There's a website called what if where
cartoonist and engineer Randall Monroe answers ridiculous hypothetical
questions with precise scientific detail. If you're like me, the family's
resident science kid who gets all the science-based Christmas gifts, then you've
probably already seen this in book form. There's one particular article I want
to draw your attention to: Droppings. If you went outside and lay down
on your back with your mouth open, how long would you have to wait until a bird
pooped in it? Monroe answers this question with a single equation containing a lot of
funny units, like poops per bird per hour, in which everything cancels out to
give a single figure of 195 years. On the topic of weird unit cancellation, Randall
then goes on a tangent into a very commonly used unit which has a similar sort of strange
cancellation: gas mileage, or fuel efficiency. I recommend just reading the article since it's
very short, but let me give a quick rundown. In the US fuel efficiency is measured in miles
per gallon; how many miles can you drive on a gallon of fuel. But in Europe and also here in
Australia it's measured in the reciprocal form, litres per 100 kilometres. That is, how many
litres do you need to drive 100 kilometres. But a litre is a unit of volume equal
to a thousand cubic centimetres, and a kilometre is a unit of length equal to
a thousand metres, so canceling things out, we find that one litre per 100 kilometres is
actually equal to 0.01 square millimetres. You can do the same thing with gallons per
mile, the reciprocal of miles per gallon, and a typical gas mileage of say 20 miles per gallon
corresponds to about 0.1 square millimetres. There's a nice physical interpretation of this
unit: imagine driving alongside a thin tube of fuel that feeds into your fuel tank. Your gas
mileage is the precise cross-sectional area of the tube chosen so that the fuel in your tank always
stays at the same level as you keep driving. Units are weird, and in my undergraduate
studies I encountered a lot of particularly cursed units. This video looks at four of
them, which gradually build in cursedness. Let's start with another unit
in common usage: kilowatt hours. When we say something like "miles per
gallon" or "metres per second" we have a simple intuition for that. How many miles
can you drive on one gallon of fuel? How many metres is something moving each second?
But just as you can divide units you can also multiply, and this has a different sort of
intuition. If your washing machine is running at a power consumption of 2 kilowatts and it
runs for 5 hours over the course of a week, the total amount of energy it uses in
that week is the product of the two, 10 kilowatt hours. Like miles per gallon though we
can do some unit cancellation. A watt is actually a joule per second – how many joules of energy
is something using every second – so we've got energy divided by time multiplied by time.
And since there's 1000 watts in a kilowatt and 3600 seconds in an hour, one kilowatt
hour is actually equal to 3.6 megajoules. You can maybe see now why I think kilowatt hours
are cursed. Let's say you wanted to estimate the distance from your house to the beach and let's
say it takes you 20 minutes to drive there, and accounting for traffic your average speed
is about 60 kilometres per hour. Since distance is speed times time, you convert the 20 minutes
into a third of an hour and multiply by 60 km/h to get 20 km. That's what a normal person would do.
Writing 'kilowatt hours' is like multiplying the 60 km/h by 20 minutes and then writing your
answer as 1200 kilometre-per-hour-minutes. We'll see more of this cancellation later
in the video, but first I want to say something about this general process
of multiplying and dividing units. Most physical constants have units. For example,
the gravitational constant which relates mass to gravitational force has units of cubic metres
per kilogram per second squared. Let's see where this unit comes from. The gravitational force
between two objects is proportional to their masses and inversely proportional to the square
of the distance between them. That means that the gravitational force between two objects is
equal to the product of their masses divided by the distance between them squared, and then
multiplied by some conversion constant G, which we don't know yet but is the same for every
pair of objects. To verify this you can either do an extremely precise measurement with extremely
precise equipment or you can do a bunch of maths to this model to derive the orbits of the planets
and then do some astronomy to see if it lines up with the actual orbits of the planets. If you
do this and you're working entirely in SI units you'll get that G is about 6.67x10^(-11). But
there's a problem: the units in this equation don't match up. We've got force on the left and
mass squared over length squared on the right, so let's rearrange things to get this constant
G by itself. Now we can see that G is actually in units of force times length squared divided by
mass squared. If we then use the fact that force is mass times acceleration and acceleration
is velocity divided by time and velocity is length divided by time, we get units of length
cubed divided by mass divided by time squared. And this isn't just a technicality; if we're
working with different units the gravitational constant will need to be converted into those
units. For example, if we were measuring mass in pounds and length in feet we wouldn't get
6.67x10^(-11). Instead we would get 1.07x10^(-9). this process of only looking at the units in
an equation is called dimensional analysis, and amazingly it can actually be used to figure
out what equation you should be using in the first place. There's a good Numberphile video
on this that I've linked In the description. Okay, those examples were both pretty quick. The
next one is going to take a bit longer to explain. It's called the Hubble constant, and it's measured
in kilometres per second per megaparsec. Oh boy. Let's start with some history. During the
1920s, physicist Alexander Friedmann derived a model for the shape of the universe based on
Einstein's field equations suggesting that the Universe could be expanding, a result supported
by the fact that distant galaxies always seem to be moving away from us. Several years later,
multiple researchers including Georges Lemaître and Edwin Hubble independently found estimates
for the rate of expansion. The idea is as follows. Take a sample of a whole bunch of distant
galaxies, and for each galaxy measure two things: how far away it is, and how fast it is moving
away. If the universe is expanding then we would expect galaxies further away to be moving
away faster, so take all the galaxies and plot them on a graph with velocity on
the y-axis and distance on the x-axis. The galaxies tend to fall along a straight line
and the slope of this line, now called the Hubble constant, describes the rate of expansion of the
universe. If you do all the measurements you get a slope of approximately 70. 70 what? 70 kilometres
per second per megaparsec! That means if a galaxy is one megaparsec away, it's moving at roughly 70
kilometres per second away from us. If a galaxy is 10 megaparsecs away it's moving at roughly
700 kilometres per second away from us, etc. Okay, but why megaparsecs? What is a parsec? It's
about 3x10^16 metres, and to explain why we're using parsecs I need to explain how you actually
measure the distance to a far away galaxy. If you look at something in the sky and it
looks small, it's hard to tell if it looks small because it's actually just small, or
if it looks small because it's far away. The way you get around this is by pointing two
different telescopes in two different locations at the same object and measuring the difference
in the angle between them. You can then do some trigonometry to figure out the distance. The way
we define a parsec is as follows: the distant object is one parsec away if the angle between the
two telescopes is one arcsecond and the distance between the telescopes is one astronomical unit.
What is an arcsecond? It's a 60th of a 60th of a degree. And what's an astronomical unit? It's the distance between the Earth and the Sun. How did we get two telescopes that far apart?
Well it's easy – you actually only need one telescope. Just take one observation in
January and a second observation in July, so you have two observations with a distance
between them of... 2 astronomical units. I don't know where the 2 factors in. Obviously
I'm brushing over a lot of details here! So, if you're an astronomer and you're doing lots
of measurements on a daily basis, it makes sense to record distances using parsecs, because it
follows directly from how you measure things in practice. The speed of galaxies on the other hand
is calculated using redshift, the Doppler effect for light, and for this there's not really any
reason to use anything other than SI units. So when we're comparing velocity and distance we get
a figure in kilometres per second per megaparsec. But once again, we can do the gallons per mile
trick! We're dividing velocity by length and velocity is length divided by time, so
if we convert the two lengths into the same unit we get that the Hubble constant
is about 2.3x10^(-18) inverse seconds. You can check this by yourself: type into
Google "70 km s^-1 Mpc^-1" in this format, and it will convert it for
you into 2.3x10^(-18)... Hertz?! Okay, I have something else to explain. Hertz is a measure of frequency, and
it's equal to inverse seconds. Why? Well, let me give an example. Let's say a sound wave
has a frequency of 100 hertz. That means 100 waves are entering your ear every second. A different
way to measure this would be to measure the time interval between two consecutive waves, which in
this example would be one hundredth of a second. So you can see that the period of a periodic
process measured in seconds is the reciprocal of the frequency measured in hertz. So
to make sense of the Hubble constant, let's flip it upside down to get a number in
seconds. Converting gives about 14 billion years. Which is a very rough approximation
for the age of the universe. Now, the intuition behind this is as follows.
Suppose a galaxy is moving away from us. If we assume it's always been moving at the same
speed, then at some point in the past it must have been right next to us. How long ago was
this? Well, time is distance divided by speed, so it will be the distance to the galaxy divided
by its velocity. But this is just the reciprocal of the Hubble constant! So the Galaxy began
its journey away from us 14 billion years ago. But this is true for every galaxy. That means
14 billion years ago, all the galaxies in the universe were on top of each other, and
that sounds an awful lot like the big bang. Now, as it turns out, this explanation
is actually completely wrong. The rate of expansion of the universe has
changed dramatically over time, so there's no reason to expect Galaxies have
always been moving at the same speed. The Hubble constant is not actually constant in
time, but rather constant across all galaxies at each fixed point in time. This is why it's
sometimes called the Hubble parameter instead. What makes this so particularly
cursed though is that the actual age of the universe is about 13.7 billion
years, which in an absurd coincidence is extremely close to the 14 billion
figure provided by the Hubble constant! Why are these two numbers similar? There is
no reason for these two numbers to be similar! Is there an intuition for the Hubble constant
measured in hertz? Sure, it's how many times the universe can expand from the big bang up to
the size of this today in the span of a second. Right... I guess some ways of writing units
just don't have a nice interpretation. Now, as we saw in the gravity example, units
sometimes just pop out of equations without any explanation. In this context it's often more
natural to write negative exponents for division, so instead of writing m^3/kg/s^2, we write
m^3 kg^(-1) s^(-2). But this raises a question: if we can get negative powers, then can we get
fractional powers? Can we get irrational powers? As it turns out, yes. Occasionally you need
to describe a relationship which follows a complicated power law, and that power
carries over to the exponent on the units. There's one particular example
that I want to go through, one which I encountered while writing
an assignment for an optics course. It's called polarisation mode dispersion and it has
units of picoseconds per square root kilometre. The speed of light in a vacuum is constant, but
inside a substance like glass light slows down, and the amount by which it slows down depends
on properties of the light like frequency and polarisation. This is why a prism can be used
to split light into its component colours – each colour is a different frequency and moves at a
different speed through the prism resulting in it bending in a different direction. This is a
big problem for optical fibres which use light to carry information. A single pulse of light
can be split up into component frequencies which all move at different speeds through the
fibre. The overall effect of this is that the pulse gradually changes shape and spreads out
as it travels, an effect called dispersion. Dispersion limits the rate of information transfer
because you need to send pulses far enough apart so that they don't blur together. Dispersion is
measured in units of time, typically picoseconds. If you send a perfectly narrow pulse through the
fibre, the dispersion is essentially the delay between the front of the pulse and the back
of the pulse reaching the end of the fibre. There are a couple of different kinds of
dispersion. The one I've just described is chromatic dispersion, where the different
frequencies move at different speeds. This builds up linearly over the length of a fibre, so
it's measured in picoseconds per kilometre – how much does the pulse broaden in picoseconds for
every kilometre traveled. Depending on the fibre there are ways to counteract this, but once you've
accounted for it you run into a different kind of dispersion called polarisation mode dispersion,
or PMD. Light is made up of oscillating electric and magnetic fields, and the angle of these
oscillations is called the polarisation. PMD occurs when different polarisations move at
different speeds through the fibre. This seems strange; if the fibre is perfectly cylindrical
then surely it doesn't matter whether the light is oscillating vertically or horizontally. But
that's the point – it's hard to make something perfectly cylindrical. Remember that optical
fibres need to stretch over many kilometres, so over that vast length there's probably
going to be some small impurities which make one polarisation slightly slower than another. But
unlike chromatic dispersion which is measured in picoseconds per kilometre, PMD is measured
in picoseconds per square root kilometre. Why is this? Let's construct a simple model and
try and understand it. Let's send a pulse of light down the fibre made of only two polarisations,
one vertical and one horizontal. Let's define the dispersion as the time taken for the vertical
one to reach the end minus the time taken for the horizontal one to reach the end. So, if the
horizontal one moves faster then the delay between them is positive and if the vertical one moves
faster then the delay between them is negative. We're interested in the absolute value of the
dispersion, but keeping track of the sign is also going to be important. Now there's going to
be random impurities in the fibre, tiny defects from the manufacturing process. Each impurity
will randomly cause the vertical and horizontal polarisations to move either faster or slower,
making the dispersion either increase or decrease. Let's be super rough and assume that the
dispersion has a 50% chance of increasing by some fixed amount every metre, and a 50% chance of
decreasing by the same fixed amount every metre. What we now have is something
called a random walk. What we're interested in is how far
this random walk moves away from zero, and the expected value of this quantity
turns out to be proportional to the square root of the length of the random walk. I'm not
going to delve into the details of why this is because this video is already long enough
as it is, but it's not too hard to prove. So what this means is that the total
absolute dispersion measured in units of time is on average equal to some
constant times the square root of the length of the fibre. in order for
the units to cancel out this constant, which is the PMD of the fibre, has to be in
units of time divided by square root length. Which is why if you look up a specification for
an optical filer online, under polarisation mode dispersion you will find a quantity measured
in picoseconds per square root kilometre. I have seen other strange exponents on units.
While doing research for this video I found a paper on pulsar scintillation (whatever
that is) which involves something called a scattering measure, measured in units of
parsecs times meters to the minus 20/3! I personally find picoseconds per
square root kilometres more cursed though, just because square roots
themselves are also kind of janky. Funnily enough the optics assignment wasn't the
only encounter I had with random walks in 2021. I was also doing a research task involving
stretching polymers, specifically DNA, and from Monte Carlo simulations I found that
the sidewards deviation of a DNA molecule also followed a random walk depending on the length
of the DNA. Which makes sense because a polymer has the literal shape of a random walk. But
what's weird about this random walk is that the length is measured in nanometres while
the sidewards deviation is also measured in nanometres. And so with a bit of hand waving
I was able to include in my research paper the most cursed unit I have ever written:
nanometres per square root nanometre. Have you encountered any other cursed units?
Put them in the comments. I want to see the most ridiculous units known to science. I think all the
places where science is mathematically beautiful are massively overrepresented, and sometimes
it's just fun to see all of the places where the universe is completely gross and inexplicably
awkward. If you share the same sentiment then subscribe, because you can expect more cursed
mathematics from this channel in the future.
Volume per …. Oh!!! Density per second!
No that’s the inverse of density… per second.
If the vertical polarization is faster, it will have a smaller period T_v, not a larger value.
This detail has no bearing on the conclusions of the video, nor even for understanding the equation for PMD as the absolute value makes this detail moot.
I am mentioning it purely for the purpose of pedantry.
Very entertaining video!
Not cursed units, but I have spent my career railing against pointless varieties of units for airflow. One supplier likes g/s, one likes kg/hr, this data sheet uses CFM, but this graph has SCFM, oh but for this calculation it must be ACFM, how about L/s, no, gal/min, nah let's use m/s, wait - ft/s.
Like can we just pick ONE unit for volumetric and ONE for mass flow rate and be done with it? (of course not, because conditions matter, but I wish).
Surely nm/√nm is just √nm?
Good stuff, I can recommend his other videos as well, unluckily for us he's too busy for a regular upload schedule