t form a
coherent system of measurement.
to make a system of measurement coherent, you just need to make sure that you have one
standard unit for each fundamental quantity, and that all the quantities that can be derived
from those are given units derived from those standard units. so like, you can just decide
that you always measure length in meters and always measure time in seconds, and then in
a coherent system you would also always measure velocity in meters per second. that way you
can avoid all the extra math.
and in the International System of Units, there’s a whole bunch of units that are
derived from the meter, kilogram, and second, the units of length, mass, and time. so like,
force is mass times acceleration, so it’s measured in newtons, equal to kilogram-meters
per second squared.
you don’t need to use MKS (meter kilogram second) units, of course. you could just as
easily use a coherent set of FPS (foot pound second) units, and measure length in feet,
mass in avoirdupois pounds, time in seconds, and-
sorry, hold on, I had to look up how to say “avoirdupois” for this video and like,
I’ve been saying it like /ˌævwɑːɹduˈpwɑː/ in my head this whole time, because this is
like the most French looking word so obviously it’s going to be pronounced like that, but
no it’s apparently usually /ˌævəɹdəˈpɔɪz/. so, I don’t wanna see anyone correcting
me in the comments saying that I should be saying it the French way, because I checked
and it is usually pronounced /ˌævəɹdəˈpɔɪz/.
where was I? right, so you can measure force in poundals, equal to pound-feet per second
squared. there’s even other coherent metric systems, like CGS (centimeter gram second)
units, where force is measured in dynes, gram-centimeters per second squared.
all of these units, the newton, poundal, and dyne, are all defined the same way, just with
different base units. force is equal to mass times acceleration, and acceleration is just
distance divided by time squared, so you could really make a valid unit of force out of whatever
length, mass, and time units you want.
you could even use the FFF (furlong firkin fortnight) system, where force is measured
in firkin-furlongs per fortnight squared. the FFF system is pretty silly, but it’s
not the joke.
here’s the thing. that statement, “force equals mass times acceleration”, doesn’t
really mean that force has to be derived by multiplying mass by acceleration, right? I
mean, you can also say that “mass equals force divided by acceleration”.
the poundal isn’t actually the most common FPS unit of force. while a pound is usually
used as a unit of mass, there’s another unit that’s also called a pound that’s
actually a unit of force. if something has a mass of one pound, then the gravitational
force between it and the Earth is one “pound of force”.
that’s what it means for something to weigh one pound on Earth: it and the Earth are gravitationally
pulled towards each other with one pound of force. using this pound of force, you can
then divide out the FPS unit of acceleration, feet per second squared, and get a unit of
mass called a “slug”, which is about 14.6 kilograms, or 32 pounds of mass.
heck, in the standard MKS system, acceleration is measured in meters per second squared,
but since acceleration equals force divided by mass, we could just as easily measure acceleration
in newtons per kilogram instead. and since it’s a coherent system of units, we know
that one newton per kilogram precisely equals one meter per second squared.
and you know what, you could go even further, because over in the metric system you have
kilograms as a unit of mass, and over in the American system you have pounds as a unit
of mass, but also separately pounds as a unit of force. so you could absolutely measure
acceleration in pounds per kilogram.
I have no idea why you would want to measure acceleration in pounds per kilogram, but you
can! and, that idea got me thinking. what other base quantities could you use to form
a coherent system of measurement?
okay, now here’s the joke.
this is my new idea for a fundamentally different kind of coherent system of measurement: the
CCC system. the three fundamental units that the majority of the CCC units are derived
from are not length, mass, and time. in fact, none of those quantities are given their own
base units at all.
first is the standard unit of velocity, the speed of light. the speed of light, commonly
abbreviated to c, is a fundamental constant of the universe; it’s the fastest possible
velocity allowed by the laws of physics. in the CCC system, rather than representing velocity
as length divided by time, it’s represented far more elegantly as some multiple, or, more
accurately, some portion, of c.
the speed of light is equal to 299,792,458 meters per second. that’s not an approximation,
by the way. the speed of light is used in the standard definition of the meter, so,
by definition, that is the exact value of the speed of light in meters per second. oh,
and since US customary units are all defined according to their relationship to SI units,
I can also say that the speed of light is equal to 670,616,629 and 539/1397 miles per
hour.
the CCC system doesn’t have a unit of mass, but that’s fine because mass is famously
equal to energy divided by the speed of light squared. the speed of light is a fundamental
unit in the CCC system, so if we just add in a unit of energy we can use that to get
a unit of mass. the unit of energy I went with is the calorie, since it’s a unit of
energy that people are generally pretty familiar with.
a calorie is the amount of energy required to raise the temperature of one gram of water
by one degree Celsius. this definition isn’t very precise, because it depends on a lot
of different factors, so the standard definition actually used for the calorie is to say that
it’s equal to exactly 4.184 joules, that is, 4.184 kilogram-square-meters per second
squared.
and, again, because E=mc², you can divide this unit of energy by the speed of light
squared and get a unit of mass out of it: the calorie per speed of light squared.
since the speed of light is very fast, a calorie per speed of light squared is a very tiny
amount of mass. it’s equal to about 46.55 femtograms, a femtogram being a unit generally
used for things the size of bacteria. there are one billion billion femtograms in a kilogram.
of course, no system of measurement is complete without a unit of length. it makes sense to
derive a unit of length from the speed of light. you can do this by multiplying it by
some unit of time. for example, the speed of light times one year is a unit of length
called a lightyear. we could do that, but that’s kinda boring.
instead, we can generate the unit of length by dividing the speed of light by a unit of
frequency, that unit of frequency being our third fundamental base unit. the unit of frequency
I chose is middle C, a pitch found in some of the most common scales used in western
music. middle C is a generally agreed “medium” pitch, around the middle of a piano’s range.
the exact value of middle C is different depending on which tuning system you use, but it’s
most commonly tuned to around 262 hertz, cycles per second. since we’re using this as a
unit of measurement, we will need to pick one precise value, and it makes sense to just
use the most common standard tuning system, where middle C is equal to exactly 220 times
the fourth root of 2 cycles per second.
I don’t want to get into different tuning systems here, but if you want to know why
it’s tuned to an irrational number of cycles per second, Minute Physics has a pretty good
video that explains what the deal is with the equal temperament tuning system. if you
want to get more into the music theory stuff of it, there’s some good videos from 12Tone
and Adam Neely that someone watching this video who is interested in the music theory
of different tuning systems has almost certainly already seen.
anyway, using middle C as our standard unit of frequency and the speed of light as our
standard unit of velocity, we can generate a unit of length: speeds of light per middle
C.
once again, the speed of light is very fast, and middle C is like, a middle frequency.
so, a speed of light per middle C is a very long distance, equal to around 1.146 million
meters, or like 712 miles.
so with our units of velocity, energy, and frequency, we have now generated units of
length and mass, which means the CCC system is now just as capable as the MKS, CGS, and
FPS systems! well, almost. we do still need a unit of time, but time is just the inverse
of frequency. our unit of time is the “per middle C”, which is like, what, 0.004 seconds?
anyway, now we can do some fun stuff. let’s pick a bunch of quantities and derive some
units for them!
I think the easiest one to start with is the unit of power, which means... power. power
is equal to energy over time, which means it’s equal to energy times frequency, so
in the CCC system power is naturally measured in middle C calories.
one middle C calorie is about 1.1 kilowatts, or one and a half horsepower. I think this
is the first derived unit in the CCC system that’s actually a convenient size, probably
just because its definition doesn’t involve the speed of light anywhere.
power is equal to energy over time, but it’s also equal to force times velocity. that naturally
means that in the CCC system, force is measured in middle C calories per speed of light.
that’s about 3.7 micronewtons.
oh, and now that we have a unit of force, we can finally figure out what this system’s
equivalent of newtons per kilogram is!
somewhat expectedly, the unit of acceleration is expressed as frequency times velocity,
which is really just another way of saying velocity over time.
there’s also some pretty obvious units for area and volume you get by just squaring and
cubing the speed of light per middle C, and since this stuff is exponential it gets pretty
huge pretty quick. the unit of volume is more than the total volume of the Earth’s oceans.
and that means anything that’s usually expressed as something per volume is going to have a
unit that’s comically small. like, let’s take density.
the unit you end up with is... this, which is about 3 times 10 to the minus 35 kilograms
per meter cubed.
oh, and using that unit of area, we can generate a unit of pressure, because pressure is force
divided by area.
the thing we get is about 2.8 attopascals. one thing that’s interesting about these
units of density and pressure is that they show a somewhat non-trivial restatement of
E=mc²: pressure is equal to density times the speed of light squared.
anyway, while there is more you can do with these base units, you can’t do everything.
there are some things you can’t do with just three C’s. the SI has units for seven
different fundamental physical quantities, so if we want this to have the same capability
as SI, we’ll need to extend it into The Seven C’s.
I’m not going to go too deep into The Seven C’s, mostly because they don’t actually
add that much to the aesthetic comedy I was going for with this system. I will define
them here, though.
for a couple of the SI quantities there’s just already common units that start with
C, so like temperature is in degrees Celsius and luminous intensity is in candelas. pretty
basic stuff.
we’re still missing a lot of units for electric stuff, so we can also add in the SI unit of
charge, the coulomb, which is one ampere times one second. we can then use the coulomb and
the other CCC units to get units for all your favorite electrical quantities.
multiplying the coulomb by middle C, we get a unit of current equal to like 262 amps.
voltage is equal to energy divided by charge, so in The Seven C system it’s measured in
calories per coulomb, which are like four volts.
you can divide charge by voltage to get capacitance, which gets you square coulombs per calorie,
around 239 millifarads.
there’s also resistance, which is voltage divided by current, giving you calories per
middle C coulomb squared, equal to about 0.016 ohms.
and of course, you can’t have resistance without conductance, so the unit of conductance
is just the reciprocal of the unit of resistance, so it’s like 62 and a half siemens.
oh, fun fact, the SI unit of conductance, the siemens, is sometimes called a “mho”
instead, which is ohm backwards, and I am 100% positive that the only reason anyone
calls it that is so they don’t have to call it a siemens.
anyway, all that’s left from the seven SI units is a unit of “amount of substance”,
which, I don’t think is entirely necessary? like, a mole isn’t really a unit, it’s
just a number. so, let’s say that the Seven C unit of amount of substance is just like,
a hundred. actually, having a hundred helps a lot for all those units that are comically
wrong sizes.
like, instead of using speeds of light per middle C as the unit of length, we could instead
use speeds of light per million middle C.
and by the way, the symbol I’m using for a hundred here is the Roman numeral one hundred,
not to be confused with the symbol for the coulomb, which is a sans serif capital letter
C.
I don’t know how to end this video.