Cursed Units

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Volume per …. Oh!!! Density per second!

No that’s the inverse of density… per second.

👍︎︎ 9 👤︎︎ u/[deleted] 📅︎︎ Jan 03 2023 🗫︎ replies

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.

👍︎︎ 8 👤︎︎ u/notquite20characters 📅︎︎ Jan 03 2023 🗫︎ replies

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).

👍︎︎ 6 👤︎︎ u/Igneous-Wolf 📅︎︎ Jan 04 2023 🗫︎ replies

Surely nm/√nm is just √nm?

👍︎︎ 3 👤︎︎ u/uffefl 📅︎︎ Jan 03 2023 🗫︎ replies

Good stuff, I can recommend his other videos as well, unluckily for us he's too busy for a regular upload schedule

👍︎︎ 1 👤︎︎ u/JWGhetto 📅︎︎ Jan 03 2023 🗫︎ replies
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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.
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Channel: Joseph Newton
Views: 860,702
Rating: undefined out of 5
Keywords: Science, Mathematics, Physics
Id: kkfIXUjkYqE
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Length: 18min 28sec (1108 seconds)
Published: Mon Jan 02 2023
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