Entropy and Einstein's turnover time

Abstract
A failed attempt at explaining Entropy,
and one zombie, coming right up...

And so it was, on a nice evening, much like this one, that we had all sat around the table, and a question popped up...

The question was along the lines of "what is the Entropic principle?", and it was asked by my brother, a brilliant man, and a science-fiction aficionado, who unfortunately for the physicist community, never had the chance to dabble with physics, and so they have to find a poor substitute in the image of your humble servant here...

At first I asked if he meant the Anthropic principle, but he just wanted to understand Entropy.

Thus I found myself trying to explain Entropy, and the 2nd law of thermodynamics to the uninitiated, in layman terms, and, after a fashion, follow the Einstein grandmother rule - "You do not really understand something unless you can explain it to your grandmother." (A.Einstein).

Incidentally , you could probably calculate the period time \(T\) of Einstein's turning over in his grave, by someone misquoting him, or otherwise deifying him, and justifying a falsehood or plain ol' stupidity by attributing something to old Einei that he never would have meant in a million light years.  

We'll start with some observational data - I have around 400 people in my human network, and on average I get an Einstein quote which falls under the aforementioned category, maybe once every two weeks.

Now, suppose only a third of the world's population leads a somewhat western lifestyle (either connected to facebook, google+, twitter etc. or alternatively reads the paper and or listens to the radio at least once a day), we have about 2.3 Billion people.

let's be harsh and assume each of the 400 people sub-networks are non-connected between them and so we neglect back-propagation we can put a lower limit of
\[\frac{2.3\cdot 10^{9}}{400}=5.75\cdot 10^{6}\,\text{instances in 2 weeks}\]
Divided by the number of seconds in a two-week period we get:
\[\frac{5.75\cdot 10^{6}}{14\cdot 24\cdot 60\cdot 60}\approx 4.75\,\text{times per second}\]

And so the period time of Einstein's turning in his grave would be \(T\approx 0.21 \,sec\).

So basically even the lower limit states that Einstein, by now, is a zombified Olympic athlete, even considering the initial rigor mortis....

By now, he would have a solid six-pack.

Anyway, I digress, I was going to explain Entropy and then a random rant stole my attention... sorry for that.

In a nutshell, Entropy is a measure of disorder, and I will explain.

<Failed attempt at an explanation : but is still worth a read>

Imagine a group of four coins, each with two sides - heads, and tails - right? (we'll have non of that Two-Face shenanigans here!)
Now suppose every coin is perfectly balanced so there's a fifty-fifty chance of getting heads or tails for each coin flip.

So, now, what are the chances of getting all 4 heads, when you flip 4 coins?
if you do the experiment enough times, you get an average of 1/16 chance.
the state of all heads, or equivocally all tails is the most "ordered" result, why?
because it is the most homogenous result (and we humans like homogeneity, symmetry and by the same token order).
Now, what's the most plausible result?
that's easy - the result where two coins are tails up, and two coins are heads up (regardless of their locations), happens ideally \(\frac{3}{8}\) of the times you flip (almost half of the times you flip the coins, you'll get this result).

That is the least "ordered" result, since we don't care about locations, and the coins show the most diversity in results.


Now, I won't go through the whole derivation, but if you'll try the same logic with 6 coins and then 8 coins you'll get a breakdown of \[\frac{1}{64},\frac{6}{64},\frac{15}{64},\frac{20}{64},\frac{15}{64},\frac{6}{64},\frac{1}{64}\,\text{for six coins}\]
and a breakdown of \[\frac{1}{256},\frac{8}{256},\frac{28}{256},\frac{56}{256},\frac{70}{256},\frac{56}{256},\frac{28}{256},\frac{8}{256},\frac{1}{256}\,\text{for eight coins}\]
And so on and so forth, the reason I'm sticking to even numbers is because it LOOKS more clear that way, but really, it makes no difference. You could go on and on until kingdom come, and you'll find the middle, most unorganized result will be the most common.

It turns out that this kind of dynamic is best approximated by a gaussian function called the \(g\) function (or the multiplicity function) and I'll spare you the details in favor of a graph:

Probable results graph

So, what you see here, is basically an overlay of 4 graphs that show the relative probability of results as they stray from the middle "disorganized" and probable result.
What is interesting, is the bigger the experiment is (i.e. instead of 8 coins, let's say a 100 or 1000 coins) the sharper the peak is, meaning it's narrower, and higher in respect to other possible results. that means by the way that the most probable result is highly probable, and the others highly improbable.  Now imagine an experiment of \(10^{23}\) coins, every result other then the most probable and it's immediate neighbors is SO improbable, it virtually is IMPOSSIBLE (in the sense that it would take a ludicrously impossible amount of experiments to perform to actually get a significant chance to get such a result).

A word of caution though - this is probability we're talking about, so in theory a highly organized result MIGHT happen, in actuality - yeah, not so much...

By the way, there are roughly no more than \(6\cdot 10^{14}\) coins in circulation today in THE WORLD, meaning even if you took all the coins in the world today you couldn't perform such an experiment, even once!

Incidentally the ridiculously high number of participants in a single experiment, makes all the difference between "hard sciences" even if they are statistically oriented, and "soft sciences".

Even if we take all the people in the world, and get them to participate in one of our experiments, the result will produce some correlation that may, or may not apply to a single participant.

In physics, while the same is true, you could say a statistic result applies and be absolutely correct on a macro level (with deviations so small as to be insignificant for most purposes), and be correct almost every time on a micro level as well!.

So anyway, Entropy is defined as the logarithm of the multiplicity function.

The reason for taking the logarithm is for the sake of defining a cumulative quantity, as opposed to multiplicative.

< /Failed attempt at an explanation : but is still worth a read> 

So anyway, obviously I failed at this attempt but let's try it in a simpler manner:

Entropy is a quantity that signifies how probable a result is.
by a fluke of chance, which isn't a fluke at all, more of a deep connection really, the most probable result is also the most diverse one, or differently put, the most disorganized.

Thus, Entropy becomes a measure of disorder of a system.

Entropy is a cumulative property in the sense, that when two non-interacting experiments are done the combined entropy is the sum.
However, when systems are allowed to interact, the combined entropy is typically larger than the sum of individual entropy.

it is by that sense, that entropy tends to increase over time (and interactions).

<example of entropy increase>

Suppose, we have two systems, each of 4 coins.

The most probable state is given by 2 heads, and 2 tails for a single experiment right?
as was explained in the above failed attempt, the chance for that happening is \(\frac{3}{8}\).

Now, what is the chance of each of the experiment to get the most probable state independently? you guessed it - the product of the two independent probabilities i.e. \(\frac{9}{64}\) right?

OK, but now, let's put all the coins in a single experiment, an flip all of them, the chance of hitting 4/4 division of heads/tails is given by \(\frac{70}{256}=\frac{35}{128}\) which is almost double the size of the product of individual probabilities.

So what happened here, really?
in essence, the combined system has more places to choose from, meaning more diversity of scenarios that lead to the same end result, thus the combined system is more "disordered" thus bigger probability that leads to bigger Entropy.

</example of entropy increase>

OK, that wraps it up for this time.
Obviously I don't understand Entropy enough, since I feel I have failed at explaining it,
but I will try again ("if at first you don't succeed" etc...).

Oh by the way, for all us \(\LaTeX\) geeks out there, isn't it cool that every time we use a fractional, we immediately get a reference to Battlestar Galactica?

Anyway, till next time...

P.S. some quick ideas to utilize Einei's incredible turnover time of 0.21 seconds :
1. Attach him to a turbine and generate electricity.
2. Display him as the 8th world wonder - the quickest man on the planet (faster than Usain Bolt).
3. Use him as the engine for a horse-ride carousel for my kid.
4. Use him to refute the 2nd law of Thermodynamics as he is both dead (Entropy supposed to get bigger), but in the greatest shape of his life (which requires work and decrease in local Entropy)...

Just sayin'...




 

Short Derivation of Archie's Law

 "captain's log: supplemental"

It's been quite a while since I wrote the previous post about Archie, his wife, a crown etc.
While I was writing that, I did not want to derive Archimedes' law on my own, so I looked around on the net, to maybe find someone who already posted that, sadly, I couldn't.

Also, this morning my wife and I had an argument about this law, and guess what, she was right, I was wrong... Isn't it weird these are the arguments my wife and I have?!? I mean, seriously, how nerdy can you get? 

So for the interest of completeness (which is actually a mathematical axiom, but let's leave that be for now), I give you a short derivation of Archie's law:

Deriving the law (I AM THE LAW)

Let us consider an infinitesimal volume element of a material with density \(\rho\) , and incident area and height \(A,h\) respectively.
So now let's suppose the material is lighter than the medium surrounding it we get a force equation that looks like this:
\[\Sigma F=A\cdot P_{up} -A\cdot P_{down}-Mg\]
Where the force equivalent (sum of all forces) is positive.

Now, let's wrap the forces that are the result of pressure and name it the Buoyancy force thus:
\[F_{Buoyancy}=A\cdot\left(P_{up}-P_{down}\right)\]
and with a little massage we get:
\[A\cdot\left(P_{up}-P_{down}\right)=A\cdot h \left(\frac{P_{up}-P_{down}}{h}\right)=-V\nabla P\]

So, now, let's take a small detour to understand who \(\nabla P\) is shall we?

suppose we are dealing with an element that is filled with the same material as the medium it's in, in that case we get a mechanical equilibrium and \(\Sigma F\) is simply zero. and so we get:
\[Mg=F_{Buoyancy}=-V\nabla P\]
and breaking up the \(Mg\) element we get:
\[V\rho_{medium}\cdot g=-V\nabla P\Rightarrow \nabla P = -\rho_{medium}\cdot g\]
and so we get quite simply:
\[F_{Buoyancy}=V\cdot \rho_{medium}\cdot g\]

Proper usage

What's useful with this representation, is that we get the force outright, thus we can make force calculations directly.

An important thing to understand is that the buoyancy force is directed not UPWARDS (which is a common misconception) but against the direction of the pressure gradient . which is quite different. that means for example that in a spinning tube of air, the buoyancy force will be mostly inwards as the pressure gradient is directed outwards (in cylindrical coordinates).

That's it.

this time it was really short.

Crowns,Boats, Subs, and other things that float in water...

Abstarct:

A short story about a friend,

and things that go "ping" in the dark.

It has been some time since last I wrote something, but life, the multiverse and everything, got in the way.

By the way, if the answer to life universe and everything is 42, then for the multiverse it should be what? a Vector? a Tensor? and what are the entries? in which linear basis?
I guess I could get cute and say that its a \(42^n\) Tensor, but that's just my regular idiotic nonsense at play...

So yes, in layman terms I was simply very otherwise occupied, but today, as my wife is away, and after almost a week of sleep deprivation and other torture methods, I found the will and the way to actually start writing stuff.

Anyway the idea for this post came from a rather dubious experience, and I might have a follow-up post on the physics of water-closets, otherwise known as restrooms, bathrooms or the unsavory "latrines".

What's the connection?

And Seeing as approximately a quarter of my life was spent on boats or in the context thereof, I deem it fit to dedicate a post to the physics of these floating wonders.

What makes things float? 

Most of us at one point or the other have had a chance to hear that quite famous cry "EUREKA" but I daresay less of us actually know the origin of this cry. so here goes...

Some 2500 years ago in the golden age of the great kingdom of Crete, there lived a king and his queen, or, more probably a queen and her lackey of a king.

Now that queen was little capricious, not unlike the queen of hearts from Lewis Carroll's Alice's adventures in wonderland, and at some point in time decided that the crown allotted to her, was not fancy enough. Thus, she ordered a new crown to be made, a crown of solid gold.

 The word spread, and a huge congregation of goldsmiths, jewelers, and their apprentices, flocked to the isle of Crete, in hopes of gaining the queen's grace and be chosen to forge the queen's new crown. 

But as is often the case with tyrannical rulers, this queen was a very suspicious being, I could theorize she had a bad experience with  goldsmiths, or might be she was simply a bitch.

In any event, she was absolutely terrified by the prospect of being swindled and as a result she wanted to make sure the crown was actually made of solid gold. 

But here's the pickle... suppose she WASN'T swindled and she melted the crown to check if it's solid gold, she's now left with a pot of molten gold, while still having to pay the goldsmith.

Thus a method had to be devised to check the crown without damaging it.

Enter stage left: Archimedes.

Archie, our friend happened to be in the vicinity and having a reputation of the genius he was, was charged with the daunting task of finding that method.

Well, Archie thought hard, maybe losing some weight (why can't I?) and some hair (why am I?) at the prospect of failing the queen and subsequently failing to breath, and at the end of 3 excruciating days, his wife decided she would have none of it anymore, "You stink!" shrilled the shrew "Go have a bath or it's the couch tonight, for you!".

Archimedes, being the thoughtful husband that he was, climbed into a warm bath, and noticed, that when he submerged more of his body, the water level rose and spilled over the sides of the tub. 

"EUREKA"

he shouted then, followed by his wife's "Shut up already you git! you'll wake the baby and then you'll have to deal with it!!" 

Anyway, I will leave you wondering as for how the story ends, did old Archie indeed wake the baby, how long was spent in the dog-house, and whether or not a goldsmith found his premature demise.

Now for the physics:

Suppose an object with a volume V is partially submerged in water - the elevation force is due to pressure differences, the partially exposed part of the object experiences just the atmospheric pressure, but the underside experiences the upward pressure from the water, so let's see what that pressure is:
Let's consider a column of water and a thin strip \(\Delta Z\) thick
\[\Sigma F=0 \Rightarrow s\cdot\left(P(z+\Delta z)-P(z)\right)-s\cdot\Delta z \cdot \rho_{water} g \\
\text{or in other words} \frac{\partial P}{\partial z}=\rho_{water}\cdot g\Rightarrow P(z)=\rho_{water}\cdot g\cdot z+ P_{atm}\]

Where z is the depth of water.

So, we have to consider \[mg=s\cdot\rho_{water}\cdot g\cdot z \Rightarrow h\cdot\rho_{object}=\rho_{water}\cdot z \]

In other words, the depth of immersion is given by the height and relative density of the object and the fluid.

Now, there's an easy way to see that this dynamic is correct, simply take a piece of wood , and see that it submerges deeper when you hold it length up, than when it's laying flat on the water.

So that takes care of boats, we just have to make sure the average density of the boat is lesser than the density of the water, mind you we're talking the average density of the space occupied by the boat meaning also the air inside the boat, unless you start to take on water, and then guess what? your downward bound.

That also might provide a hint why we sometime encounter "unflushables"...

That also might provide an insight as to how submarines stay submerged at a constant depth:
When the sub is at "bubble up" state, basically the density of the sub is lower than the surrounding water thus the sub tends to float up. to hasten the process the sub might or might not apply it's propeller or other means of propulsion.

When the sub is at "bubble down" state, the average density of the sub is higher than that of the surrounding water making the sub "heavier" and thus sinks down, again applying propulsion or not is at the captain's discretion.

By the way, subs mostly have compressed air tanks, which they discharge into  ballast sections, to change the average density of the vessel, and then use compressors to re-compress said air to the tanks, evacuate the ballast air ballast sections to increase the average density (water then flood the ballast sections).

So it seems fairly simple right? WRONG, we actually took the water's density to be constant where it really isn't, cold water is denser than hot water, and deep water is a tad denser than shallow, so what's the deal? 

Well the physics for this is fairly complicated in terms of the math involved, but the IDEA is fairly simple, water density is a product of the mutual forces between water molecules, that are essentially electric in nature, and so external pressure is somewhat involved in this, but even more so, temperature.

So with pressure \(\rho_{water}\) rises linearly at first but pretty quick stabilizes to a constant.

Like so:

Courtesy Windows To the Universe (NESTA)

With temperature the change is much more pronounced, but still pretty much the same applies, A linear rise in density when temperature drops, and then exponential decay to a constant.
I suspect somewhere in the middle there's actually a point where it all turns to ice...

That remind me of the cool Thermometer where there are different glass bells with different nifty colored liquids in a glass water tube, and when the water in the tube is in thermal equilibrium with the area, some bells float up, some sink down, and the one left in the middle shows the right temperature on it.... pretty cool if you ask me...

Told you water density changes with temperature!

So anyway making the calculations needed to predict the density at a certain depth and temperature is a pretty nasty undertaking thus usually subs employ feedback loop mechanisms to apply the right density. either that or they do it by hand and eye i.e. "bubble up"\"bubble down" mechanism.

By the way, remember the couple of pictures in the beginning? well there you go:

Baby Ruth is swimming pool - not quite what you think...

I could go on and on about this, about weighing ships, (as opposed to sheep), and using partially submerged sonar buoys, Thermocline, and using different water densities to mislead enemy vessels as to your true location etc. etc. But I'm pretty sure if you read Clancy's "The hunt for Red October" you'd learn all this and have great time doing so...

a short post about Wind Power and shams

ABSTRACT: 

Wind power, commercial jets, and politics.

And almost no humor this time...

A couple of weeks ago, maybe 3~4 days after posting the little thingy about Solar energy, two things happened.

1. I was a guest at an eminent solar scientist BBQ lunch, and we had a little chat about what I wrote.
   He said he'd send me a paper he wrote, so I could avoid making the same mistakes as everybody, and go on to make some mistakes of my own... :)
   
   I promise when I finish reading it, I'll atone for my sins, stop eating meat, cease all gasoline use and switch over to the green side. Well no. but I will try to rework my logic and share my thoughts to the extent possible, since it is article material and I don't want this blog to jump the gun on hard work, not my own.
   
2. I came across this nice video :
   
   
   
   Now, a word of caution, this video is very political, and actually, the field of environmental science is highly political in nature, which by the way brings the question whether Thomas Kuhn's largely misinterpreted  view on science is actually correct - i.e. science is as politicized as everything and social trends actually effect scientific truth.
   
   Kuhn probably never intended to say or write that, but that's a different story altogether.
   
   Anyway my only intention here is to derive the part of how much energy an electro-windmill can produce.

Since wind have several sources, I can't simply rely on Ideal gas equation plus the Euler equation (though I've done the math, it actually accounts for very little of the wind regiment in the world), so I'd have to start from observational data...

After digging a bit, I found several sources, but I'm going to be extremely optimistic and take the average wind velocity, averaged over the globe over a period of about 50 years to be about 7.5 m/s.

So, the energy is given by \(\frac{1}{2}mv^2\) but we need a way to evaluate the total mass of the moving gas that comprises the wind right?

Let's do that real quick shall we?
the volume of air passing through an incident area \(A\) in one second is basically given by \(A\cdot v\), thus the wind power that goes through a windmill with incident area \(A\) is simply:
\[P=\frac{1}{2}\rho_{air}Av^3\]
with \(\rho_{air}\) being roughly \(1.25 kg\cdot m^{-3}\), we get that the power as a function of incident area is given by:
\[P_{\text{wind averaged}}\approx 264\cdot A\,\,\, W\]

So supposedly its enough for every person to have about 9 square meters to his name in the constellation of a private windmill to account for all our earthly needs.

Enter windmill efficiency at a staggering  36% (the theoretical boundary is actually about 60% but wind drag, mechanical stress etc. account for further efficiency loss), so every person must now have about 27 square meters, which is not that much right?

Really? 2% of the energy a commercial aircraft needs to take off ?

The Energy needed for an empty Boeing 747 at 170 tons to take off and achieve cruise altitude of about 10km is about \(2\cdot 10^{10}\,\,\,W\) which means we need a windmill with a blade length of about 5km or about 1000 150m blade turbines, or 10,000 50m blade turbines.

What about just take-off speed? average takeoff speed for a 747 is about 80 m/s, and you'd have to maintain that speed for the duration of the takeoff, but let's calculate just for one second shall we?
\[E_{\text{one second}}=10^9\]
So we would need about 1km long blade to account for that plane taking off, and a 50-store blade spells about 150 meters blade which means about \(1.9\cdot 10^{7}\, W\)  and that really is about 2% of the energy needed for an empty 747 to take-off...
So you see? the guy in the video was right!!! or was he? 

Cheap rhetoric or outright sham?

well no, that argument is just throwing sand in the public's eyes because supposedly if we had enough turbines to cover the world's energy needs, a spike in demand as reflected by a taking-off event would not even show on the overall scale, with about 4 to 5 orders of magnitude difference...

The real consideration has to be this one:
in order to provide for humanity's power needs, we need about \(2\cdot 10^{11}\) square meters dedicated to power production. factor in spacing issues (a factor of 6 times the rotor blade length means about 36 times the area). thus we get about \(8\cdot 10^{12}\,m^2\) which means about 5% of the land area available on earth.

Again, this is about the size of Europe, and I didn't even go into wind regiment as a function of height consideration, or considering blackout times and backup requirements as the wind DOES tend to stop at rather inconvenient times (for instance when it's hot as hell outside and there's no wind,   and you want to turn on the AC, but guess what - no wind means no electricity means no AC!!! damn you green energy advocates!!)

I read somewhere (an incomplete) analysis that concluded it calls for covering the surface of the earth with some kind of a wind capturing apparatus a 150 meters high in order to manufacture enough energy for us to  live the way we do. I'm not that pessimistic, but let's just say wind power is no more a true messiah than solar power is...

So you see, this field is indeed highly politicized, the green energy advocates would have you believe that so called green energy is a magic pill solution, with no downsides, when really, at best it's a small part of the solution.

While the people who basically side up with the petroleum industry will try to sell you imaginary numbers, or otherwise hammer you down with irrelevant comparisons, conclusions and data.

If I might add my 2 cents - Wind power as well as Solar power, are part of the solution and it's very important for us to develop these further. a 100% efficiency is impossible, but imagine what could happen if we had 80% efficiency on Solar&Wind energy production? we could maybe minimize (not eliminate mind you!) our trace on the environment, while also lessening our demand for energy production dedicated land. together with another 2-3 sources of clean energy, we may very well create a society that is both abundant in food,water and energy and as non-obtrusive as can be.


I know, this post might have bored you to death, I was sick when I wrote it so it might be lacking in humor, wits or logic, but I do hope later posts will be better that way...

Physics of Environment and lies

Abstract

IQ loss, Solar power, and misconceptions 

and an egg?!?

"Ladies and gentlemen of the class of 2012, wear sunscreen... "

Well, again it's been quite a while since last I wrote something. Mostly the reason is I was in Louisville KY visiting my daughter's great-grandma which is incidentally indeed great, and I enjoy calling her "Savta" (which is Hebrew for grandma and implies her actually being MY grandma), partly because the position was unmanned, i.e. I have no grandparents of my own, but also partly due to said greatness...

During that visit the internet connection was tenuous at best, and so I wasted away my time (and energy and brains) by zombiing-out in front of a relatively old computer game - Elder-scrolls IV: Oblivion, which is basically a fantasy game that does exactly that - throws your real life into oblivion.

The main straightforward result of this is ,basically now I am in search of an additional ~40-50 IQ points to compensate for my lost mind (I seem to recall a song about that... hmmm....).

Additionally someone told me it must not be as bad as I think and that I'm pretty smart as it is. enter this vain attempt at a blog as a failing effort to prove myself wrong and them right :)

So anyway, one of the courses I'm supposed to be taking this semester is a course about environmental physics, which for this course is really a misnomer. it should have been called plain "Mental Physics" since the person giving the lectures is a bit of a cognitive dissonance. the guy has credentials as long as the great wall of china, and obviously the guy's bright as all hell, but he has some kind of an issue that prevents him from writing anything on the whiteboard, so the whole course is conducted without writing down a single physical formula.

I don't mean to be disrespectful, mind you, I'm just ranting a bit about the fact that I'm having difficulty following his train of thought. oh by the way, at this point in time another student and myself comprise the totality of the survivor list, that started some 15-strong.   

Anyway, so in an effort to basically teach myself the whole (nonexistent) syllabus I am writing this post that will have to do with environmental physics and ultimately some of the lies people tell you about green-energy.

How much energy does it take to live anyway?

 

Well, the above subtitle or subsection or dissection or C-section or whatever, is actually the first and foremost question we have to answer upon entering this game of energy production vs. consumption, which is really what's it all about anyway, right?

by the way, I am going to be EXTREMELY optimistic this whole analysis  just to underscore the seriousness of the situation we're in and the blatancy of the lies all of us are being told...

So let's approach this question by breaking it down to small bits:

1. how much energy is spent by simply surviving? (on average! on average! sheesh... don't kill me YET)

2. how much energy is spent on living i.e. driving to work and back, computers, home appliances etc..

So in order to answer the first question, let's take the average age across the world's population, find out what the average calorie intake at that age and that will be our rule of thumb, assuming basically what goes in goes out in terms of energy.

So for some cold hard data:

• Median age of the population is 28.4 years overall, and since roughly 70% of the people in the world age anywhere between 14 to 65 we can't be too off the mark by saying the average age is probably somewhere between 23-33 years.
• Now the average male weight around the world ranges from 65-87 kg.
• the same for women ranges 56-75 kg.

Now as I said before I'm going to be EXTREMELY optimistic so I'll take the lower entries, make the assumption that women and men distribute roughly 50-50 of the worlds population.

The calorie intake of an average man at 65kg, no physical exercise, at age 23 where metabolism hadn't yet gone too far down, comes out about 1550 Kcal/day. 

For women it turns out to be about 1350 Kcal/day.

Now if you happen to be a large person, reading this, DON'T BE ALARMED, again, I am optimistic to a fault here, and I couldn't survive on 1500 Kcal/day even if I wanted to... 

Well, let's average this out to about 1450 Kcal/day and find out that:

\[1 cal \approx 4.1 J\\ 1450 Kcal = 1450\times 10^3\times4.1\\ P= 5945000 \text{ Joul per day} \Rightarrow P_{\text{average person}}\approx 70 W
\]

Ok, so this answers part 1 of our question but what about part 2? well this could get messy now, cause navigating the sea of data on average gas consumption, power consumption per capita etc. is all but impossible,so I'll do what most physicists do, either ignore the problem or invent some lame excuse why it's insignificant...

for myself I think the honest thing to do here is say that I just don't know, and am to lazy to calculate it right now so let's just go with surviving for now...

Solar energy - not quite what it's cracked up to be

 

One not so sunny day at my campus, a clean energy activist came up to me and asked if I could sign a petition to support green energy, mainly raising 2-3 more solar farms in our sun-scorched state, and erecting wind-turbine fields in the Negev etc. 

I said I was willing to sign it, but first I would like to hear some facts and data about the cost-effectiveness  of such endeavor, and if he could please tell me what are some of the adverse affects, and what is intended by way of offsetting those. 

I don't quite remember if I ended up signing it or not, but that's beside the point. the point being we rarely, if ever, get educated about the downside of so called green energy, and for the most part most of us buy into the idea of traceless energy production hook, line and sinker. 

In a previous post I mentioned the amount of solar radiation the earth's surface experiences, is given by \(S_{\oplus}\approx1360\, \frac{J}{sec\cdot m^2} \).

so the good news are there's enough solar energy to go around for all of us to survive, as one might suspect... after all indirectly that's what's happening anyway...
But here's the shocker:
\[\text{Earth's surface} \equiv A_{\oplus}=4\pi r_{\oplus}^2 = 4\pi \times 6400,000^2\approx 5\cdot 10^{14}\, m^2 \\
\text{Total solar energy on earth, per second} \equiv E_{\oplus}=A_{\oplus}\times S_{\oplus}\approx 7\cdot 10^{17} W\\
\text{Total Energy used (surviving mode)}=E_{humen\, race}=7\cdot 10^9\cdot 70 \approx 5\cdot 10^{11} W \]

and even if we only use Earth's available land, forget about covering the sea with mirrors we get about \( 0.29 \times E_{\oplus}\approx 2 \cdot10^{17} W \) .

So indeed the good news is that if we were all blown back into the stone-age, lived life simple and basic, human population will never exceed the ability of the sun to supply energy, mainly in the form of food and warmth...

if we choose to be egomaniacs (as a race, which we probably are), and annihilate every other non-beneficial life form on this planet, and limit ourselves to living off sugarcane (8% efficiency), wheat, and stuff like that we could very well multiply earth's population by a couple orders of magnitude and still be OK.

but wait a minute! that doesn't get me where I intended to go, so again like every physicist does every once in a while... remember that pesky thing I was too lazy to evaluate? namely the energy cost of living (as opposed to survival)? well the data doesn't support the intended conclusion so let's get different data!!!

I'm largely joking around, but I shit you not, this sort of thing happens all the time in hard science, and don't even get me started on "soft science" as they call it (no offense but really to me these are better named "non-science", and I might write a post about that at some point when I feel sufficiently antagonistic).

Anyway, so I looked around and found an amazing piece of information. it turns out that the average energy consumption per capita in the world, in 2008 was 21,228kWh, that is, after quick unit conversion:
\[E_{year}=21,228\cdot 10^3 \frac{J}{sec} \cdot hour= 21,228\cdot 10^3\cdot 60\cdot 60\\
\Rightarrow P_{\text{average, 2008}}= \frac{21,228\cdot 10^3}{365\cdot 24}\approx 2400=2.4kW \]

Which is dire news indeed, since the total available solar energy on the planet was about \(10^{18}\,W\) ,
and now the total consumption of the human race turns out to be around \(1.6\cdot 10^{13} W\).
or in other words, each person, on average needs 2 square meters of land to his name in order to sustain a mostly modern way of life.

And again, after checking my math, again and again, I am shocked with the fact that presumably if the United States so wished it could have easily supplied the whole world with enough power 5 times over, by simply transforming Arizona into a huge mirror-field!! this is how:
\[A_{AZ}\approx 3\cdot 10^{11}m^2\\
\frac{A_{AZ}}{(2m^2)\cdot(Population)}\approx \frac{3\cdot 10^{11}}{15\cdot 10^9}=20\]

Enter the current efficiency of Solar power plants at about 44% cutting edge, but let's be cynical and take what's out there on the market right now at about 25% efficiency and still we get that the US could have easily supplied the world with it's power needs about 5 times over.


Damn it! did I just convert -myself- to the dark side?.... ummm light side? err.. sunny-side-up? whatever... I think I might join solar research.

 

Sunny side (up)

Dark side

Wow, hold yer horses boys, and let's get \(\mathcal{R}\)eal. it seems that we have some more calculation to make.
First off, the earth's solar constant, as it happens, relates to the solar radiation the earth experiences as a black body meaning that up in the upper atmosphere, beyond the clouds and ozone, when facing the sun directly we get \(1360\,\frac{W}{m^2}\). So we have to average it over night and day, plus we took the whole surface of the earth when really, we should have taken incident area meaning:
\[A_I=\text{Incident area}=\pi r_\oplus^2\]
So, we have to take what we got and divide it by 4, average it over a cosine squared function (dividing it in 2 again), AND to top it all off we have to account for cloud cover, so let's divide it by another 2. thus we get:
\[E_\oplus\approx 4\cdot 10^{16}\]
Now apparently, when using mirror collectors, we need to cool them down, so the total available area drops down to about a fifth, let's factor in the 25% efficiency factor we talked about earlier and get to about 5% of the available energy, that gets us to
\[E_\oplus\approx 2\cdot 10^{15}\]
Multiply that by the percentage of available land i.e. 29% to get:
\[E_\oplus\approx 6\cdot 10^{14}\]
So we get to the conclusion that we need about \(\frac{1}{40}\) of our land in order to account for the global power demand. Mind you, that all of these calculation were done disregarding prevalent trends, meaning the average person in the US uses about 5.5 times the average power consumption, and the current trend is towards that way of life. so with significant technological progress things are bound to get worse in that respect.  

As it is forget about using Arizona as a giant solar farm, try Greenland and India combined.

And I didn't even get started! I mean, try factoring in dust!
as it so happens, the mirrors need to be constantly cleaned, as dust quickly builds up, and when it does, efficiency goes WAY down -  try 1% percent efficiency instead of 20%.

You know what? I promised to be optimistic though,so let's take cutting edge solar technology at about 44% efficiency, meaning we narrowed it down from India and Greenland to oh, let's say JUST Greenland. Let's just say I doubt the good people of Greenland are that accommodating, and I don't think you'll get a better response from the people of India.

Of course that's a fallacy right there, as it is not required for all collectors to be at the same place but in terms of land per capita it means that  each person has to own about 267 square meters, which is quite a lot, it's actually about 50'X50' area.

So unless we act and make Lennon's vision of no countries real, it's a bit of a toughie.

Hey - what was that about traceless energy production?

All we've done as of now is just talk physics, but what about environmental impact?

The common (mis)conception is that solar energy, being renewable (which it is) also has little to no impact on the environment. 

that's only partly true, what's true is there's no airborne pollution, no solid waste, trash etc. BUT what's there is oh, I don't know, thermal pollution? light pollution?  displacement of wild-life, and the possible destruction of unique species. and that's just off the top of my head and remember I'm NOT an environment scientist/wild-life expert/forester or any other kind of nature geek. 

Now I'm all for green and clean energy, and it's a noble calling to further develop solar energy production, hey, I might even go into solar energy research myself. 

YES IT'S THAT IMPORTANT (that I might actually dedicate my life to this).

but, do not ever fall for something that sounds too good to be true, it usually just ain't.

and green energy,if not carefully and thoughtfully developed might be almost or as bad as 

conventional energy.

Again I'm no expert in ecology, so take what I say with a grain of salt as far as natural ramifications, but trust me, on the sunscreen... err physics.

Next time: Wind energy or Wormholes

 

Physics of Water polo

Abstract:

Water-Copters, Beckham and dirty playing 
oh right, and drowning just a little bit 

It has been some time now since I've written my last real post, even though the "passing remark" about butterfly flatulence was rather lengthy...

When I was fifteen or so I played Water polo at a semi-pro league, which is to say the only non-pro league in Israel. Now, for those of you who are not familiar with that sport, you don't have to be ashamed, this is NOT as embarrassing as for instance not knowing who Michael Jordan is, since basically water polo is as obscure as for instance Tolkien's "Mr. bliss". that is to say, in essence only people who play water polo or have an immediate relative that does, actually know what water polo is.

If you ask the average Schmoe what water polo was, you'd get something like "err... a bunch of people in water passing a ball around?", or otherwise simply - "you mean water basketball, right?"

Consequently, it's not surprising that if you search the web for "physics of water polo" you'd get very few resources and pretty lame at that. for instance the "eggbeater kick" entry in Wikipedia explains a little what that move is, but none of the physics involved is explained.

So, in a lame effort to contribute something unique to the web, and maybe get a higher rank in google's algorithms, this post will deal with water polo and physics.

The Water-Copter
Water polo players are probably among the best cardiovascular athletes in the world, that due to the fact that this game has 4 quarters, each quarter is theoretically 8 minutes long but about 12-15 minutes of real time, meaning the players are in the water for almost an hour.

That in itself is meaningless but you have to take into account the  "water-copter" move or the eggbeater kick, which is - moving your legs in two separate counter circles, which creates a constant water current directed downwards.

The "water-copter" technique

this move acts on the same premise as your basic helicopter, hence - the water-copter.

So how a helicopter works? simple! (well not that simple but whatever...).
each second the helicopter is supposed to accelerate downwards in a constant acceleration \(g\), that is, the gravitational acceleration. so starting from a stationary state in one second it is supposed to develop a downward momentum of \(m_{\chi}g\), where \(m_{\chi}\) denotes the helicopter's mass. so, in order to keep the chopper flying, the rotor moves the air downwards in a momentum that is equal to the downward momentum, thus momentum-wise we even out. more explicitly:
\[\left\{\begin{array}{l}m_{\chi}\cdot g \cdot 1sec =m_{air}\cdot V_{air}\\
m_{air}=\underset{Surface}{S}\cdot\underset{height}{V_{air}\cdot 1sec} \cdot
\underset{density}{\rho}\\
\Rightarrow V^{2}_{air}= \frac{m_{\chi}\cdot g}{S\cdot\rho}\end{array}\right.\]

So for a chopper that weighs about 7 tons (for instance AH-64 Apache) this means the air velocity going down should be about \(18 \frac{m}{sec} \approx 67 \frac{km}{hr} \approx  41 mph\) which is quite amazing.

and if you've ever seen a Black-hawk approaching a ship's aft for a landing or extracting approach, the sheer awesomeness of beholding this 10-ton helicopter "raising the sea" is simply beautiful, with a downward wind-blast equivalent to ~ 45 mph.


Anyway, I digress...
with the same kind of dynamics happening underwater, the water-polo player's mass is effectively decreased by buoyancy and thus we get that the water velocity under the player amounts to about  \(6.5 \frac{m}{sec} \approx 23\frac{km}{hr} \) which is formidable indeed. especially when you consider top running speeds.
this means that these guys get to underwater velocities that are equivalent to top 5k competitive runners.
taking into account water resistance being lower then track resistance we get to the inevitable conclusion that these guys develop speeds that are probably comparable to top track runners of 800 to 1000 meter runs.

the thing is, these guys have to do this just to stay afloat, not even mentioning lifting up to deliver a shot, so you can imagine how hard this sport is, when these guys are required to do the equivalent of running 4 5k runs back to back at an Olympic pace, to do nothing more then "stand" in the water, like sitting ducks if you will...

Bend it like Azevedo (or Sapic)
There's actually a movie I never saw that's called "bend it like Beckham", I don't know, maybe it was the stupid name, or maybe it's just the fact that I never really liked soccer, much to my father's disappointment.
he sometimes says "God! how is it possible that I raised two boys that don't like soccer?!?" and proceeds with "Are you guys sure your mine?" anyway, my sister actually used to like soccer very much, but me and my older brother - not to much, we enjoyed causing chaos and mayhem while trying to break each other... by the way over the years we've gotten pretty good at this, so now we don't fight as much...

Anyway, Azevedo and Sapic are both professional water-polo players, Azevedo was called at one point "the Michael Jordan of water polo" while Sapic was hailed as the best water polo player ever for some time.

But this part will deal more with the "bend it" and less with the "Beckham" -

If you ever played table-tennis, or plain ol' tennis or even basketball, you know that when you pass or shoot or whack away at the tennis ball, while applying SPIN, the ball behaves funny when bouncing off the ground...

A similar thing happens when applying spin to a ball simply going through the air!!!

Warning: Physics ahead

<physics>
Consider a ball going through the air in a velocity \(V\), and spinning away at angular velocity \(w\), like this here poorly executed diagram:

Well, there are a couple of things happening in this diagram, and I'll spare you the Navier-Stokes equations, cause they're a drag... literally :)

what happens is this:
essentially drag is proportional to the velocity, and so the drag on the lower side of the ball in this diagram is more pronounced than the drag on the upper side, ultimately, this means that the ball is turning downwards (in this diagram), or more generally in the direction of \(\overrightarrow{V}\times \overrightarrow{w}\).

the other thing that happens is the air in front of the ball is a tad denser than the air behind it and so the drag on the leading edge of the ball is more pronounced than the one on the late edge thus creating an additional effect in the \(\overrightarrow{V}\times \overrightarrow{w}\) direction.
This corresponds to the \(\eta\)  factor in the N-S equation, which is the factor that embodies the density of the medium, but the ball has to fly in really high velocities for this dynamic to be anywhere near being pronounced.

So that's what happens to a ball spin-flying mid-air, but what happens to a spinning ball when bouncing of a hard surface?

Well, for starters, if the ball bounces off a surface such that the spin is not perfectly perpendicular to the plane of incidence, that fraction of spin will simply give the ball a momentum component that is opposite in direction to the direction of the spin on the hitting edge.
For instance - if you spin the ball clockwise and it hits a wall the ball will bounce back and to the left, or in another instance like in this video, you apply a spin that goes in the direction of the ball on the upper side of the ball, and opposite on the lower side, thus when hitting the court the spin component donates additional speed to the ball after bouncing back from the court.
This move serves to trick the opponent, since we constantly gauge the approach of the ball and extrapolate where we should hit, but when the ball accelerates mid-shot, it throws off your intuition.

But, what if the spin component is perfectly perpendicular to the incident plane?
Well then, this case get's more interesting doesn't it? in this instance we have to turn to analytical mechanics and the deep understanding of a two-spring system which isn't that complex but I wont bore you with the details...

suffice to say that if we have a system of two springs, like so:

Two spring system

 We have two separate modes of oscillations, one is the combined up&down oscillation that would happen if a fat kid stands smack in the middle of this contraption and jumps up and down, and the other is the normal seesaw action we all know and love.

And so with that in mind, and the understanding that this is a nice model of what happens to the hitting edge of the elastic ball that bounces, we approach this dilemma.

So the ball approaches the incident plane in some kind of an angle, meaning that the leading edge hits the plane first and starts the double spring system, so we have the first mode contracting, and the second mode starts with the front "spring" contracting. since we have the same spring constant for both our springs (in this model) the cycle-time for both are the same, so the leading edge experiences the overall contraction (1st mode) PLUS the seesaw contraction (2nd mode).
the back edge experiences the seesaw contraction (2nd mode) when the overall mode (1st mode) is basically starting to extend thus the pressure on the back edge is significantly lower than that on the leading edge.

thus the normal force experienced by the edges are different accordingly, and thus friction isn't uniform leading to an overall addition of momentum in the direction of \(V\times w\).
<\physics>

So all in all the conclusion is that if you spin a ball in a right-hand spin (rightmost edge going forward) you get a swerve right, and if you spin left, you get a swerve left.

AND THAT"S HOW YOU BEND IT!!!!


Plain Ol' dirty playin'
Another great experience that is (for me) connected with water polo is the one of almost dying (again).
it turns out that water polo is also one of the most violent sports ever. it's probably comparable only to rugby.

We'll start with the amazing fact that, whatever the ref doesn't see, doesn't exist. bearing in mind the refractive nature of even still water, and taking into account these are less than still waters,  well, let's just say most fouls and injuries do NOT get noticed.

A typical water polo foul

and we'll finish it off with the unwritten law of water polo: if you dunked the ball, i.e. pushed the ball into the water, well, basically they dunk you. by the way, don't matter if it's your team or not, they'll dunk you!

Now, I didn't know that, and as a young kid, tired from all the egg-beating, I hung down on the ball for a second and dunked it. just a tad bit! I swear!!! when all of the sudden my own team member came up on me from the back and dunked me like there was no tomorrow, resulting in my almost dying, yet again....

Water polo at it's best :)

Fun fact: usually when people die from drowning they don't die from water filling their lungs, what usually happens is called "Laryngospasm" meaning the muscles in your throat contract violently, and you actually die from asphyxia.

This usually happens either from panic or due to swallowing hefty amounts of salt water...

So there you go, a "short" post with very few equations and a lot of physics, and best of all - another almost death for me! :)

Next time: Physics of floating stuff - submarines, boats, crowns and other stuff :)

A short remark on the significance of butterflies and their respective effects.

Abstarct:

"You keep using that word. I do not think it means what you think it means. "

Inigo Montoya

"The butterfly effect" is called that, not because of some cockamamie story about a butterfly farting somewhere to create the next hurricane Catrina..

I was looking at a friend's FB wall today when I realized they made a reference to the so called "butterfly effect", I want to set some things straight regarding that effect.

Usually when people use the term "butterfly effect" they use it without actual knowledge about what it really stands for, and where and when it was coined.

So, a little history lesson is probably called for here:

Of strange attractors, and non linear effects:

Some of you may have at some point or the other, had a chance to have a fatal brush with physics studies, be it in high-school, university or college or you simply had a pesky friend who was into that kind of kinky stuff and insisted on explaining, oh I don't know, general relativity to you while insisting you most certainly possess the means to understand and assimilate everything he "taught" you right there and then. well that's me.

Warning: physics rant here:
<Physics rant>
Well it just so happens, that MOST physics areas we encounter, even as PHD students or actual researchers, are areas of LINEAR PHYSICS. for instance quantum physics is strongly embedded within the framework of linear algebra, with operators that are linear by definition, another field which is strongly linear is electrodynamics, which for the most part (but not always) relies heavily on linear algebra tools ,methods and operators to provide refutable results.
</Physics rant>.

The world isn't linear, or in the words of my namesake, "it ain't necessarily so".
Looking for the linear approximation of some dynamic, is not entirely different than looking for the proverbial coin under the street-light.

It so happened that in a cold day in the late 1950s an egghead, or otherwise called a scientist, sat in his lab, punched numbers in his then top-of-the-line computer, and got meters upon meters of output sheets, containing numbers that would represent a meteorological system.

By a fluke, the researcher went out for coffee, while the computer heaved and clicked and buzzed and chirped and spewed out sheets of numbers.

When he came back, the computer was silent. no more chirping, just plain cold data. but the experiment was not over!!! since in those days the internal memory of a computer was very limited it became necessary to punch in the last output you got,and so proceed with the number spewing.

But alas, the variables were saved in the computer to a precision of six decimal places, yet the output was given in only 3! So by default the researcher rounded off the last output and introduced it into the computer.

What he got was very strange, he projected a certain result but got a result differing greatly from expectation.
So like every respectable researcher he had tried to reproduce the experiment... getting results that differ greatly from both expected values, AND previous results!!!

See, if it were me in my first year at the academy I would have just drawn the damn graph I knew was expected, shrugged my shoulders, and go to sleep.

But not our epic protagonist! He set out to delve deeper into what had happened and stumbled upon a great discovery!!!

The protagonist was called Edward (Norton) Lorentz and he found out, that in some systems, an infinitesimal change in starting conditions, translate very quickly into a gross divergence in results.

Not the same Edward Norton

To illustrate  here is a picture of a Strange attractor:

Strange attractor

Weird Al in: Strange repellent

Just to explain what's going on here, I'll disregard the right hand picture as the freak of nature that it is, and concentrate on the left hand picture.

This is a graph generated by this applet *, the applet simulates a Lorentz' system which is a meteorological model.
If you want to actually see it in action, just click anywhere on the white field, twice at the same place.

You will have generated 2 plots, or more if you have Parkinson's disease, that started at the exact same spot - or did they?
Actually no they didn't. see the computer saves the mouse pointer in a "float variable", thus it saves it to a degree of precision that we can't see, and I'm pretty sure the applet rounds off a couple of places after the decimal - thus recreating Lorentz' experiment actually...

So if you wait a while, you will see a divergence in paths taken by the two differently colored plots.

This graph is called "Lorentz' Butterfly" and THIS is why they call the effect "the butterfly effect", not because of some cockamamie story about a butterfly farting somewhere to create the next hurricane Catrina...


So this concludes the short history lesson, and here I go into proving the world is evil.

Well, no it's not, but I do have to make an attempt to show it is...

OK, so a lot of people use the so called "butterfly effect" to justify a line of reasoning that says something like this:

If you do something good now, for a certain person, it will generate more good deeds like the ripples on a pond, and like the butterfly effect, this will cause a major force for good in the world... or something like that.

Now let me be absolutely clear about this:
I DO subscribe to the notion of doing at least one random act of loving kindness a day, I think it's good for you, and if only I would remember this every day I would probably be a better human being right now.

BUT! (and it's a big butt) this line of reasoning sucks.
it's flawed in so many ways I can't even start counting all of them so I will just give up and point the one I was aiming at.

First off the butterfly effect is all about a small change in starting condition having a huge effect later on, what is described in the above dynamic is called a "chain-reaction".

A chain reaction is in essence a LINEAR dynamic, it has absolutely NOTHING to do with the butterfly effect (as far as I know).

Secondly and more importantly is this:

Invoking the butterfly effect assumes a non-linear system.
Assuming that if we do something good things get better in the world assumes strong linearity of the system.

These two assumptions are mutually exclusive and while the former is actually well documented, experimented and reproduced, the latter is a figment of our wishful thinking (that may be true in a convoluted way as far as I know, but is neither proven nor reproduced).

In general MOST dynamics in the world are non-linear, and we physicists make assumptions, and approximations to linear or at most quadratic cases in order to get prevailing approximate results we can later compare with experimental results.

For the most part I think most of us would agree that human behavior is non-linear as well thus, I tend to lean towards the first assumption rather than the second.

And if we take in account Murphy's law we instantly come to the grim realization that it is much more probable that if you do a good deed, something will go terribly wrong.
or differently put:
"No good deed goes unpunished" in-deed (see what I did there?).

But also, at the same time no bad deed goes unpunished, as almost EVERY change in starting conditions results in a huge divergence in results thus I propose that:
"No deed goes unpunished" -  and so we are all constantly punished... whether we do something or not - thus the world is an evil place.
QED.

But to end this on a lighter note, two things.
the first: if we all do random acts of loving kindness every day, the whole graph jumps up on the scale of good vs. evil, and so the whole dynamic is shifted up by a constant, it might be just the constant that may save us as a species.
so don't stop doing good things!!!

the second is this...
If I learned anything in my 30 odd years of life experience it is that when it comes to physics I am usually wrong the first time around and I always need further study.

oh and something else - you can argue with your wife, but ultimately, she's right and you know it.

And I mean this in the best possible way.... *GULP*

again, this was longer than expected... bummer.

I also almost died -again- yesterday, maybe I will tell you all about it in a future post...


* with permission. Copyright 1996, James P. Crutchfield. All rights reserved.

Physics of Lenses and Idiots (part II)

 Abstract:

Burning ships, perfect lenses, Red shirts

and weird Al

Hello again folks...

Just for spite. this post is just for spite.
well, not really but it's fun to say it is.

As some of you might remember I wrote an inhumanely long post, and my liege-wife told me to cut it short.
so I did, but no good deed goes unpunished or as theorized by the great-but-not-overly-sane Newton "To every action there is always opposed an equal reaction", and so it is with great yet perverse pleasure, that I give you the second part of the Lenses & Idiots Trilogy(!!!)

Not quite the shilling, though he probably minted some shillings...

 
Anyway, last time I was reminded of two nice stories....

The first about Archimedes and the Siege of Syracuse, where supposedly the defending forces used about 300 round bronze shields, polished to a shine, in order to concentrate sunlight, and burn the marauding fleet to a crisp.did anyone say he loved the smell of napalm in the morning?
well let's put that theory to the test:

groceries:
1. 300 bronze body shields of area 1.3m x 0.7 m, polished to an "\(\varepsilon\)" sheen.
2. the sun - Earth's solar constant. - 1360 \(\frac{J}{sec\cdot m^2}\)
3. auto-combustion temperature of wood - at the most \(~ 450^{\circ}c\)
4. a  distance estimate say at 500 meter.

and now:

300 shields yield an incident area of about 273 square meters. thus according to our previous calculations we simply substitute the respective areas and get the temperature at a 1 square meter of surface on the wood to be about  1600K at  \(\varepsilon=1\), which means this should burn like kindle wood but experimental results here assert differently... so what went wrong?

well, a couple of things:

first off, bronze is not a mirror, meaning for starters, \(\varepsilon\neq1\) in fact not even close.
secondly, we assumed the image area of the shields on the side of the boat is at the same size of the shield. this is miserably wrong.

Consider a piece of shiny metal or a wrist-watch you use to blind the lecturer at a "Mechanics and Special Relativity" course, it is pretty obvious that the area of light traced by the image of the watch's surface is larger than the actual surface of the watch right?

The added length of a line image is given by
\[L_{image-line}=L_{original-line}\left(1+\sin(\alpha)r\right)\]
so basically the area increases like
\[A=A_{original}\left(1+2\sin(\alpha)r+\sin^2(\alpha)r^2\right)\]
So even though the flux is multiplied 300-fold, by all the shields reflecting the sun at the same site, still the area of incident is also multiplied, and even if we take only the linear approximation and not the whole deal we STILL get  the incident area to grow like \(1+cr\) thus:
\[\Phi_{boat}=1360\cdot 300 \cdot \frac{1}{1+c500}\]
And even taking the  angle to be fairly small let's say \(15^{\circ}\) we get the temperature to be about 410K which is about \(110^{\circ}c\) and that with an \(\varepsilon\) of 1!! its enough that we assume \(\varepsilon\) to be 50% which is  of course a gross over estimate, we get a temperature of about \(50^{\circ}c\).
That kind of a temperature is not enough to boil water, much less  trigger wood auto-combustion ,in fact, even if we were talking about perfect reflective surfaces at vacuum conditions this just ain't enough or differently put: "I just can't do it captain, I don't have the power" - Hmm... maybe if the wood was treated with a combustion agent first...

So the obvious conclusion here is that the people of Syracuse had an inside agent!!!
They had a traitor in their midst!!!!

Or most likely this never actually happened....

Warning- computer geek humor below:

<Computer geek humor>

try{kill_Redshirts();}

Throw Exception("attempted divide by zero.");

By the way, did you ever notice that while redshirts are being slaughtered by the dozens in the original star trek series, Scotty actually wears a red shirt, but is immune to the redshirt-death-rule?
hmm... makes me want to throw an exception....
</Computer geek humor>

Anyway...

Let's see what happens when we apply a lens at the 1 square meter target area, with such accuracy as to concentrate the rays at an area no bigger than 1 square centimeter, we get with the same type of calculations, even taking into account the \(\frac{1}{r}\) factor, a temperature of about 4100K, and if we have for example a solar tower, surrounded by perfectly reflective parabolic mirrors, that span an area of,say, 500 square meters, all directed at a Zeiss Parabolic lens of a perfect nature, with the longest mirror to lens distance of about 200 meters, and an \(\varepsilon\) factor of 70% we get, about 5400K.

Again what melts at 5400K?
hmm a short list that includes about everything...
a short list of things that actually BOIL at 5400K includes among others:
Carbon, Platinum, Rhodium, Titanium, Silicon, Palladium, Cobalt, Nickel, Iron etc...

For a reference the surface temperature of the SUN is evaluated at about 5800K.


But, interestingly enough, a star-ship that tries to show-off and make a run near the sun will have to withstand the ludicrously high temperature of about  5 million Kelvin of the sun's corona, about 3 orders of magnitude higher then the surface temperature - thus while bathing in the sun's surface sea of fire might be enjoyable, getting there could prove messy.
Of course this goes against star-trek TNG's episode  "redemption II" story-line, when a Klingon Bird Of Prey survived the corona only to explode on the surface...

A Klingon BOP taking a nice warm bath in a sun.

The second story goes something like this:
I have a friend who's father is the head of solar energy research at a notable institute.
He told me the story of acquiring an almost perfect Zeiss lens. and it goes something like this:
One day, he and one of his colleagues were wondering the streets of Dresden Germany,  biding their time in between lectures, when they saw a group of kinder-garden age kids, playing around with a large lens, having fun with reflections and images.
This physics professor immediately recognized the lens to be of tremendous quality, and approached the kinder-garden staff with an offer to buy the lens.
They really didn't know what they had in their hands, or simply didn't care too much, but they pretty much GAVE the lens away to that professor, with a simple demand - give us a lens that will do what this one know how to do, so the kids will be able to continue playing. He gave them an ordinary, albeit good quality lens, and basically got this magnificent lens for a song...
It is that lens that is still on the top of the solar tower at his laboratory, and through which they get temperatures high enough to burn through steel as if it was butter.

Now as promised a short mass on Fresnel Lenses vs. parabolic lenses:
In short - a parabolic lens has the interesting feature where every light beam coming straight from infinity hits the same focal point, and vise-verse, if you put a light source at the focal point of a perfect parabolic lens, you can be damn sure that all the light goes straight ahead - in fact that is how the high beams on your car works, in other words - know the mechanism of the bastard that blinds you!!!


To show that, we take a simple parabola - \(f(x)=\alpha x^2\), and we will take a beam that comes from straight up and hits the parabola at \(x=x_0\).
the tangent to the graph at that point is given by \(y=2\alpha x_0 x+b\), and the beam that hits the parabola at that point is diverted to a line that looks like \(y=-\frac{1-(2\alpha x_0)^2}{4\alpha x_0}x +c\) Where calculating \(c\) yields \(c=\frac{1}{4\alpha}\).

The fact that where the diverted light ray cuts the "y" axis, doesn't include \(x_0\) as an argument already shows that all rays meet at the same place i.e. at the same focal point at \(\left(0,\frac{1}{4\alpha}\right)\).

A Fresnel lens is a neat way to manufacture a closely approximated parabolic lens, while reducing physical bulk and bill of materials. What you want to do is take a regular parabolic lens, slice it in regions, and piece together what you got. I found a nice illustration of this on the net, here is the picture...

Fresnel Vs. Parabolic lens, notice the corresponding regions.

Thus it is pretty clear, that while Parabolic lenses capture all light rays, it is heavier and bulkier than the corresponding Fresnel lens.
On the other hand Fresnel lenses capture MOST light ray, but not all, so while being lighter and more compact, it creates some distortion in the image, depending of course on the quality of material used, and the "resolution" or the density of "cut&paste" done to the original Parabolic lens to get the Fresnel lens.

Of course this is a very low concern for lighthouses, and other navigational signs and lights, like for instance masthead, port and starboard lights and also port entry and hazard lights.

And so we see that using Fresnel lenses is more efficient in the long run, because it takes up more energy to rotate or move a bulky and cumbersome lens then a lightweight one, and of course reducing weight and energy immediately corresponds to corrosion and mechanical faults ratio reduction.

So why not use a parabolic reflective surface? this could be lightweight as well as cheap AND efficient in terms of light reflection?
Well I think today most mechanisms where we have enough space to imbed that surface, actually DO use reflective parabolic surfaces, another good example of this would probably be the lenses and mirrors inside a telescope.

As for why NOT to use parabolic lenses and mirrors? - mainly physical space considerations probably, but also, in older parabolic mirrors I THINK there might have been significant energy losses via a nifty little mechanism called the "skin effect" but that is all for now, I may discuss this effect in a different post.


Wow, this was long, I hope you enjoyed this...

If indeed you have, you have the making of a true nerd. Oh and sorry I didn't get to almost die in this post, I have plenty of cases in which I did, and I hope they will always continue to be of the "almost" type in order for me to continue recounting them...

a true nerd...

By the way that's Time Independent Schrodinger Equation (a.k.a TISE) for a particle in central E&M potential given by a point charge plastered there behind weird Al. It's used ,for instance, for calculations regarding the Hydrogen atom .... - there I proved I'm a nerd, even though I'm not fluent neither in JavaScript or Klingon, and I hate mayonnaise. 

As always, next time : oh... why bother...