tag:blogger.com,1999:blog-48270591515702534282024-02-07T07:00:19.302+02:00House of physics funThoughts of physics and other stuff, while waiting for shit to compileAnonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.comBlogger16125tag:blogger.com,1999:blog-4827059151570253428.post-29971368289814608522019-01-15T19:30:00.000+02:002019-01-15T19:30:43.736+02:00Procrastinating, super-powers and caloric intake<div style="text-align: center;">
<u><b>Abstract</b></u></div>
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Things that keeps me up at night, how much calories in<br />a short plasma burst, and things I am <br />doing instead of writing articles.</div>
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If you are a member of the male caste of the human race, it is very probable that at some point or the other you have dreamt or even wished that you had super powers. Some of us (gee I wonder who...) still do. By the way in the spirit of equality, this is not to say that being a female precludes you from having the same super-aspirations, it's just that men are basically over-grown kids, thus we tend to sustain this dream longer into adulthood.<script type="text/javascript">
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Who among us haven't mused about having superman's awesome super-power arsenal. But most of us would settle even for the modest powers of, eh, let's say, Cypher which is essentially that he is a polyglot. </div>
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At any rate, this past week I was a bit under the weather, and one of my computers crashed, which gave me a good excuse to procrastinate. One of the things I did this week was binge on the first season of 'Titans' - the DC young superhero team comprised of Dick Grayson aka NightWing, Raven, StarFire, and Beast Boy. </div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLzFmwqCm-ySL9rLU1gx9qe62PeiEnZilYDERR8MGpAtjbj7yQ_d69FnVuE4RiU1LvwsSyXqwJgcZKu2-H5yKQMgrT7mF5aFUGWbbVkqoN8Cu31HCkZY4XFhEZdHtI0ISzZrIDf4WYSaY7/s1600/StarFire1.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1076" data-original-width="1600" height="267" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLzFmwqCm-ySL9rLU1gx9qe62PeiEnZilYDERR8MGpAtjbj7yQ_d69FnVuE4RiU1LvwsSyXqwJgcZKu2-H5yKQMgrT7mF5aFUGWbbVkqoN8Cu31HCkZY4XFhEZdHtI0ISzZrIDf4WYSaY7/s400/StarFire1.JPG" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12.8px;">Burning daddy to a crisp at a gas station, with super-heated plasma bursts</td></tr>
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The thing about being both a geek AND a physicist is that they sometime interfere. destructively.</div>
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This happened to me last night, as after watching two episodes of the series, I got to thinking, other than being a full on badass, does StarFire really work? I mean physics-wise? </div>
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Her superpowers are (at this point in the plot) having the ability to discharge super heated plasma (I guess?) from her body, and a slight healing factor. It is also implied that her powers are dependent on some inner reservoir of energy. Which got me thinking. What might be the energy budget of such a thing, I mean at some point we see her burning a bad guy to a crisp, and at no point in the series do we actually see her eat anything. She drinks some hard liquor sometimes (tequila, whiskey shots etc.) but she doesn't seem to eat anything. This begs the question - since she exerts bursts of energy, how does she replenish? </div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjliwiBmFcJNj2fXktO1dy6RLnhFeyi57fAVp_QSaCcIcsRis32ouBUaf9wWvhg3D4vwAA6gpO-VEaufGoq9hoxm_w2W-ak1sgntCsLoTvO6dFaBjcLVDXYmqe95fAZcHGIbdOOvo6HAJvX/s1600/StarFire2.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="774" data-original-width="1600" height="192" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjliwiBmFcJNj2fXktO1dy6RLnhFeyi57fAVp_QSaCcIcsRis32ouBUaf9wWvhg3D4vwAA6gpO-VEaufGoq9hoxm_w2W-ak1sgntCsLoTvO6dFaBjcLVDXYmqe95fAZcHGIbdOOvo6HAJvX/s400/StarFire2.JPG" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Well Done, extra crispy.</td></tr>
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Supposedly she gets her powers from the sun, like good ol' Kal, and and some point she says she needs several hours in the sun to fully recharge, but that math doesn't really work at all.</div>
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<span style="color: red;"><Warning: Physics ahead></span></div>
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At one point we see her showcasing her powers by burning the body of a tractor. Usually cars and other heavy machinery's body is made out of steel. In fact the average automotive contains about 55% steel, which breaks down to chassis and body, the latter is the majority of the percentage. Some high end cars use aluminum for the body, but this is both rare and expensive. This is mostly done to decrease total weight in favor of energy conservation.</div>
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So she probably burnt through steel or iron, and since we don't see any spill-over it makes sense to assume she actually vaporized the iron. The latent heat for the vaporization of iron from room temperature is roughly given by 350 kJ/mol. From the picture we can assess the vaporized area as about \(\pi\;(m^2)\) and the average sheet metal used in cars bodies is about 1.6 mm and mass per square meter of 11.84 \(\frac{kg}{m^2} \) so the total mass of the evaporated material turns out to be about \(37.2\; kg\) which is about 666 moles of Iron. This means that this short burst of about ~3 seconds delivered at least 233 MJ, which is a whopping 55712 kcal. </div>
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Let me illustrate this by using food.</div>
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This is the equivalent of 24 Large cheesy bite pies from Pizza Hut. Not slices. Whole pies.</div>
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This is the equivalent of 30 Big Mac meals, complete with a large order of fries and a big shake.</div>
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In 3 seconds.</div>
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And she never eats a darn bite.</div>
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So let's examine the story of solar charging.</div>
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The solar constant, which is the mean solar energy density flux on earth's surface is roughly \(1.362 \frac{kW}{m^2}\), let's call the human body surface at about \(2\; m^2\) thus it will take StarFire about \[ \frac{233 000}{2\cdot 1.362}\simeq 85,500\; sec \simeq 24\; hrs \]</div>
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<span style="color: red;"></Warning: Physics ahead></span><br />
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Which means that for that paltry 3 seconds of showing off she should have sun-bathed for that whole day. Assuming obviously she absorbs everything. With 100% efficiency. And a 100% efficiency discharge.</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEht_Ciq8zJUPa2KPwLyppRnKrFAg76FlhS1OfMPwiF9P6p6dZmSOIk_tLxsF1xBwObgYBYoiMx_Ha2A7aVPApYYMSAOfqqtaD9UJI2ToxLI6051mD7RmRbazoPmTW6HBbo5zsfWZalT4JDF/s1600/StarFire3.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="929" data-original-width="785" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEht_Ciq8zJUPa2KPwLyppRnKrFAg76FlhS1OfMPwiF9P6p6dZmSOIk_tLxsF1xBwObgYBYoiMx_Ha2A7aVPApYYMSAOfqqtaD9UJI2ToxLI6051mD7RmRbazoPmTW6HBbo5zsfWZalT4JDF/s320/StarFire3.JPG" width="270" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Lightly toasted tractor anyone?</td></tr>
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Anyway, with a steady diet of maybe 15 large burger meals a day and a strict sunning regiment of say all the available sun per day, she might actually be dangerous for more than 3 seconds a day.</div>
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Now I get why she's such a devout martial arts student... </div>
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She only has so much juice in her.</div>
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By the way, this makes me curious about scenes of female super-heroes eating. For some odd reason I can't seem to recall to many of those. Most female super-beings that were shot eating are either villains or straight up horribly mutated beings. hmmm....</div>
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Having said that, it was fun to watch the first season. And as an added bonus we get to see Deanna Troy smashed by a car.... which is positive in a way. I never cared for her, nor Riker for that matter. </div>
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Be that as it may, the last Purim party my wife and I attended, we went as Riker and Troy, we had to be a couple. She was pregnant. No way was she pulling anything but the infamous Troy dress. </div>
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Plus, I wasn't nearly as Picard-esque as I wanted to be, and I don't think she would have gone for the Crusher red hair look. Oh, right, and being that tool's mom. Awful.</div>
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Anyway, now you know what keeps me awake at night. Energy calculations and super heroes. I know. Lame.</div>
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I need to go back to work. Damn.</div>
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Oh right, another instance of superheroes charging up....</div>
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Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-85106338621526546672018-09-30T14:26:00.000+03:002018-09-30T14:26:20.602+03:00Physics of overeating and spaghetti (well, actually strings) <br />
Jewish holidays are a horrible thing. Horrible. Especially so for the spatially challenged (fatso-es, like yours truly). Among those of Jewish origins, it is a known fact that every Jewish holiday could basically be described thus: "Hey look! they tried to kill us, we survived, let's eat!!!". Even in the Jewish day of atonement, where you are supposed to fast, if for any reason you are not allowed to - guess what: you need to EAT!<br />
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<tr><td><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgC1xyM0IBSMAdXYynzROQSoVTzYZMkt4vUzF596sC46d1Qfpa-Nx0Pj1Xhg2flSjzR_7CavPpFleq1Rzi_-I1c6gtdXaR5PbRwEvmUko-OxHN3t-2t8L2AeZffoF6dM6jsPPU9BDfvH7q7/s1600/tjones6.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgC1xyM0IBSMAdXYynzROQSoVTzYZMkt4vUzF596sC46d1Qfpa-Nx0Pj1Xhg2flSjzR_7CavPpFleq1Rzi_-I1c6gtdXaR5PbRwEvmUko-OxHN3t-2t8L2AeZffoF6dM6jsPPU9BDfvH7q7/s200/tjones6.jpg" width="185" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Before the holidays</td></tr>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH8LCsrBE-KjTJrpTxriobX0z6Sl9g5vIttNijnFN69tQZNOBD8p2wNd5-ay4yRBL67E9PXMwrODy-rSAgMUL4lJHbmlFo3NLd4pzDqvVwIzMC9vb8DnZj999SrW0PiwlTT6e9S9lUbEFO/s1600/terryJonesFat.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="151" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhH8LCsrBE-KjTJrpTxriobX0z6Sl9g5vIttNijnFN69tQZNOBD8p2wNd5-ay4yRBL67E9PXMwrODy-rSAgMUL4lJHbmlFo3NLd4pzDqvVwIzMC9vb8DnZj999SrW0PiwlTT6e9S9lUbEFO/s200/terryJonesFat.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">After the holidays</td></tr>
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Well, I promised at one point to elaborate on at least one way of how being fat can save your life. This might be actually a reference to string theory (!!). Let's look at the energy that is present in a standing wave. Oh shucks!, some of us might need an explanation of standing waves - so here goes: </div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8ySEHEZNPcP0JClt2btH6EQyM_gr20WABI7UDC8jxt9AGKAi1SjQ4j-mas6v0Bo8Zga_l3rAEadxzXVIJauY2Lj3yihC8iTxzseK5ZMzNKyM9rL-VAxUeADeRE76WInoKWRpHT1mD5o32/s1600/DD.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="267" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8ySEHEZNPcP0JClt2btH6EQyM_gr20WABI7UDC8jxt9AGKAi1SjQ4j-mas6v0Bo8Zga_l3rAEadxzXVIJauY2Lj3yihC8iTxzseK5ZMzNKyM9rL-VAxUeADeRE76WInoKWRpHT1mD5o32/s400/DD.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A game of DoubleDutch - standing waves in action!</td></tr>
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Take a rope, tie one end to something very heavy (say a refrigerator handle), and wiggle the other end. You will notice that as you wiggle the rope's end a "hump" in the rope is created, it travels to the refrigerator door, and back again.</div>
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In purely ideal conditions, the hump would travel back and forth forever. However the world, much like my bank account situation, is NOT ideal. This means that <b><u>dispersion</u></b> and <b><u>energy loss</u></b> occurs, which is a main theme in our story. </div>
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So, for the potentially stray physicist that may have come across my blog, here is the full derivation of string dynamics and (average) energy in standing waves.</div>
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<span style="color: red;">< warning - physics ahead> </span></div>
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We can do it in several ways, but we'll take a shortcut:</div>
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We will first assume we have a wave, derive the wave equation, and find the associated energy. Here goes:</div>
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The wave equation reads:</div>
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\[\frac{\partial^2 \psi}{\partial t^2}=v^2 \frac{\partial^2 \psi}{\partial x^2} \]</div>
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Which could be solved by method of variable separation or by ansatz. Let's go with variable separation:</div>
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\[\psi=X(x)\cdot T(t) \Rightarrow \frac{\ddot{T}}{T}=v^2\frac{X''}{X}\] </div>
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Well, now we have two a-priori uncorrelated functions of different variables, which are equal regardless of time and space, thus they both equal a constant.</div>
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And so, we get a general solution: \[T=\tilde{A}e^{Ct}\;\; ; \;\; X=\tilde{B}e^{Dx},\] with the connection C^2=v^2 D^2. Now we \(C\) has dimensions of reciprocal time thus we may call it \(\omega\) up to a numeric constant. Also \(D\) has dimensions of reciprocal length, thus we can call it \(k\) such that \(\omega^2\propto v^2 k^2\).</div>
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Now by process of eliminating the constants being purely real, or zero, we are left with a solution of the form: \[\psi=Ae^{i(wt-kx)}+Be^{i(wt+kx)}\] which correspond to right propagating waves and left propagating waves. </div>
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In order to completely solve for \(\psi\) we now need to apply boundary conditions. For people interested in string theory this is basically the whole story behind "Branes" (i.e. D-Branes etc...), meaning the applying of boundary conditions forces some interesting physical behavior of these objects.</div>
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BUT! in order to get what we want which is the average energy per wave solution. We can get that by realizing that in a wave dynamic energy is interchanged between kinetic and potential terms, so if we are only interested in a rough estimate we need only to consider the \("\frac{mv^2}{2}"\) term. Meaning, for a small section of the rope with \(\mu_0\) mass density per unit length we have:</div>
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\[E_k=\mu_0 \frac{v_{\perp}^2}{2}=\frac{\mu_0}{2}\left(\frac{\partial \psi}{\partial t}\right)^2\] </div>
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This means that the total energy for a string vibrating with a frequency \{\omega\) is roughly:</div>
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\[E_k\propto \mu_0 L A^2\omega^2\]</div>
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For an open or closed string there is a connection between the length of the string and the frequency: \(\omega_n \propto n\) in other words: </div>
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\[E_n\propto \mu_0 L A^2 n^2\]</div>
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Now, there is a nifty principle in physics called equipartition. It's not always true, the theorem that actually holds is called the "virial theorem", but equipartition is a close and inaccurate relative.</div>
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Equipartition states that given a total quantity of energy, the partition of energy to the different constituents is basically equal. This means that plucking a guitar string will start all possible frequencies.</div>
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This means that if all active modes have the same amount of energy \(\epsilon\), the amplitude of each mode goes like the reciprocal of the wave number. This gives us a nice "tool" - If we know the amplitude of the "slowest" mode activated, we only need to count modes until the mode amplitude is comparable to "noise". The mean free path of water is around 3 angstrom. So assuming we can treat human bodies like "ugly bags of mostly water" ,given the initial amplitude is \(A\) the number of activated modes are about \[N=\frac{A}{3\cdot 10^{-8} cm} \].</div>
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<span style="color: red;">< /warning - physics ahead> </span></div>
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Suppose now, that you <a href="http://beastraban.blogspot.co.il/2011/12/physics-of-falling-from-second-floor.html" target="_blank">fall from a second floor window</a> only to land squarely on your back. Let's assume the displacement of the compressed body is about a \(cm\). this means that by the previous "back of the envelope" assessment we have about \(10^7 \sim 10^8\) activated modes. the weight of a water molecule is about \(3\cdot 10^{-26}\; kg\), the speed of sound in water is about 1500 m/s, so the basic frequency is given by \(\omega=\frac{c_s}{2\pi \lambda}\simeq \frac{236}{\lambda}\) where we can substitute the basic wavelength \(\lambda\) with the length of the body \(L\). Thus the basic energy quote \(\epsilon\) is given by:</div>
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\[\epsilon = \frac{3\cdot 10^{-26}}{2}\cdot L \cdot 10^{-4}\left(\frac{236}{L}\right)^2 \simeq \frac{8.5}{L}\cdot 10^{-26} J\].</div>
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We will now suppose that \(L\simeq 2\), thus the characteristic energy quanta of energy is given by \[\epsilon \simeq 4 \cdot 10^{-26} J\]. So naively we get the energy in the form of wiggling to be comparable to \[E=4 \cdot 10^{-19} J\]. However! We are dealing with a 3 dimensional body, so mode degeneracy kicks in, as well as some other considerations.</div>
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I won't go into detail, rather I will state that the number of activated modes in a 3 dimensional body goes like \(N^3\) (for high numbers). So let's get a minimal assessment for the total energy: \[E=4 \cdot 10^{-5} J\] This seems like a very small number.</div>
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A different assessment is given by understanding that in the above equation \(\mu_0 \cdot L\)</div>
is the total string mass, we can replace it with the body mass ~ 100kg.<br />
<div class="separator" style="clear: both; text-align: left;">
Calculating this in this way we get: \[E \simeq 1100 J\] which is about a \(\frac{1}{6}\) of the energy deposited in that crushing interaction.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
What I believe ACTUALLY happens is that the initial blow activates only the first mode, and then by a process of interference, the energy gets distributed to the other modes. This energy then is released into heat (and sound, and screams, and pain) which leaves the effective blow half as deadly.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
How does being fat comes into all of this? Being fat means that the actual volume you have for containing wiggling movement is a lot larger, thus a lot more modes can reside in the same space. It also means that the initial displacement will be times 2 or 3 than if your a skinny dude. This means that the amount of energy that can go into that is bigger from the get go.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
and that's my two cents about how being fat can actually protect you and delay your meeting with John Cleese.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDHzM_BLuMU6URclKjNMIu-nbE1eukKf-x4xkH2BjCpUbOEzMgHL4CS6q8xeTBKGssJDbIcKXzeNChFf54znXt9XbYM8myOKYEtOMvimFNWclA7BPQcv8ruV1SPQab8lMZaeiJFk-4UTvf/s1600/cleese.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="325" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDHzM_BLuMU6URclKjNMIu-nbE1eukKf-x4xkH2BjCpUbOEzMgHL4CS6q8xeTBKGssJDbIcKXzeNChFf54znXt9XbYM8myOKYEtOMvimFNWclA7BPQcv8ruV1SPQab8lMZaeiJFk-4UTvf/s400/cleese.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">"The salmon mousse!!! Darling you didn't use canned salmon did you?</span></td></tr>
</tbody></table>
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<br /></div>
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<br /></div>
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<br /></div>
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<br /></div>
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Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-20653639404672814532013-06-23T16:59:00.000+03:002013-06-23T16:59:11.429+03:00Prof. Avi Schiller's departure<div style="text-align: center;">
<h3>
A post will be dedicated to his memory,</h3>
this will take some time but I hope in a couple of days<br />
something nice will be written.<br />
<br />
<div style="text-align: left;">
This is by no means a eulogy, but I feel it is fitting to dedicate something to this great man.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
It is customary to dub people "great" for their accomplishments and otherwise professional stature.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Avi Schiller was a great man, simply because he was a "mench". One of the few. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: center;">
<i>He would also say: A boor cannot be sin-fearing, an ignoramus cannot be
pious, a bashful one cannot learn, a short-tempered person cannot teach,
nor does anyone who does much business grow wise. <b>In a place where
there are no men, strive to be a man.</b> </i></div>
<div style="text-align: center;">
</div>
<div style="text-align: center;">
-Old Hillel, Ethics of the Fathers, 2,5.</div>
</div>
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</script>Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-52170599659949423032013-05-03T08:56:00.001+03:002013-07-01T13:21:38.345+03:00Vikings, Relativity and Couch potatoes<h2 style="text-align: center;">
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</script><span style="font-size: large;">Abstract: It's not my fault I'm lazy, <br />Physics made me this way</span></h2>
<h2 style="text-align: center;">
<span style="font-size: x-small;">dedicated to my sister-in-law and her fresh hubby's mawwiege</span></h2>
<h2 style="text-align: center;">
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<tr><td class="tr-caption" style="text-align: center;">Mazal Tov!!!</td></tr>
</tbody></table>
</h2>
<div style="text-align: left;">
Recently I attended my sister-in-law's wedding...</div>
<div style="text-align: left;">
After a full month of being forgotten, taken as granted and ultimately being treated as some ungainly babysitter to my own girl, it has arrived...<br />
<br />
The event dreaded and anticipated, probably by the bride and groom, but also by yours truly - <br />
<div style="text-align: center;">
<br />
<h4>
<span style="font-size: large;">THE WEDDING</span></h4>
<br /></div>
Awesome and gruesome, with a multitude of family-related potential calamities, just waiting to happen, hanging on the off chance of someone sitting in the wrong seat or uttering the wrong sentence, incurring a death sentence, with the obvious exception of the crazy and otherwise fully-certifiable-bunkers pseudo-relative (She can say whatever the hell she wants, nobody's home thus no-one really listens or cares what she says).<br />
<br />
The food was good, the company better, and the music divine. well, not really... the first two statements are correct whereas the third - not so much....<br />
<br />
The DJ was poor, probably not in the fiscal sense after the presumed hefty sum he was paid, but poor taste in music, poor know-how of his trade, poor execution, and ultimately, the only reason his music was anywhere near palatable, was this was their wedding and we were so damn happy for them, that everything else vanished, not unlike the 10th order Taylor term for a quadratic potential function. <br />
<br />
At some point the music was pounding so hard, I literally saw hordes of vikings working at a huge smithy, pounding away with all their might to mold some unfathomably hard metal into some equally mysterious shape. </div>
<div style="text-align: left;">
<br />
Now, if you follow my posts you know full well what happens to me when faced with potential demise - physics comes into play!<br />
<br />
I immediately went on to contemplate vikings and physics, and stumbled upon Dichroism, which is basically when light polarized in different directions go through a certain special material, only one polarization survives (the otherwise polarized rays are reduced to a point of vanishing altogether).<br />
<br />
Now since sunlight is non-polarized, this normally doesn't mean squat, but when there's heavy cloud cover, the light rays disperse over the clouds so you get a general bright haze, but no obvious direction for the sun.<br />
<br />
Enter dichroism: by using dichroic glass the light that passes through the stone is brightest in the direction of the sun, because right in front of the sun (through the clouds) the light is differently polarized.<br />
<br />
The trick is you have to find the right direction to hold the dichroic stone in, else you'll basically get nothing.<br />
<br />
Anyway, this dichroism trick is how vikings were able to sail across wide stretches of open sea, since they didn't have land in sight for coastal navigation, and either they didn't know their celestial-nav basics, or the skies were constantly cloud-covered.<br />
<br />
BOOM!!<br />
<br />
One of the loud pounding beats threw me back into consciousnesses, in time to realize how horribly out of shape I was...<br />
You see, I was actually dancing (well, more like contorting) on the dance floor, sweating and panting, and the all too familiar "why am I so lazy?" question kept burning in my mind...<br />
<br />
Which again set a chain of events in my mind culminating with physics:<br />
<div style="text-align: center;">
<br /></div>
<div style="text-align: center;">
<h4>
<span style="font-size: large;">It's not my fault!!! Even physics is lazy!!!</span></h4>
</div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: left;">
And I will prove this immediately:</div>
<div style="text-align: left;">
First off, remember old Newt's law: </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
</div>
<div style="text-align: center;">
<b style="font-family: inherit;">"</b><span style="font-family: inherit; line-height: 19.1875px;"><b>Every body persists in its state of being at rest or of moving uniformly straight</b></span><b font-family:="" inherit="" style="font-family: inherit; line-height: 19.1875px;">forward, except insofar as it is compelled to change its state by force impressed"</b></div>
<div style="font-family: inherit; font-weight: bold; line-height: 19.1875px; text-align: center;">
<b style="font-family: inherit;"><br /></b></div>
<div style="font-family: inherit; line-height: 19.1875px; text-align: left;">
<span style="font-family: inherit;">Translated: every body is basically a fat blob. you really have to exert force to change its current state.</span></div>
<div style="font-family: inherit; line-height: 19.1875px; text-align: left;">
<span style="font-family: inherit;"><br /></span></div>
<div style="font-family: inherit; line-height: 19.1875px; text-align: left;">
<span style="font-family: inherit;">I will now delve into a bit of Lagrangian physics to prove a point:</span></div>
<div style="text-align: left;">
<br />
<span style="color: red; line-height: 19.1875px;"><Warning: Physics ahead></span><br />
<span style="line-height: 19.1875px;">This is a simplification of deriving Lagrangian Mechanics but here goes:</span><br />
<span style="line-height: 19.1875px;">Consider Newton's 2nd Law:</span><br />
\[F=ma\]<br />
And if you're a physicist you know this is a simplification of:<br />
\[\frac{dP}{dt}=F\]<br />
<span style="line-height: 19.1875px;">So, now, applying D'Alambert pricnciple we can constrict ourselves to "external" forces, and taking F to always mean a conservative force (i.e. arising from some scalar potential function) we can write:</span><br />
<span style="line-height: 19.1875px;">\[F=-\nabla \left(U\right)\] where U is just the potential function otherwise known by the name "Potential Energy". </span><span style="line-height: 19.1875px;">If we constrict ourselves to one dimension we can see:</span><br />
<span style="line-height: 19.1875px;">\[F=-\frac{\partial U}{\partial x}\] </span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="line-height: 19.1875px;">Looking at the left hand side of the above equation where we treat \(P=mv\) let's explore the relations between the kinetic energy term - \(\frac{mv^2}{2}\) and the above term.</span><br />
<span style="line-height: 19.1875px;">It's fairly easy to see that:</span><br />
<span style="line-height: 19.1875px;"> \[ mv=\frac{\partial}{\partial v}\left(\frac{mv^2}{2}\right) \]</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="line-height: 19.1875px;">Let's call the kinetic energy term T and the potential energy term U, so we now get:</span><br />
<span style="line-height: 19.1875px;">\[\frac{d}{dt}\left(\frac{\partial T}{\partial v}\right)=\frac{\partial (-U)}{\partial x}\]</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="line-height: 19.1875px;">Understanding kinetic energy to be non-dependant on location, and potential energy to be solely location dependant we can add expressions to both sides that vanish in derivation thus defining:</span><br />
<span style="line-height: 19.1875px;">\[\mathcal{L}=T-U\]</span><br />
<span style="line-height: 19.1875px;">to get the Euler-Lagrange equations</span><br />
<span style="line-height: 19.1875px;">\[\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial \dot{q}}\right)=\frac{\partial\mathcal{L}}{\partial q}\] </span><br />
<span style="line-height: 19.1875px;">Where here I've also switched to some generalized coordinates \(q,\dot{q}\).</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="line-height: 19.1875px;">Now, defining the action \(\mathcal{A}\) as the time integral of the Lagrangian \(\mathcal{L}\), or in other words:</span><br />
<span style="line-height: 19.1875px;">\[\mathcal{A}=\int\mathcal{L}dt\]</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="line-height: 19.1875px;">We have now successfully exchanged Newton's rules formulation by the<b> "Principal of least action", </b></span><span style="line-height: 19.1875px;">since it follows that demanding minimal Action, actually yields Euler-Lagrange equations which are the equations of motion.</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="line-height: 19.1875px;">A word of caution though: </span><br />
<span style="line-height: 19.1875px;">First off this derivation is a sketch, the true process is a lot more rigorous, but I assure you, it works.</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="line-height: 19.1875px;">Secondly, Lagrangian Mech is a heavy canon, all you need to do is describe the problem in some sensible coordinates, find the kinetic energy and potential energy, perform some derivations...</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="line-height: 19.1875px;">and BAM!!! </span><br />
<span style="line-height: 19.1875px;"><br /></span><span style="line-height: 19.1875px;">you get equations of motion, as well as conserved quantities and symmetries of the system like magic!!</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="line-height: 19.1875px;">But, since we use a couple of assumptions in deriving the Lagrangian, this actually covers LESS general physical situations. In reality there's ALWAYS friction, which is by no means a conserving force, thus adaptation to Lagrangian Mech are needed and they are not always as simple.</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="color: red; line-height: 19.1875px;"><Warning: Physics ahead/></span><br />
<br />
<span style="line-height: 19.1875px;">The obvious conclusion of all this mess, is that Newton laws of motion are equivalent (under some basic assumptions) to a first principle that states the following:</span><br />
<br />
<div style="text-align: center;">
<span style="line-height: 19.1875px;"><b>"Every physical system aspires to minimize the action taken"</b></span></div>
<div style="text-align: center;">
<span style="line-height: 19.1875px;"><br /></span></div>
<div style="text-align: left;">
for instance, water will flow in the path of least resistance, and light will travel the path of least (optical) distance, and I, ladies and gentlemen will aspire to do practically NOTHING if I can possibly get away with it.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<span style="line-height: 19.1875px;">This means I will absolutely abhor every single time I really have to get up, for instance to go to the fridge and get some grub, I'd much rather my lovely wife make lunch for me and serve it while I leisurely sit back and enjoy the latest mind-numbing episode ofsome stupid comedy show.</span></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<span style="line-height: 19.1875px;">By the way, this is EXACTLY why light bends in the presence of gravitation. Not to watch stupid comedy shows, the other thing - being lazy.</span></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<span style="line-height: 19.1875px;">See, gravitation can be thought of as a property of space(-time), rather than some force or field that occupies said space. by comrade Einstein's equivalency principle you can not tell whether you are being pulled by gravity, or being accelerated in the opposite direction.</span></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<span style="line-height: 19.1875px;">And by that virtue, even light itself (having no mass is irrelevant here) bends to accommodate for this equivalency.</span> </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<span style="line-height: 19.1875px;">One might even say (and be wrong) that light is the submissive partner in the energy-gravitation relationship... But as I said, this is not correct, since light is a form of energy, and gravitation is a product of energy concentrates. So really, it's a marriage of energy and gravity, or in oriental philosophy terms, marriage of heaven and earth, or ying and yang, or horse and carriage (err... that's from a different philosophy methinks).</span><br />
<br />
<span style="line-height: 19.1875px;">To conclude, marriage is such a blessing in one hell of a disguise, since left to our own devices we will do absolutely nothing until some external force propels us to action. When in a relatively successful marriage, the same is true, only the external force is called a wife, and hopefully, most of the times she kicks you in the gonads, it's for your own good.</span><br />
<br />
<span style="line-height: 19.1875px;">This also enables two people to become a system of coupled equations, thus redefining the least action to actually produce some interesting dynamics, at least until a third little equation comes into play to dominate the hell out of their lives... if you're a parent, you know what I mean, right?</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOMuZQmmvSoubrItu9Fj0azt2u4NtoD8jWHmsbtV5aiwUPpKFz0ZFh35XbZCR_c7wL65YX_MG41ceBCyqGrubuGZ7JXFpoQuVFMEYb0Fe3RPJmx2k2xIeIwLAx3GnYjprP1wgbJF_loyIV/s1600/TamchuAlabamchu.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOMuZQmmvSoubrItu9Fj0azt2u4NtoD8jWHmsbtV5aiwUPpKFz0ZFh35XbZCR_c7wL65YX_MG41ceBCyqGrubuGZ7JXFpoQuVFMEYb0Fe3RPJmx2k2xIeIwLAx3GnYjprP1wgbJF_loyIV/s1600/TamchuAlabamchu.JPG" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">My own sweet third coupled equation </td></tr>
</tbody></table>
<br />
<span style="line-height: 19.1875px;">Alternatively, if your marriage sucks, well, this explains why people stay in dysfunctional relationships for too long - it simply is to much ACTION to get out of one, better (by physical standards, not mine), to become mentally dislodged and otherwise apathetic to a fault.</span><br />
<br />
<span style="line-height: 19.1875px;">But we know it's the former rather than latter. we really do.</span><br />
<br />
<span style="line-height: 19.1875px;">Anyway we love you guys very much.</span><br />
<span style="line-height: 19.1875px;"><br /></span>
<span style="line-height: 19.1875px;">Mazal Tov!! </span></div>
<div style="text-align: left;">
<span style="line-height: 19.1875px;"></span> </div>
</div>
</div>
Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-56047821455253162332013-01-28T20:28:00.001+02:002013-07-01T13:23:21.595+03:00Inflations, Oscillations, and Weight loss... <br />
<h2 style="text-align: center;">
<span style="font-size: large;">
Both Cosmic and otherwise</span></h2>
<div>
<br /></div>
<div>
To those who had the (mis)fortune of having me as a part of their lives, weight gain, and weight loss are all too familiar. if not by first experience, they have probably witnessed my own physique change, all across the range spanning from wirey-thin to brown dwarf ( a so called Failed-Star, or UltraCool Dwarf, which is a stellar body of roughly 13-85 Jupiter masses), and hopefully, one day, back again.<br />
Since my travels through mass-space have been oscillatory in nature (for the most part), it is with pain-laced mirth I give you the following analysis and analogy of myself, and the Cosmos.</div>
<div>
<br /></div>
<div>
I know, I know, this seems a bit megalomanic on my part, but I assure you... I am. </div>
<div>
Well really, since my current field of study is Cosmology, and specifically inflation models, I found it might be funny to recount my thoughts and findings and relate it to the human condition, or rather, the fatso condition...</div>
<div>
<br /></div>
<div>
To any and all cosmologists out there, who might be reading this right now, be gentle, I am a mere neophyte to this field, and so I appeal to your sense of humor, and sympathy (enter violins please).</div>
<div>
<br /></div>
<div>
In order to first pique your interest, please listen to this number by Monty Python:</div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/JWVshkVF0SY?feature=player_embedded' frameborder='0'></iframe></div>
<div>
Which, by the way have some major faults in cosmological terms, for instance, the universe (acording to leading theories) actually does NOT expand at the speed of light. our EVENT-HORIZON (or simply horizon) does. It's meaningless to ascribe velocity to the expansion of the universe, since very close to us, the universe recedes from us in small velocities and far from us the universe seems to recede with greater velocity, as reflected by Hubble's law - velocity is proprtional to distance from us... \(\left(v=H_{0}D\right)\) .<br />
<br />
By the way, this also implies that Galaxies which are far enough from us, recede at velocities greater than the speed of light (yes it IS possible), and thus they "drop off" our horizon, since they fade away faster than their light travels to us.<br />
<br />
This also foretells a dark and lonely eventual demise, for our universe.<br />
<br />
At any rate, I would like at this juncture, to acquaint the reader with Friedmann's equation, which I will first write down and derive LATER (in two ways by the way...)<br />
\[\left(\frac{\dot{R}}{R}\right)^{2}=\frac{8\pi G \rho}{3}-\frac{\kappa}{R^2}\]<br />
Where \(R\) is the radius of an arbitrary sphere in space. this equation describes the evolution of \(R\) due to gravitational forces alone.<br />
<br />
<h3>
<div style="text-align: center;">
In layman terms:</div>
<div style="text-align: center;">
The universe was born, got fat, tried dieting for a bit, got frustrated and ultimately gave up, becoming ever so fat.</div>
</h3>
<br />
<br />
For those who have some interest of keeping their sanity, you are invited to skip these derivations.<br />
For those of brave soul and not so sound mind, please take special interest in the general relativity case...<br />
(Important conclusions, in green)</div>
<div>
<span style="color: red;"><br /></span>
<span style="color: red;"><derivation - Newton style></span><br />
from Newt's 2nd law: \[F=ma\\<br />
\Rightarrow F=m\ddot{R}\]<br />
Gravitational force due to enclosed mass, on the perimeter of a sphere:<br />
\[F=-\frac{GMm}{R^2}\]<br />
Equating these yields:<br />
\[\ddot{R}=-\frac{GM}{R^2} \Rightarrow \dot{R}\ddot{R}=-\frac{GM\dot{R}}{R^2}\]<br />
Integrating, we get:<br />
\[\dot{R}^2=\frac{2GM}{R}-\kappa\]<br />
So, now, for matter in a sphere with a radius of \(R\), \(M=\frac{4\pi R^3 \rho}{3}\), thus:<br />
\[\dot{R}^2=\frac{8\pi G \rho R^2}{3}-\kappa \Rightarrow \left(\frac{\dot{R}}{R}\right)^2=\frac{8\pi G \rho}{3}-\frac{\kappa}{R^2}\]<br />
taking \(R=R_0\cdot a(t)\) we get:<br />
\[\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G \rho}{3}-\frac{\tilde{\kappa}}{a^2}\]<br />
<span style="color: red;"></derivation - Newton style></span><br />
<span style="color: blue;">It's fairly easy to see that, according to Newton, the term \(\frac{\kappa}{R^2}\), is simply a term that is a result of integration, i.e. an integration constant. And so according to this derivation \(\kappa \) is simply given by initial conditions, and given no other incentive, we can gauge it away.</span><br />
<br />
Let's see what old Einei have to say about this...<br />
<span style="color: red;"><derivation - Einstein style></span><br />
I'll try to be brief yet informative:<br />
The metric for a spherical symmetric curved space is given by:<br />
\[ds^2=-dt^2+a(t)^2\left[\frac{dr^2}{1-\kappa r^2}+r^2d\Omega\right]\]<br />
Or in matrix form:<br />
\[g_{\mu\nu}=\left(\begin{array}{cccc}<br />
-1&&&\\<br />
&\frac{a^2}{1-\kappa r^2}&&\\<br />
&&r^2&\\<br />
&&&r^2\sin^2(\theta)<br />
\end{array}\right)\]<br />
From here it's relatively easy to find the Christoffel connections:<br />
\[\Gamma^{0}_{11}=\frac{a\dot{a}}{1-\kappa r^2}\;;\;\Gamma^{0}_{22}=a\dot{a}r^2\\<br />
\Gamma^{0}_{33}=a\dot{a}r^2\sin^2(\theta)\;;\;\Gamma^{i}_{0j}=\delta_{ij}\frac{\dot{a}}{a}\\<br />
\Gamma^{1}_{11}=\frac{\kappa r}{1-\kappa r^2}\;;\;\Gamma^{1}_{22}=-r(1-\kappa r^2)\\<br />
\Gamma^{1}_{33}-r(1-\kappa r^2)\sin^2(\theta)\;;\; \Gamma^{2}_{12}=\Gamma^{3}_{13}=\frac{1}{r}\\<br />
\Gamma^{2}_{33}=-\sin(\theta)\cos(\theta)\;;\; \Gamma^{3}_{23}=\cot(\theta)\]<br />
all other symbols vanish.<br />
<br />
The Ricci scalar is then given by:<br />
\[R=6\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^2+\frac{\kappa}{a^2}\right]\]<br />
The Einstein equation:<br />
\[G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G T_{\mu\nu}\]<br />
or, equivalently :<br />
\[R_{\mu\nu}=8\pi G \left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)\]<br />
Taking the 00 term we get:<br />
\[3\left[\left(\frac{\dot{a}}{a}\right)^2+\frac{\kappa}{a^2}\right]=8\pi G\rho\Rightarrow \left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G\rho}{3}-\frac{\kappa}{a^2}\]<br />
And, if we use the other notation also we get:<br />
\[\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}<br />
(\rho+3p)\]<br />
<br />
<br />
<span style="color: red;"></derivation - Einstein style></span><br />
For now, we'll leave the second equation be, and look at the first.<br />
<span style="color: blue;">We'll notice a couple of things first - Although it SEEMS as though the equations (Newt's, and Einei's) are the same, really the real Friedmann equation (derived from General Relativity) is the more general case. that is due to the former derivation relying on dynamics of MATTER only, and the latter does not, though, a careful massage of the former with reletavistic ideas, might give the correct equation for radiation, and other cosmic stuff...</span><br />
<span style="color: blue;">The second thing I wish to emphasize is the existence of \(\kappa\) and it's meaning:
In the Newtonian analysis, it was simply an integration constant, but in the Einsteinian analysis this is an intrinsic factor to the very fabric of the universe, this is the CURVATURE signature of the metric, that governs whether the cosmos is positively, negatively or null curved (i.e. flat) like in this picture:</span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJuA4Li0YEVP5KAbd1nFseW9QbLRLhHRP8-bUOt__4vEbZHhBMmqdzjmrnt2zkPSHDCroieFF2DVGYBQRAHhB4L8hfJcU2oHPw0qIN8DBfJGjeKo8yVbJc_3pyht_LQuX1gWeKt5Ffauxa/s1600/loadBinary.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJuA4Li0YEVP5KAbd1nFseW9QbLRLhHRP8-bUOt__4vEbZHhBMmqdzjmrnt2zkPSHDCroieFF2DVGYBQRAHhB4L8hfJcU2oHPw0qIN8DBfJGjeKo8yVbJc_3pyht_LQuX1gWeKt5Ffauxa/s320/loadBinary.gif" width="248" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Curvature types: positive, negative, and null.</td></tr>
</tbody></table>
<span style="color: red;"><br /></span>
<span style="color: red;"><derivation - Gangnam style></span><br />
And this is just for fun:<br />
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<br />
<span style="color: red;"></derivation - Gangnam style></span><br />
<span style="color: red;"><br /></span>
<br />
<h3 style="text-align: center;">
Admission of Guilt, errr... Ignorance</h3>
<span style="color: red;"><br /></span>
But wait!! What does all of this have to do with inflation?<br />
Well, in fact, suppose space is sufficiently dillute, the matter density is sufficiently close to zero, and if it's sufficiently cold, radiation density drops to zero (almost), with an (almost) flat curvature, we are then left with some quantity we'll call \(\rho_0\), ("Rho naught"), and we get a dynamic equation, with an inflationary/deflationary solution:<br />
\[\left(\frac{\dot{a}}{a}\right)=\pm\sqrt{\frac{8\pi G\rho_0}{3}} \Rightarrow a=A\exp\left(\sqrt{\frac{8\pi G\rho_0}{3}}t\right)+B\exp\left(-\sqrt{\frac{8\pi G\rho_0}{3}}t\right)\]<br />
The second part drops off rapidely, and so we are left with an inflationarry solution.<br />
<br />
"What is this witchcraft?!?!" you ask - where does this \(\rho_0\) come form?<br />
well, suppose you're a fat guy, and you go into a fasting mode, note that you are actually GAINING weight (at least in the short run)... and by the time you've lost the battle against hunger, and went on a burger binge, guess what? you've now underwent inflation.<br />
<br />
Well, actually this has nothing to do with \(\rho_0\), here's the REAL explanation for this:<br />
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
WE DON'T KNOW!!!!</div>
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<param name="movie" value="http://cdn.hark.com/swfs/player_bar.swf?pid=fkjzxckpht" />
</object>
<br />
<br />
<br />
Oh my god, I can't believe I said that!!! this is the absolute NO NO for physicists!!!<br />
To actually state that I don't know something? to recognize that everything we *THINK* we know is simply an approximate modeling of the awsome and complex reality we live in???!?<br />
<br />
I should stop. NOW!!! I hear them knocking already... don't let them take me! NOOOOOOOOOOOOOOOOO!!!!<br />
<br />
Ok, done with that gag, are we?<br />
<br />
Moving on, there are several possible explanations for this, most are problematic to say the least, but hey, this is what it means to be in the front lines of science. you either get promoted or get diced....<br />
<br />
At any rate, Inflation is a widely accepted theory of the adolescent universe, whatever mechanism manifests it.<br />
<br />
What about oscillations?<br />
In a nutshell - at the onset of the early universe, after inflation, some major ocillations occured in matter density, affected by ordinary (barionic) matter as well as dark matter, these oscillations are known as BARIONIC ACOUSTIC OSCILLATIONS (or modes) , and they could be seen quite nicely in analysis of the cosmic background radiation.<br />
<br />
Moreover, when the universe was matter dominated, the same dynamics might have happened on a cosmic scale - Suppose matter is the dominant part of the universe, the dominant force then is gravitation, thus the universe itself, aspires to CONTRACT, offset only by radiation pressure, there MIGHT have been an epoch of slight contraction on the universe's part.<br />
<br />
Sadly I didn't find a sufficiently fascinating animation to show, of the acoustic oscillations, but maybe some other time...<br />
<br />
In conclusion, much like myself, the universe was "born", and began inflating.<br />
Undergoing some oscillations, and at a certain point (just about.... now!) the universe moved from matter driven dynamics into moderate inflationary epoch.<br />
<br />
In layman terms:<br />
The universe was born, got fat, tried dieting for a bit, got frustrated and ultimately gave up, becoming ever so fat.<br />
<br />
In the words of our mutual friend:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
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<br />Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-90818327061033492412012-10-10T19:09:00.000+02:002019-01-14T21:00:39.952+02:00Entropy and Einstein's turnover time<script type="text/javascript">
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<br />
<h2 style="text-align: center;">
<u>Abstract</u><br />A failed attempt at explaining Entropy,<br />and one zombie, coming right up...</h2>
<br />
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...<br />
<br />
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...<br />
<br />
At first I asked if he meant the Anthropic principle, but he just wanted to understand Entropy.<br />
<br />
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).<br />
<br />
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. <br />
<br />
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.<br />
<br />
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.<br />
<br />
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<br />
\[\frac{2.3\cdot 10^{9}}{400}=5.75\cdot 10^{6}\,\text{instances in 2 weeks}\] <br />
Divided by the number of seconds in a two-week period we get:<br />
\[\frac{5.75\cdot 10^{6}}{14\cdot 24\cdot 60\cdot 60}\approx 4.75\,\text{times per second}\]<br />
<br />
And so the period time of Einstein's turning in his grave would be \(T\approx 0.21 \,sec\).<br />
<br />
So basically even the lower limit states that Einstein, by now, is a zombified Olympic athlete, even considering the initial rigor mortis....<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIpJ8ce7G0oNnIGS5WnCwroWdVjfPzdrRWX7IQlots9nIHuZhXzCors3x-nJuv6uZteWTdpQRW-8pb5gqGHEDkZAx4Qden2s-h0jPiYnwgqPx1vWm-1whNYiFwq8eELhMtXEad5odxxg3z/s1600/tumblr_m6pnozCO1a1qcz7qio1_500.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIpJ8ce7G0oNnIGS5WnCwroWdVjfPzdrRWX7IQlots9nIHuZhXzCors3x-nJuv6uZteWTdpQRW-8pb5gqGHEDkZAx4Qden2s-h0jPiYnwgqPx1vWm-1whNYiFwq8eELhMtXEad5odxxg3z/s400/tumblr_m6pnozCO1a1qcz7qio1_500.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">By now, he would have a solid six-pack.</td></tr>
</tbody></table>
<br />
<br />
Anyway, I digress, I was going to explain Entropy and then a random rant stole my attention... sorry for that.<br />
<br />
In a nutshell, Entropy is a measure of disorder, and I will explain.<br />
<br />
<span style="color: red;"><Failed attempt at an explanation : but is still worth a read> </span><br />
<br />
Imagine a group of four coins, each with two sides - heads, and tails - right? (we'll have non of that Two-Face shenanigans here!) <br />
Now suppose every coin is perfectly balanced so there's a fifty-fifty chance of getting heads or tails for each coin flip.<br />
<br />
So, now, what are the chances of getting all 4 heads, when you flip 4 coins?<br />
if you do the experiment enough times, you get an average of 1/16 chance.<br />
the state of all heads, or equivocally all tails is the most "ordered" result, why?<br />
because it is the most homogenous result (and we humans like homogeneity, symmetry and by the same token order).<br />
Now, what's the most plausible result?<br />
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).<br />
<br />
That is the least "ordered" result, since we don't care about locations, and the coins show the most diversity in results. <br />
<br />
<br />
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 \[\frac{1}{64},\frac{6}{64},\frac{15}{64},\frac{20}{64},\frac{15}{64},\frac{6}{64},\frac{1}{64}\,\text{for six coins}\]<br />
and a breakdown of \[\frac{1}{256},\frac{8}{256},\frac{28}{256},\frac{56}{256},\frac{70}{256},\frac{56}{256},\frac{28}{256},\frac{8}{256},\frac{1}{256}\,\text{for eight coins}\]<br />
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.<br />
<br />
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: <br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRRv0rBgVcVf4QA7qq3QOk3IB8OJMFo14Jz3mTsJwx5tejqM6MWDiiOXPEV6U4GggA9AD38E0_1bna4_LTDDdgXtT0t5oehZzbrmibkqrcehXei02PeCyyTgWQtolgsWphtUDPpwjQKHzy/s1600/save.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="245" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRRv0rBgVcVf4QA7qq3QOk3IB8OJMFo14Jz3mTsJwx5tejqM6MWDiiOXPEV6U4GggA9AD38E0_1bna4_LTDDdgXtT0t5oehZzbrmibkqrcehXei02PeCyyTgWQtolgsWphtUDPpwjQKHzy/s400/save.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Probable results graph</td></tr>
</tbody></table>
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.<br />
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 <u>im</u>probable. 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).<br />
<br />
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... <br />
<br />
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!<br />
<br />
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".<br />
<br />
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.<br />
<br />
In physics, while the same is true, you could say a statistic result applies and be <u>absolutely correct</u> 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!.<br />
<br />
So anyway, Entropy is defined as the logarithm of the multiplicity function.<br />
<br />
The reason for taking the logarithm is for the sake of defining a cumulative quantity, as opposed to multiplicative.<br />
<br />
<span style="color: red;">< /Failed attempt at an explanation : but is still worth a read> </span><br />
<br />
So anyway, obviously I failed at this attempt but let's try it in a simpler manner:<br />
<br />
Entropy is a quantity that signifies how probable a result is.<br />
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. <br />
<br />
Thus, Entropy becomes a measure of disorder of a system.<br />
<br />
Entropy is a cumulative property in the sense, that when two <u>non-interacting</u> experiments are done the combined entropy is the sum.<br />
However, when systems are allowed to interact, the combined entropy is typically <u>larger</u> than the sum of individual entropy.<br />
<br />
it is by that sense, that entropy tends to increase over time (and interactions).<br />
<br />
<span style="color: red;"><example of entropy increase></span><br />
<br />
Suppose, we have two systems, each of 4 coins.<br />
<br />
The most probable state is given by 2 heads, and 2 tails for a single experiment right?<br />
as was explained in the above failed attempt, the chance for that happening is \(\frac{3}{8}\).<br />
<br />
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?<br />
<br />
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.<br />
<br />
So what happened here, really?<br />
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.<br />
<br />
<span style="color: red;"></example of entropy increase> </span><br />
<br />
OK, that wraps it up for this time.<br />
Obviously I don't understand Entropy enough, since I feel I have failed at explaining it,<br />
but I will try again ("if at first you don't succeed" etc...).<br />
<br />
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?<br />
<br />
Anyway, till next time...<br />
<br />
P.S. some quick ideas to utilize Einei's incredible turnover time of 0.21 seconds :<br />
1. Attach him to a turbine and generate electricity.<br />
2. Display him as the 8th world wonder - the quickest man on the planet (faster than Usain Bolt).<br />
3. Use him as the engine for a horse-ride carousel for my kid.<br />
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)...<br />
<br />
Just sayin'...<br />
<br />
<br />
<br />
<br />
Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-28872330419276798522012-09-23T15:37:00.000+02:002013-07-01T13:26:31.553+03:00Short Derivation of Archie's Law<h2 style="text-align: center;">
"captain's log: supplemental"</h2>
It's been quite a while since I wrote the <a href="http://beastraban.blogspot.co.il/2012/08/crownsboats-subs-and-other-things-that.html" target="_blank">previous post about Archie</a>, his wife, a crown etc.<br />
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.<br />
<br />
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? <br />
<br />
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:<br />
<br />
<h3 style="text-align: center;">
Deriving the law (I AM THE LAW)</h3>
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Let us consider an infinitesimal volume element of a material with density \(\rho\) , and incident area and height \(A,h\) respectively.<br />
So now let's suppose the material is lighter than the medium surrounding it we get a force equation that looks like this:<br />
\[\Sigma F=A\cdot P_{up} -A\cdot P_{down}-Mg\]<br />
Where the force equivalent (sum of all forces) is positive.<br />
<br />
Now, let's wrap the forces that are the result of pressure and name it the Buoyancy force thus:<br />
\[F_{Buoyancy}=A\cdot\left(P_{up}-P_{down}\right)\] <br />
and with a little massage we get:<br />
\[A\cdot\left(P_{up}-P_{down}\right)=A\cdot h \left(\frac{P_{up}-P_{down}}{h}\right)=-V\nabla P\]<br />
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So, now, let's take a small detour to understand who \(\nabla P\) is shall we?<br />
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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:<br />
\[Mg=F_{Buoyancy}=-V\nabla P\]<br />
and breaking up the \(Mg\) element we get:<br />
\[V\rho_{medium}\cdot g=-V\nabla P\Rightarrow \nabla P = -\rho_{medium}\cdot g\]<br />
and so we get quite simply:<br />
\[F_{Buoyancy}=V\cdot \rho_{medium}\cdot g\]<br />
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<h3 style="text-align: center;">
Proper usage</h3>
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What's useful with this representation, is that we get the force outright, thus we can make force calculations directly.</div>
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An important thing to understand is that the buoyancy force is directed not UPWARDS (which is a common misconception) but <b>against the direction of the pressure gradient </b>. 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).</div>
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That's it.</div>
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this time it was really short. </div>
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</script>Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-43902367489389077832012-08-14T11:38:00.002+03:002013-07-01T13:28:19.252+03:00Crowns,Boats, Subs, and other things that float in water...<div style="text-align: center;">
<script type="text/javascript"><span id="goog_1110712728"></span><span id="goog_1110712729"></span>'גגvar _gaq = _gaq || [];
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})();</script><script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script><span style="font-size: large;"><u>Abstarct:</u></span></div>
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<span style="font-size: large;">A short story about a friend,</span></div>
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<span style="font-size: large;">and things that go "ping" in the dark.</span></div>
<br />
<br />
It has been some time since last I wrote something, but life, the multiverse and everything, got in the way.<br />
<br />
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?<br />
I guess I could get cute and say that its a \(42^n\) Tensor, but that's just my regular idiotic nonsense at play... <br />
<br />
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.<br />
<br />
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".<br />
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<br /></td><td><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpCo7fYWQest4qPZM-0fSa232PJT1guMmEaTJT02QRNjX4RSzVEzHegpgqGalICrsD9SFWfwjVk7T236mri6xNKqgrtjlDj9PFmiGaed1fXG59Tf8-tYsIO9UcIaRPACoixST-Qz6nWeh6/s1600/swimming-pool-03.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="149" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpCo7fYWQest4qPZM-0fSa232PJT1guMmEaTJT02QRNjX4RSzVEzHegpgqGalICrsD9SFWfwjVk7T236mri6xNKqgrtjlDj9PFmiGaed1fXG59Tf8-tYsIO9UcIaRPACoixST-Qz6nWeh6/s200/swimming-pool-03.jpg" width="200" /></a></td></tr>
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What's the connection?</div>
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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.<br />
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<h3 style="text-align: center;">
What makes things float? </h3>
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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...</div>
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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. </div>
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Now that queen was little capricious, not unlike the queen of hearts from Lewis Carroll'<i>s Alice's adventures in wonderland, </i>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.</div>
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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. </div>
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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.</div>
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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. </div>
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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.</div>
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Thus a method had to be devised to check the crown without damaging it.</div>
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Enter stage left: Archimedes.</div>
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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.</div>
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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!".</div>
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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. </div>
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<h2 style="text-align: center;">
"EUREKA"</h2>
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he shouted then, followed by his wife's "Shut up already you git! you'll wake the baby and then <u>you'll</u> have to deal with it!!" </div>
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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.</div>
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Now for the physics:</div>
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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:<br />
Let's consider a column of water and a thin strip \(\Delta Z\) thick <br />
\[\Sigma F=0 \Rightarrow s\cdot\left(P(z+\Delta z)-P(z)\right)-s\cdot\Delta z \cdot \rho_{water} g \\<br />
\text{or in other words} \frac{\partial P}{\partial z}=\rho_{water}\cdot g\Rightarrow P(z)=\rho_{water}\cdot g\cdot z+ P_{atm}\]<br />
<br />
Where z is the depth of water.<br />
<br />
So, we have to consider \[mg=s\cdot\rho_{water}\cdot g\cdot z \Rightarrow h\cdot\rho_{object}=\rho_{water}\cdot z \]<br />
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In other words, the depth of immersion is given by the height and relative density of the object and the fluid.<br />
<br />
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.<br />
<br />
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. <br />
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That also might provide a hint why we sometime encounter "unflushables"... </div>
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That also might provide an insight as to how submarines stay submerged at a constant depth:<br />
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.<br />
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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.<br />
<br />
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).<br />
<br />
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? <br />
<br />
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.<br />
<br />
So with pressure \(\rho_{water}\) rises linearly at first but pretty quick stabilizes to a constant.<br />
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Like so:<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5AqtJPbobZHwY_wa5YvxuWHThByiSnZ4cd04mUbphSeocmSMiJRKNLAxH6ZwCbaFz53sCqoRja7tXMjjr_WD-MIDJ6oCN81RnOIRzhqw9nTYJbhkNIcymvjVYz2j7hrxs-GAP0_nKnCH-/s1600/density_depth.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5AqtJPbobZHwY_wa5YvxuWHThByiSnZ4cd04mUbphSeocmSMiJRKNLAxH6ZwCbaFz53sCqoRja7tXMjjr_WD-MIDJ6oCN81RnOIRzhqw9nTYJbhkNIcymvjVYz2j7hrxs-GAP0_nKnCH-/s400/density_depth.jpg" width="327" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Courtesy Windows To the Universe (NESTA)</td><td class="tr-caption" style="text-align: center;"></td></tr>
</tbody></table>
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.<br />
I suspect somewhere in the middle there's actually a point where it all turns to ice...<br />
<br />
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...<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKeh_jtClv-fjbMbYck1RqD4O1rO_laj-ZCr0DYx3mgiJOUSvExJlroRoeVUAVufr671lMU9Ct7vc_1a2TzEJfndJppKWn5aoqeYDCNlt-AZ-czWQYL-k5YceN78kbaemBrLP6lEeDtoa-/s1600/F0029467.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKeh_jtClv-fjbMbYck1RqD4O1rO_laj-ZCr0DYx3mgiJOUSvExJlroRoeVUAVufr671lMU9Ct7vc_1a2TzEJfndJppKWn5aoqeYDCNlt-AZ-czWQYL-k5YceN78kbaemBrLP6lEeDtoa-/s200/F0029467.jpg" width="102" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Told you water density changes with temperature!</td></tr>
</tbody></table>
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.</div>
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By the way, remember the couple of pictures in the beginning? well there you go:</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFOPGNCOtDuMLNdg0vokHOQJIbU0G-YqBkYqocB5KxkHp-hv8i3O3NHQcTPRvB8uCCxNiwXEoc8psjHTErRWFVncZ-0_3qCRXrX-lk3RTVasws7qX2JhWO_yF1RW0bxTDwZ76Tyh45wQ52/s1600/BabyRuthInPool.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFOPGNCOtDuMLNdg0vokHOQJIbU0G-YqBkYqocB5KxkHp-hv8i3O3NHQcTPRvB8uCCxNiwXEoc8psjHTErRWFVncZ-0_3qCRXrX-lk3RTVasws7qX2JhWO_yF1RW0bxTDwZ76Tyh45wQ52/s1600/BabyRuthInPool.jpg" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Baby Ruth is swimming pool - not quite what you think...</td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
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</div>
<div style="text-align: left;">
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...</div>
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</div>
Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-22841666763877806952012-05-14T17:16:00.001+03:002013-07-01T20:42:35.901+03:00a short post about Wind Power and shams<div style="text-align: center;">
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</script> <span style="font-size: large;"><u>ABSTRACT:</u> </span></div>
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<span style="font-size: large;">Wind power, commercial jets, and politics</span><span style="font-size: large;">.</span></div>
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<span style="font-size: small;">And almost no humor this time...</span></div>
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<br />
A couple of weeks ago, maybe 3~4 days after posting the little thingy about Solar energy, two things happened.<br />
<br />
<ol>
<li>I was a guest at an eminent solar scientist BBQ lunch, and we had a little chat about what I wrote.<br />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... :)<br /><br />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.<br /> </li>
<li>I came across this nice video :<br /><br />
<iframe allowfullscreen="" frameborder="0" height="250" src="http://www.youtube.com/embed/nJcjgAdsS1k" width="440"></iframe><br /><br />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.<br /><br />Kuhn probably never intended to say or write that, but that's a different story altogether.<br /><br />Anyway my only intention here is to derive the part of how much energy an electro-windmill can produce.</li>
</ol>
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...<br />
<br />
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.<br />
<br />
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?<br />
<br />
Let's do that real quick shall we?<br />
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:<br />
\[P=\frac{1}{2}\rho_{air}Av^3\]<br />
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:<br />
\[P_{\text{wind averaged}}\approx 264\cdot A\,\,\, W\] <br />
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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.<br />
<br />
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?<br />
<br />
<h3 style="text-align: center;">
Really? 2% of the energy a commercial aircraft needs to take off ? </h3>
<br />
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.<br />
<br />
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?<br />
\[E_{\text{one second}}=10^9\]<br />
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...<br />
So you see? the guy in the video was right!!! or was he? <br />
<br />
<h3 style="text-align: center;">
Cheap rhetoric or outright sham?</h3>
<br />
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...<br />
<br />
The real consideration has to be this one:<br />
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.<br />
<br />
Again, <u><b>this is about the size of Europe</b></u>, 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!!) <br />
<br />
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... <br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
<br />
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...<br />
<br />
<br />
<ol>
</ol>
Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-6192290721485043832012-04-19T11:15:00.003+03:002013-07-01T20:44:49.548+03:00Physics of Environment and lies<script type="text/javascript">
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<span style="text-decoration: underline;"><span style="font-weight: bold;"></span></span></h2>
<h3>
</h3>
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<h3 style="text-align: center;">
<span style="font-size: large;"><u>Abstract</u></span></h3>
<div style="text-align: center;">
<span style="font-size: large;">IQ loss, Solar power, and misconceptions </span></div>
<div style="text-align: center;">
<span style="font-size: large;"><span style="font-size: small;">and an egg?!?</span></span></div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: center;">
</div>
<br />
"Ladies and gentlemen of the class of 2012, wear sunscreen... "<br />
<br />
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...</div>
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<br /></div>
<div style="text-align: left;">
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. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
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 <a href="http://www.youtube.com/watch?v=qrdpliMfoAM" target="_blank">song</a> about that... hmmm....).</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
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 :)</div>
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<br /></div>
<div style="text-align: left;">
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.</div>
<ol style="text-align: left;"></ol>
<div style="text-align: left;">
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. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
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.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: center;">
<h3>
How much energy does it take to live anyway?</h3>
<h3>
</h3>
<div style="text-align: left;">
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?</div>
<div style="text-align: left;">
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...</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
So let's approach this question by breaking it down to small bits:</div>
<div style="text-align: left;">
1. how much energy is spent by simply surviving? (on average! on average! sheesh... don't kill me YET) </div>
<div style="text-align: left;">
2. how much energy is spent on <u>living</u> i.e. driving to work and back, computers, home appliances etc..</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
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. </div>
<div style="text-align: left;">
So for some cold hard data:</div>
<ul style="text-align: left;">
<li>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.</li>
<li>Now the average male weight around the world ranges from 65-87 kg.</li>
<li>the same for women ranges 56-75 kg.</li>
</ul>
<div style="text-align: left;">
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.</div>
<div style="text-align: left;">
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. </div>
<div style="text-align: left;">
For women it turns out to be about 1350 Kcal/day.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
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... </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Well, let's average this out to about 1450 Kcal/day and find out that:</div>
<div style="text-align: left;">
\[1 cal \approx 4.1 J\\ 1450 Kcal = 1450\times 10^3\times4.1\\ P= 5945000 \text{ Joul per day} \Rightarrow P_{\text{average person}}\approx 70 W<br />
\]<br />
<br />
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...<br />
<br />
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...<br />
<br />
<h3 style="text-align: center;">
Solar energy - not quite what it's cracked up to be</h3>
<h3 style="text-align: center;">
</h3>
<div style="text-align: left;">
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. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
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. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
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. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
In a previous post I mentioned the amount of solar radiation the earth's surface experiences, is given by \(S_{\oplus}\approx1360\, \frac{J}{sec\cdot m^2} \).<br />
<br />
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...<br />
But here's the shocker:<br />
\[\text{Earth's surface} \equiv A_{\oplus}=4\pi r_{\oplus}^2 = 4\pi \times 6400,000^2\approx 5\cdot 10^{14}\, m^2 \\<br />
\text{Total solar energy on earth, per second} \equiv E_{\oplus}=A_{\oplus}\times S_{\oplus}\approx 7\cdot 10^{17} W\\<br />
\text{Total Energy used (surviving mode)}=E_{humen\, race}=7\cdot 10^9\cdot 70 \approx 5\cdot 10^{11} W \]<br />
<br />
and even if we only use Earth's available <u>land</u>, forget about covering the sea with mirrors we get about \( 0.29 \times E_{\oplus}\approx 2 \cdot10^{17} W \) . </div>
<div style="text-align: left;">
<br />
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...<br />
<br />
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.<br />
<br />
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!!!<br />
<br />
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).<br />
<br />
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:<br />
\[E_{year}=21,228\cdot 10^3 \frac{J}{sec} \cdot hour= 21,228\cdot 10^3\cdot 60\cdot 60\\<br />
\Rightarrow P_{\text{average, 2008}}= \frac{21,228\cdot 10^3}{365\cdot 24}\approx 2400=2.4kW \]<br />
<br />
Which is dire news indeed, since the total available solar energy on the planet was about \(10^{18}\,W\) ,<br />
and <u>now</u> the total consumption of the human race turns out to be around \(1.6\cdot 10^{13} W\).<br />
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.<br />
<br />
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:<br />
\[A_{AZ}\approx 3\cdot 10^{11}m^2\\<br />
\frac{A_{AZ}}{(2m^2)\cdot(Population)}\approx \frac{3\cdot 10^{11}}{15\cdot 10^9}=20\]<br />
<br />
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.<br />
<br />
<br />
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.<br />
<br />
<br /></div>
</div>
</div>
<h2>
<table>
<tbody>
<tr>
<td><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEid3Cs6YQIJ8gsWAGK7fJJUOX0IOA2QeaTGtgsgkr9dUU8jLqlvhZpYebJNfWgrUMJhsRu8nsiCT2OoPtnTVwbH5hr2vpqV-QH8tgLToe15PY0PoWdhivjjwIGkmAirjrN1W8mzueP02tmj/s1600/Fried_egg%252C_sunny_side_up.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="172" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEid3Cs6YQIJ8gsWAGK7fJJUOX0IOA2QeaTGtgsgkr9dUU8jLqlvhZpYebJNfWgrUMJhsRu8nsiCT2OoPtnTVwbH5hr2vpqV-QH8tgLToe15PY0PoWdhivjjwIGkmAirjrN1W8mzueP02tmj/s200/Fried_egg%252C_sunny_side_up.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Sunny side (up)</td></tr>
</tbody></table>
</td>
<td><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUoMT0uJt_r_6_wSq-w_WDintN6cHVrMH8RkwHnhPlAdTF-wr52oxbQ5VydbMXEdUIdxKW2fOZ6MNAqI_LawC3y0KlabrRUbeUnibRXsm7C6RZOrnxQW5aO7vfdxT9edsm7BT2B1DH3j7s/s1600/ord_vader_1024.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUoMT0uJt_r_6_wSq-w_WDintN6cHVrMH8RkwHnhPlAdTF-wr52oxbQ5VydbMXEdUIdxKW2fOZ6MNAqI_LawC3y0KlabrRUbeUnibRXsm7C6RZOrnxQW5aO7vfdxT9edsm7BT2B1DH3j7s/s200/ord_vader_1024.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Dark side</td></tr>
</tbody></table>
</td></tr>
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</h2>
<div style="text-align: left;">
<br />
Wow, hold yer horses boys, and let's get \(\mathcal{R}\)eal. it seems that we have some more calculation to make.<br />
First off, the earth's solar constant, as it happens, relates to the solar radiation the earth experiences as a <u>black body</u> 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:<br />
\[A_I=\text{Incident area}=\pi r_\oplus^2\]<br />
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:<br />
\[E_\oplus\approx 4\cdot 10^{16}\]<br />
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<br />
\[E_\oplus\approx 2\cdot 10^{15}\]<br />
Multiply that by the percentage of available land i.e. 29% to get:<br />
\[E_\oplus\approx 6\cdot 10^{14}\]<br />
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. <br />
<br />
As it is forget about using Arizona as a giant solar farm, try Greenland and India combined.<br />
<br />
And I didn't even get started! I mean, try factoring in dust!<br />
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%.<br />
<br />
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.<br />
<br />
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.<br />
<br />
So unless we act and make Lennon's vision of no countries real, it's a bit of a toughie.<br />
<br />
<h3 style="text-align: center;">
Hey - what was that about traceless energy production?</h3>
</div>
<h2>
<table><tbody>
<tr></tr>
</tbody></table>
</h2>
<div style="text-align: left;">
All we've done as of now is just talk physics, but what about environmental impact?</div>
<div style="text-align: left;">
The common (mis)conception is that solar energy, being renewable (which it is) also has little to no impact on the environment. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
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. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
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. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
YES IT'S THAT IMPORTANT (that I might actually dedicate my life to this).</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
but, do not ever fall for something that sounds too good to be true, it usually just ain't.</div>
<div style="text-align: left;">
and green energy,if not carefully and thoughtfully developed might be almost or as bad as </div>
<div style="text-align: left;">
conventional energy.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Again I'm no expert in ecology, so take what I say with a grain of salt as far as natural ramifications, <a href="http://www.youtube.com/watch?v=OojsLDYr7RY" target="_blank">but trust me, on the sunscreen</a>... err physics. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Next time: Wind energy or Wormholes </div>
<h2>
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Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-23375266884821021702012-03-03T19:11:00.011+02:002013-07-01T20:46:53.514+03:00Physics of Water polo<script type="text/javascript">
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<br />
<br />
<div style="text-align: center;">
<span style="font-size: large;"><u>Abstract:</u></span></div>
<div style="text-align: center;">
<span style="font-size: large;">Water-Copters, Beckham and dirty playing<span style="font-size: small;"> </span></span><br />
<span style="font-size: large;"><span style="font-size: small;">oh right, and drowning just a little bit</span> </span> </div>
<br />
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...<br />
<br />
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.<br />
<br />
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?"<br />
<br />
Consequently, it's not surprising that if you search the web for "<a href="http://www.google.co.il/search?hl=iw&client=firefox-a&hs=cL1&pwst=1&rls=org.mozilla:en-US:official&sa=X&ei=V_M8T6SlIoOu8AO7luG3CA&ved=0CCEQvwUoAQ&q=physics+of+water+polo&spell=1&biw=1280&bih=895" target="_blank">physics of water polo</a>" 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.<br />
<br />
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.<br />
<br />
<span style="font-size: large;"><u>The Water-Copter</u></span><br />
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.<br />
<br />
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.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSOsL582PY3LQ7kuH0S9czo8wlY9nsXJE3DC1jaoC0Xpy2grG0KOKMqOzabqc6Dctjci2HlMLaYRDnPOnzX6cDixfkt2AuCKiYadLzjruGU1OJdb4dQrG63SLycf8GVdmPHU-V4YvBOsH1/s1600/7_8_eggbeater_kick.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="217" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSOsL582PY3LQ7kuH0S9czo8wlY9nsXJE3DC1jaoC0Xpy2grG0KOKMqOzabqc6Dctjci2HlMLaYRDnPOnzX6cDixfkt2AuCKiYadLzjruGU1OJdb4dQrG63SLycf8GVdmPHU-V4YvBOsH1/s320/7_8_eggbeater_kick.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The "water-copter" technique</td></tr>
</tbody></table>
<br />
<br />
this move acts on the same premise as your basic helicopter, hence - the water-copter.<br />
<br />
So how a helicopter works? simple! (well not that simple but whatever...).<br />
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:<br />
\[\left\{\begin{array}{l}m_{\chi}\cdot g \cdot 1sec =m_{air}\cdot V_{air}\\<br />
m_{air}=\underset{Surface}{S}\cdot\underset{height}{V_{air}\cdot 1sec} \cdot<br />
\underset{density}{\rho}\\<br />
\Rightarrow V^{2}_{air}= \frac{m_{\chi}\cdot g}{S\cdot\rho}\end{array}\right.\]<br />
<br />
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.<br />
<br />
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.<br />
<br />
<br />
Anyway, I digress...<br />
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.<br />
this means that these guys get to underwater velocities that are equivalent to top 5k competitive runners.<br />
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.<br />
<br />
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...<br />
<br />
<span style="font-size: large;"><u>Bend it like Azevedo (or Sapic)</u></span><br />
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.<br />
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...<br />
<br />
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.<br />
<br />
But this part will deal more with the "bend it" and less with the "Beckham" -<br />
<br />
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...<br />
<br />
A similar thing happens when applying spin to a ball simply going through the air!!!<br />
<br />
<div style="color: red;">
Warning: Physics ahead</div>
<physics><br />
Consider a ball going through the air in a velocity \(V\), and spinning away at angular velocity \(w\), like this here poorly executed diagram:<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3mzLjHLU71Z7z01ZFNs8p__UTA5NffEJQQxeCSCWWVCAgl6aPPVpal1V6r_uOhyvVSzP1ZqfTNeZ6V81ko_iYLpLlKvMMcy5TFwIyqV_hljXHvXnmlILJKBVPNnCrzfMpjywvwr8mZBbS/s1600/ball1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="251" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3mzLjHLU71Z7z01ZFNs8p__UTA5NffEJQQxeCSCWWVCAgl6aPPVpal1V6r_uOhyvVSzP1ZqfTNeZ6V81ko_iYLpLlKvMMcy5TFwIyqV_hljXHvXnmlILJKBVPNnCrzfMpjywvwr8mZBbS/s320/ball1.jpg" width="320" /></a></div>
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 :)<br />
<br />
what happens is this:<br />
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 \(\overrightarrow{V}\times \overrightarrow{w}\).<br />
<br />
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.<br />
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. <br />
<br />
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?<br />
<br />
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.<br />
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<a href="http://www.youtube.com/watch?v=5nt2g3EgSmk" target="_blank"> in this video</a>, 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.<br />
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.<br />
<br />
But, what if the spin component is perfectly perpendicular to the incident plane?<br />
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...<br />
<br />
suffice to say that if we have a system of two springs, like so:<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDeOvln3j3_eejuwZ4eOIuTpolFzIEOVmoD3lR3mizLXsmuY7680HxvRHF6CB29xQNxTRgAQHMT-PBtwZzwOa9oUUC2UlbfVg6EGcxKaJ8KEpmqJdpoS8Tb4CFjXcCrmdbY83JRuBFMj0N/s1600/img_4432.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDeOvln3j3_eejuwZ4eOIuTpolFzIEOVmoD3lR3mizLXsmuY7680HxvRHF6CB29xQNxTRgAQHMT-PBtwZzwOa9oUUC2UlbfVg6EGcxKaJ8KEpmqJdpoS8Tb4CFjXcCrmdbY83JRuBFMj0N/s320/img_4432.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Two spring system</td></tr>
</tbody></table>
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.<br />
<br />
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.<br />
<br />
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).<br />
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.<br />
<br />
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 \(V\times w\).<br />
<\physics> <br />
<br />
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.<br />
<br />
AND THAT"S HOW YOU BEND IT!!!!<br />
<br />
<br />
<span style="font-size: large;"><u>Plain Ol' dirty playin' </u></span><br />
Another great experience that is (for me) connected with water polo is the one of almost dying (again).<br />
it turns out that water polo is also one of the most violent sports ever. it's probably comparable only to rugby.<br />
<br />
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.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0B_9KDaHPVmijAA85VfYjv06bW9_8Hv9MZRZpCqRhWR7FXSL_9DRx2Ljheji0YF8sSx2UvBsuS0cHn7v8GbNAa_WUb2r_h-uoksQUFZFh55c-fCQ7o67NTevMC4qT428PnmhsALb15FLH/s1600/water-polo-fouls.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="278" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0B_9KDaHPVmijAA85VfYjv06bW9_8Hv9MZRZpCqRhWR7FXSL_9DRx2Ljheji0YF8sSx2UvBsuS0cHn7v8GbNAa_WUb2r_h-uoksQUFZFh55c-fCQ7o67NTevMC4qT428PnmhsALb15FLH/s400/water-polo-fouls.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A typical water polo foul</td></tr>
</tbody></table>
<br />
<br />
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!<br />
<br />
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....<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiLlRZKY4ILeteRNyCAt5b1UjSuMuG53qCHBjQR-FImcowETunRNEQwPJhYDITWoxN3XJ5Gc1JcDFVEYMgWX6mb7ByC00bLIADYTP5uIO2nB7QqetUCxQSg-MEv8MyVvb1zO_pHNFFAbWq/s1600/cartoon.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="328" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiLlRZKY4ILeteRNyCAt5b1UjSuMuG53qCHBjQR-FImcowETunRNEQwPJhYDITWoxN3XJ5Gc1JcDFVEYMgWX6mb7ByC00bLIADYTP5uIO2nB7QqetUCxQSg-MEv8MyVvb1zO_pHNFFAbWq/s400/cartoon.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Water polo at it's best :)</td></tr>
</tbody></table>
<br />
<br />
Fun fact: usually when people die from drowning they don't die from water filling their lungs, what usually happens is called "<a href="http://en.wikipedia.org/wiki/Laryngospasm" target="_blank">Laryngospasm</a>" meaning the muscles in your throat contract violently, and you actually die from asphyxia.<br />
<br />
This usually happens either from panic or due to swallowing hefty amounts of salt water... <br />
<br />
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! :)<br />
<br />
Next time: Physics of floating stuff - submarines, boats, crowns and other stuff :)Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com4tag:blogger.com,1999:blog-4827059151570253428.post-15591014945380529532012-02-03T15:59:00.000+02:002013-07-02T09:06:15.157+03:00A short remark on the significance of butterflies and their respective effects.<div style="text-align: center;">
<span style="font-size: large;"><u>Abstarct:</u></span></div>
<div style="text-align: center;">
<span style="font-size: large;">"You keep using that word. I do not think it means what you think it means. "</span></div>
<div style="text-align: center;">
Inigo Montoya<br />
<br />
<div style="text-align: left;">
"The butterfly effect" is called that, not because of some cockamamie story about a butterfly farting somewhere to create the next hurricane Catrina..</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
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.<br />
<br />
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.<br />
<br />
So, a little history lesson is probably called for here:<br />
<br />
<span style="font-size: large;">Of strange attractors, and non linear effects:</span><br />
<br />
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.<br />
<br />
<span style="color: red;">Warning: physics rant here:</span><br />
<Physics rant><br />
<i>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.</i><br />
</Physics rant>.<br />
<br />
The world <u>isn't</u> linear, or in the words of my namesake, "it ain't necessarily so".<br />
Looking for the linear approximation of some dynamic, is not entirely different than looking for the proverbial coin under the street-light.<br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
What he got was very strange, he projected a certain result but got a result differing greatly from expectation. <br />
So like every respectable researcher he had tried to reproduce the experiment... getting results that differ greatly from both expected values, AND previous results!!!<br />
<br />
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.<br />
<span style="font-size: large;"></span><br />
But not our epic protagonist! He set out to delve deeper into what had happened and stumbled upon a great discovery!!!<br />
<br />
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.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8jIfNMPFaLKnqHdbTkgEb-5i90yd-mMCqI8WzwIIOn9gvXtEicm4TfBv9KnDFa-7Fo8BKLwNiYfV4uZw-EYxsUX4l4aRo0lI9rhX0vHrXSJuYnKhw0-9D3La-Hn1TXo-ooN3W1NPaRRte/s1600/Enorton.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8jIfNMPFaLKnqHdbTkgEb-5i90yd-mMCqI8WzwIIOn9gvXtEicm4TfBv9KnDFa-7Fo8BKLwNiYfV4uZw-EYxsUX4l4aRo0lI9rhX0vHrXSJuYnKhw0-9D3La-Hn1TXo-ooN3W1NPaRRte/s1600/Enorton.jpg" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Not the same Edward Norton</td></tr>
</tbody></table>
<br />
<br />
To illustrate here is a picture of a Strange attractor:<br />
<table><tbody>
<tr> <td><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcN0ZXqoZEI5COS8HTvvK73WEdx6KL5T_szOTRWQ-LYHvgf9NrsXmJ_u17Lw75zMydQ-2bSEuiZb5_7iMPlDTLMO40YsYo36h6I2uUr_52xeah6YglpGQx6xYjHKwhquP7i2HSlZzjjN_3/s1600/5.jpg" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="148" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcN0ZXqoZEI5COS8HTvvK73WEdx6KL5T_szOTRWQ-LYHvgf9NrsXmJ_u17Lw75zMydQ-2bSEuiZb5_7iMPlDTLMO40YsYo36h6I2uUr_52xeah6YglpGQx6xYjHKwhquP7i2HSlZzjjN_3/s200/5.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Strange attractor</td></tr>
</tbody></table>
</td> <td><div style="text-align: left;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo-VBSbDxUVSL-GBOjECbcL236IkXIMumRdO3niUkINzP649zPXf9WEibaqeiFNtYbe57H4tft0x1TXebaFcdKt6aN23kk0K6VAOpyF471AxyEjoCVW6W70-u_PN9PV986LbhOHAIl3ldZ/s1600/weird-al-meat-dress.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="143" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo-VBSbDxUVSL-GBOjECbcL236IkXIMumRdO3niUkINzP649zPXf9WEibaqeiFNtYbe57H4tft0x1TXebaFcdKt6aN23kk0K6VAOpyF471AxyEjoCVW6W70-u_PN9PV986LbhOHAIl3ldZ/s200/weird-al-meat-dress.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Weird Al in: Strange repellent</td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
</div>
</td></tr>
</tbody></table>
</div>
</div>
<br />
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.<br />
<br />
This is a graph generated by <a href="http://www.exploratorium.edu/complexity/java/lorenz.html" target="_blank">this applet</a> *, the applet simulates a Lorentz' system which is a meteorological model.<br />
If you want to actually see it in action, just click anywhere on the white field, twice at the same place.<br />
<br />
You will have generated 2 plots, or more if you have Parkinson's disease, that started at the exact same spot - or did they?<br />
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... <br />
<br />
So if you wait a while, you will see a divergence in paths taken by the two differently colored plots.<br />
<br />
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...<br />
<br />
<br />
So this concludes the short history lesson, and here I go into proving the world is evil.<br />
<br />
Well, no it's not, but I do have to make an attempt to show it is...<br />
<br />
OK, so a lot of people use the so called "butterfly effect" to justify a line of reasoning that says something like this:<br />
<br />
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.<br />
<br />
Now let me be absolutely clear about this:<br />
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.<br />
<br />
BUT! (and it's a big butt) this line of reasoning sucks.<br />
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.<br />
<br />
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".<br />
<br />
A chain reaction is in essence a LINEAR dynamic, it has absolutely NOTHING to do with the butterfly effect (as far as I know).<br />
<br />
Secondly and more importantly is this:<br />
<br />
Invoking the butterfly effect assumes a non-linear system.<br />
Assuming that if we do something good things get better in the world assumes strong linearity of the system.<br />
<br />
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).<br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
or differently put:<br />
"No good deed goes unpunished" in-deed (see what I did there?).<br />
<br />
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:<br />
"No deed goes unpunished" - and so we are all constantly punished... whether we do something or not - thus the world is an evil place.<br />
QED.<br />
<br />
But to end this on a lighter note, two things.<br />
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.<br />
so don't stop doing good things!!!<br />
<br />
the second is this...<br />
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.<br />
<br />
oh and something else - you can argue with your wife, but ultimately, she's right and you know it.<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMv8UT5aUTn7FKoBGs4sxSNjwl-oq5CnYkD3hSGCdZRvIkgGRgmhlH3bUcRtUm7gd42EqIMqNet7JzA5AHDgx3v8t5HWiblTaBev7gyzYf53qix3K6TQ5NbNUOZ6vqSEL0Kc1Zke_UVYJn/s1600/our-marriage-wedding-bride-groom-oops-demotivational-poster-1275276206.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="316" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMv8UT5aUTn7FKoBGs4sxSNjwl-oq5CnYkD3hSGCdZRvIkgGRgmhlH3bUcRtUm7gd42EqIMqNet7JzA5AHDgx3v8t5HWiblTaBev7gyzYf53qix3K6TQ5NbNUOZ6vqSEL0Kc1Zke_UVYJn/s400/our-marriage-wedding-bride-groom-oops-demotivational-poster-1275276206.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">And I mean this in the best possible way.... *GULP*</td></tr>
</tbody></table>
<br />
again, this was longer than expected... bummer.<br />
<br />
I also almost died -again- yesterday, maybe I will tell you all about it in a future post...<br />
<br />
<br />
* with permission. Copyright 1996, James P. Crutchfield. All rights reserved.Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com3tag:blogger.com,1999:blog-4827059151570253428.post-57753759489769751452012-01-25T19:20:00.000+02:002013-07-03T21:22:11.280+03:00Physics of Lenses and Idiots (part II)<div style="text-align: center;">
<u><span style="font-size: large;"> Abstract:</span></u></div>
<div style="text-align: center;">
<span style="font-size: large;">Burning ships, perfect lenses, Red shirts</span></div>
<div style="text-align: center;">
and weird Al</div>
<br />
<br />
Hello again folks...<br />
<br />
Just for spite. this post is just for spite.<br />
well, not really but it's fun to say it is.<br />
<br />
As some of you might remember I wrote an inhumanely long post, and my liege-wife told me to cut it short.<br />
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(!!!)<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipMQeUCJ4iFHm6x3-b1G5YylQBTDiLTwKj_fIo8iyvh6CPSlj9mt_onYSJKZTfINLVxpGv9lZhdd6mrzhJ0NB6YxVbSbjW2hBKGlzQRR4iGh-ic-BIXA2o4K28pqj64xAbLTO7axkul6no/s1600/demotivational-poster-14899.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="312" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipMQeUCJ4iFHm6x3-b1G5YylQBTDiLTwKj_fIo8iyvh6CPSlj9mt_onYSJKZTfINLVxpGv9lZhdd6mrzhJ0NB6YxVbSbjW2hBKGlzQRR4iGh-ic-BIXA2o4K28pqj64xAbLTO7axkul6no/s400/demotivational-poster-14899.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Not quite the shilling, though he probably minted some shillings...</td></tr>
</tbody></table>
<br />
Anyway, last time I was reminded of two nice stories....<br />
<br />
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?<br />
well let's put that theory to the test:<br />
<br />
<u>groceries:</u><br />
1. 300 bronze body shields of area 1.3m x 0.7 m, polished to an "\(\varepsilon\)" sheen.<br />
2. the sun - Earth's solar constant. - 1360 \(\frac{J}{sec\cdot m^2}\)<br />
3. auto-combustion temperature of wood - at the most \(~ 450^{\circ}c\)<br />
4. a distance estimate say at 500 meter.<br />
<br />
and now:<br />
<br />
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 <a href="http://en.wikipedia.org/wiki/Archimedes#The_Archimedes_Heat_Ray" target="_blank">experimental results here</a> assert differently... so what went wrong?<br />
<br />
well, a couple of things:<br />
<br />
first off, bronze is not a mirror, meaning for starters, \(\varepsilon\neq1\) in fact not even close.<br />
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.<br />
<br />
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?<br />
<br />
The added length of a line image is given by<br />
\[L_{image-line}=L_{original-line}\left(1+\sin(\alpha)r\right)\]<br />
so basically the area increases like<br />
\[A=A_{original}\left(1+2\sin(\alpha)r+\sin^2(\alpha)r^2\right)\]<br />
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:<br />
\[\Phi_{boat}=1360\cdot 300 \cdot \frac{1}{1+c500}\]<br />
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 \(50^{\circ}c\).<br />
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...<br />
<br />
So the obvious conclusion here is that the people of Syracuse had an inside agent!!!<br />
They had a traitor in their midst!!!!<br />
<br />
Or most likely this never actually happened....<br />
<br />
<span style="color: red;">Warning- computer geek humor below:</span><br />
<br />
<Computer geek humor><br />
<table><tbody>
<tr> <td><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZY56ZlsFKPTIPE1wVuzwjCjsri0gsk6b9EoHGaGsCspHfGIt_GgGOXZnSYBrLUUcHpFB_dZguxC9De6i3EfHDzlHqGqBGAAteD7GwRb8mA7EKPqmHbDVFhm5TYkIDJyJKTEqNNGqt9OAa/s1600/6643_a203.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZY56ZlsFKPTIPE1wVuzwjCjsri0gsk6b9EoHGaGsCspHfGIt_GgGOXZnSYBrLUUcHpFB_dZguxC9De6i3EfHDzlHqGqBGAAteD7GwRb8mA7EKPqmHbDVFhm5TYkIDJyJKTEqNNGqt9OAa/s200/6643_a203.jpeg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">try{kill_Redshirts();}</td></tr>
</tbody></table>
</td> <td><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtV1hNsBusveG2J2BhssZi2rm7UPkLzKMq1N3cvtrhcP4L5GX1UH2Fk677OIcqOaqGNbdTfKc1XrnD7lPg-NDFoJkCmJCJ-sWUbL6Y6tbJA8U1e6yBPrHQvmYSZSFVxpCJ5KRH_bbaYgQ1/s1600/scott.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="151" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtV1hNsBusveG2J2BhssZi2rm7UPkLzKMq1N3cvtrhcP4L5GX1UH2Fk677OIcqOaqGNbdTfKc1XrnD7lPg-NDFoJkCmJCJ-sWUbL6Y6tbJA8U1e6yBPrHQvmYSZSFVxpCJ5KRH_bbaYgQ1/s200/scott.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Throw Exception("attempted divide by zero.");</td></tr>
</tbody></table>
</td> </tr>
</tbody></table>
<br />
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?<br />
hmm... makes me want to throw an exception....<br />
</Computer geek humor><br />
<br />
Anyway...<br />
<br />
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.<br />
<br />
Again what melts at 5400K?<br />
hmm a short list that includes about everything...<br />
a short list of things that actually BOIL at 5400K includes among others:<br />
Carbon, Platinum, Rhodium, Titanium, Silicon, Palladium, Cobalt, Nickel, Iron etc...<br />
<br />
For a reference the surface temperature of the SUN is evaluated at about 5800K.<br />
<br />
<br />
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.<br />
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... <br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgraZ7ou68exxehcprHMhby3Amz4yTcg-dpHUZmPovj_qFVYFzZD4-UA759Dv68n0mFKCgjMqgwEnPIZ_2m1g4tjyX6JKodbQJSkOYFZo_edpzug-gDGen5F_QpdNixy0G5mP_wYHzgOQ_-/s1600/GKlingonWar4.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="261" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgraZ7ou68exxehcprHMhby3Amz4yTcg-dpHUZmPovj_qFVYFzZD4-UA759Dv68n0mFKCgjMqgwEnPIZ_2m1g4tjyX6JKodbQJSkOYFZo_edpzug-gDGen5F_QpdNixy0G5mP_wYHzgOQ_-/s400/GKlingonWar4.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A Klingon BOP taking a nice warm bath in a sun.</td></tr>
</tbody></table>
<br />
<br />
The second story goes something like this:<br />
I have a friend who's father is the head of solar energy research at a notable institute.<br />
He told me the story of acquiring an almost perfect Zeiss lens. and it goes something like this:<br />
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.<br />
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.<br />
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...<br />
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.<br />
<br />
Now as promised a short mass on Fresnel Lenses vs. parabolic lenses:<br />
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!!!<br />
<br />
<br />
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 \(x=x_0\).<br />
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 \(c=\frac{1}{4\alpha}\).<br />
<br />
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 \(\left(0,\frac{1}{4\alpha}\right)\).<br />
<br />
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...<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAtNW5XMov5LuXsxR9WbfwAzpyNOUKcMFzKLd-0wzJKphk36g3WQSkxQdexpDUavPCGgcu2VxAOrinBQFtHQS_JV5faIkk_KRluf2Am1-JKphnDckVQ09jVtZ1md3fr6wp-iq-UxzdQPMs/s1600/Fresnel+vs+parabolic.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAtNW5XMov5LuXsxR9WbfwAzpyNOUKcMFzKLd-0wzJKphk36g3WQSkxQdexpDUavPCGgcu2VxAOrinBQFtHQS_JV5faIkk_KRluf2Am1-JKphnDckVQ09jVtZ1md3fr6wp-iq-UxzdQPMs/s320/Fresnel+vs+parabolic.jpg" width="225" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Fresnel Vs. Parabolic lens, notice the corresponding regions.</td></tr>
</tbody></table>
<br />
<br />
Thus it is pretty clear, that while Parabolic lenses capture all light rays, it is heavier and bulkier than the corresponding Fresnel lens.<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
So why not use a parabolic reflective surface? this could be lightweight as well as cheap AND efficient in terms of light reflection?<br />
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.<br />
<br />
As for why NOT to use parabolic lenses and mirrors? - mainly physical space considerations probably, but also, in older parabolic mirrors I <u>THINK</u> 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.<br />
<br />
<br />
Wow, this was long, I hope you enjoyed this...<br />
<br />
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...<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjc7BCjXNqXFWtD6KJEInyoCNSu6uX2EgSHq6pklqA5WKnc0-0RXZh95PoD37ZfnI5pMyZ1OcNCD8xtwFfnrjpUTUTBA66gQadepbV52brLnCj7b8dIcXevEGdQL6jH3K6qQ4L6-g47mK7g/s1600/White%2526nerdy.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="197" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjc7BCjXNqXFWtD6KJEInyoCNSu6uX2EgSHq6pklqA5WKnc0-0RXZh95PoD37ZfnI5pMyZ1OcNCD8xtwFfnrjpUTUTBA66gQadepbV52brLnCj7b8dIcXevEGdQL6jH3K6qQ4L6-g47mK7g/s400/White%2526nerdy.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">a true nerd...</td></tr>
</tbody></table>
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. <br />
<br />
As always, next time : oh... why bother...<br />
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<div style="text-align: center;">
<span style="font-size: large;"><u>Abstract:</u></span></div>
<div style="text-align: center;">
<span style="font-size: large;">Found a magnificent Lens, Got the story.</span></div>
<div style="text-align: center;">
<span style="font-size: small;">and thoughts about Idiocy in organizations </span></div>
<br />
This time I didn't lie!<br />
<br />
How's about me keeping y'all on your toes ha?<br />
<br />
Well, as promised this <a href="http://beastraban.blogspot.com/2012/01/physics-of-lenses-and-idiots-part-i.html">post</a> will deal with lenses, and idiots.<br />
Following our recent brush with hypoxia, my lovely wife and myself proceeded to tour the coastline of California, we drove through San Francisco, met with a distant relative, and all in all had a great time.<br />
<br />
At some point we drove through Cambria, which is a quaint little town somewhere on the coastal road between Carmel-by-the-sea and L.A.<br />
<br />
<div style="color: red;">
WARNING: next is a tacky description fitting of a less-than-mediocre dungeon master:</div>
<br />
<div style="font-family: "Courier New",Courier,monospace;">
<i><span style="font-size: small;">"It was a cool morning. the air was still. thick fog covered the valley and wayward sounds made their eery way to our ears. the cold ate at our bones, and it was difficult to see anything further then the tip of your nose. </span></i></div>
<div style="font-family: "Courier New",Courier,monospace;">
<i><span style="font-size: small;"><br />
</span></i></div>
<div style="font-family: "Courier New",Courier,monospace;">
<i><span style="font-size: small;">Somewhere to the left you hear the crack of dried bones, and you think you might have felt something brushing against your boot, or maybe it was your imagination...."</span></i></div>
<div style="font-family: "Courier New",Courier,monospace;">
<br /></div>
Ok, sorry, got carried away there just a tad....<br />
Anyway, it was cold that morning so the town was shrouded in mist. we took a little stroll down to the waterfront, where it was rather stinky as the sea washed ludicrous amounts of kelp to the shore and it just lay there rotting.<br />
<br />
Well we needed to refuel so we pulled over, and while we were in the station, after grabbing a cup of coffee something caught my eye.<br />
<br />
by that time the sun was about half way between horizon and zenith, and the glint of a glass house - the kind you should not throw stones from - was pulling at me.<br />
<br />
We went to investigate only to find something incredible.<br />
there is was, encased in a glass gazebo, welded shut and otherwise unapproachable, like a queen enthroned in a crystalline palace - The most wonderful Fresnel Lens I have ever laid eyes upon....<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-ml_Koix6LSc/TJI_jSLBhII/AAAAAAAABYw/mhVXL_TwO4o/s1600/IMG_2330.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="640" src="http://2.bp.blogspot.com/-ml_Koix6LSc/TJI_jSLBhII/AAAAAAAABYw/mhVXL_TwO4o/s640/IMG_2330.JPG" width="360" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Piedras Blancas lighthouse Fresnel Lens at Cambria</td></tr>
</tbody></table>
I was psyched.<br />
We went to find someone who could maybe open the locked carousel, so I could take a closer look at it, and found an elderly fellow who was kind enough to tell us the wonderful story of this lense.<br />
<br />
In his crackled yet fiery voice he told the story of the lens.... <br />
<br />
It seems that ever since a couple of decades ago, as the US coastguard decommissioned several lighthouses along the US coastline, this lens was all but forgotten by all, laying embedded in a non-active lighthouse's torch.<br />
<br />
Imagine this untouched treasure trove, laying partly in the sand, covered by rotting kelp, the sand and salt slowly eating away at the tempered glass, scratching it, scarring it....<br />
oh the humanity!!! <br />
<br />
The good people of Cambria led by local coastguard veterans secured the consent of the US coastguard, took the lens, fixed it, re-polished it, and enshrined it in the glass carousel in town,<br />
by the <a href="http://maps.google.com/maps?q=Veterans+Memorial+Building,+1000+Main+St,+Cambria,+California+93428,+United+States&hl=en&ll=35.565009,-121.097057&spn=0.001807,0.002114&sll=31.046051,34.851612&sspn=7.789981,8.657227&vpsrc=6&hq=Veterans+Memorial+Building,&hnear=1000+Main+St,+Cambria,+California+93428&t=h&z=19&layer=c&cbll=35.565009,-121.097057&cbp=12,0,,0,0&photoid=po-25256426" target="_blank">veterans center</a>.<br />
<br />
Now as it turns out, the US coastguard, found out that this kind of lens is not manufactured anymore. kind of like the precise German lenses and mirrors used in WWII to blind British air-fighters on their runs over bombarded Berlin. it just costs too damn much to manufacture these kind of lenses especially when there are "better" alternatives to be found, more on that later...<br />
<br />
That no one manufacture these anymore is a nicer way of saying these babies cost a small fortune each!!! and now, the US coastguard found out about that and wanted to put their hands on this lens.... snatch it away from the land lubbers' keep.<br />
<br />
After a long and epic struggle, almost as epic as Gandalf's struggle with the Balrog at the bridge of Khazad-dum (and later falling, and in the waters, and on mountains... sheesh he must've been really tired that night), or the no-less epic struggle of scraping myself out of bed each morning....<br />
the land lubbers won the lens and it can still be seen lit on special events and festivals of Cambria <br />
<br />
<table><tbody>
<tr> <td><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzsOcNBir5dmPHz6lcQpc0jqvL_liR4-vJvV8OVjtE6-n6DI8L3BWHfVaKsLEXrsuOYFv0fQVlinFu6MjLgXXiwFp6LKcVBocy80g3rYhDdNx3314RrhW2feAdtoMqShaTLnj5N6yWlSx-/s1600/Lord+of+the+Rings+Gandalf+Falls+with+the+Balrog+%25281%2529.jpg" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzsOcNBir5dmPHz6lcQpc0jqvL_liR4-vJvV8OVjtE6-n6DI8L3BWHfVaKsLEXrsuOYFv0fQVlinFu6MjLgXXiwFp6LKcVBocy80g3rYhDdNx3314RrhW2feAdtoMqShaTLnj5N6yWlSx-/s320/Lord+of+the+Rings+Gandalf+Falls+with+the+Balrog+%25281%2529.jpg" width="203" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">An epic struggle</td></tr>
</tbody></table>
</td> <td><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8_x7XCEeDjBt90NSvDPP_seSjZqb4aKQADx3xNjBMKdemh7vnQNzUmKDKSjLs9VG8KgTtgniIq_y1B_heedDpE8bHAu78Edh_SGGOyy7UbFaTtyPXAIDyEWRdGBHKrbkB8862t-N8ENRG/s1600/IMG_20111113_205055.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="191" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8_x7XCEeDjBt90NSvDPP_seSjZqb4aKQADx3xNjBMKdemh7vnQNzUmKDKSjLs9VG8KgTtgniIq_y1B_heedDpE8bHAu78Edh_SGGOyy7UbFaTtyPXAIDyEWRdGBHKrbkB8862t-N8ENRG/s200/IMG_20111113_205055.jpg" width="200" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">An equally epic struggle</td></tr>
</tbody></table>
</td> </tr>
</tbody></table>
<br />
<br />
Another nice story about this lens is this:<br />
<br />
When first they brought the lens to town, they put a regular old incandescent light bulb at the focus of this lens, only to find out that the housing for the light bulb melted completely!!!<br />
<br />
it took them a while (and several light bulbs and housings) to understand that at the focal point things get heated. so much so in fact that the whole thing simply melted.<br />
<br />
<br />
Warning: physics ahead-<br />
<br />
Let's try to understand how hot it gets at the focal point shall we?<br />
<br />
Let's say the incident area of the lens is about 1 square meter.<br />
assuming the light bulb is approximately round with a radius of say 5 cm the incident area it presents is about \(\frac{\pi}{4}\cdot 10^{-3}\,m^2 \).<br />
<br />
Now the light flux that goes through the lens goes to the focal point with a fractional portion accounting for flux losses due to diffraction, interference and other energy losses (crepuscular rays for instance) so basically:<br />
\[\Phi_{bulb}=\varepsilon\Phi_{lens}\]<br />
<br />
Now the flux per area averaged over all earth's incident area, over all emission wave-lengths, and over night-day at the mean earth-sun distance is called the earth's <b>solar constant</b>, and the observed value is \(S_{\oplus}\approx1360\, \frac{J}{sec\cdot m^2} \) , so:<br />
\[J_{bulb}=\frac{\Phi_{bulb}}{A_{bulb}}=\varepsilon\frac{\Phi_{lens}}{A_{bulb}}=\varepsilon S_{\oplus}\frac{A_{lens}}{A_{bulb}}\]<br />
<br />
Where \(J\) denotes the energy flux per area unit, which is proportional to the temperature in the 4th power by Stefan-Boltzman's law:<br />
\[J=\sigma T^4\] <br />
In which \(\sigma\) is the Stefan-Boltzman constant which is given in SI units by \(\sigma=5.67\cdot 10^{-8} \frac{Jouls}{sec\times m^{2}K^{4}}\)<br />
And so if we substitute all the constant and data we get that:<br />
\[T_{bulb}=\sqrt[4]{\varepsilon 3.05\cdot 10^{12}}\]<br />
<br />
Where providing no energy loss whatsoever (\(\varepsilon=100\%\)) we get the temperature there to be about 1300K or about \(1000^{\circ}c\).<br />
Let's say the situation is pretty bad and we have \(50\%\) energy loss we then get "only" 1100K<br />
which is about \(800^{\circ}c\)... you get the picture.<br />
<br />
Just for proper comparison purposes:<br />
Aluminum melts at about 900K, Copper at about 1350K, Lead and Zinc at 600K and 700K respectively and Iron at about 1800K.<br />
So obviously if the bulb housing was made with aluminum wiring it is fairly easy to see the whole thing would simply melt even in relatively cold days, not to mention the plastic in the housing...<br />
after a fairly short research it seems most plastics melt at a temperature no higher than about \(200^{\circ}c\), while even temperature and fire resistant brands do not fair well above \(400^{\circ}c\) to say the least. so really the lens could be in a horrendous state or simply our assumption for \(\varepsilon\) could be way off, since even at \(\varepsilon=5\%\) it turns out the plastic housing would have melted...<br />
<br />
Ok, so what they did in the end is lower the bulb housing by maybe ten centimeters, so now, while the lighthouse does not yield such illumination as it used to, it is now operable.<br />
<br />
So my lovely wife went through this post and had several remarks:<br />
1. This post is too long.<br />
2. I am a complete and utter nerd, but she loves me still.<br />
3. it is late and I should go to bed if I prefer hell not to be risen....<br />
<br />
Thus I am at this point apologizing, and cutting this post in two.<br />
<br />
Let's conclude for now by deriving an interim equation for the ambient stupidity in an organization...<br />
<br />
It is obvious that the bigger an amount of money is involved in a project, the more foreign consideration (i.e. bribes, conflicting interests etc. ) will come into play.<br />
Also there is a pretty known principle where a chain or for our purposes a net is only strongest as it's weakest link, this also applies to teams, where a team is only strongest as its weakest member.<br />
and for our purposes a project team is only as capable as it's dumbest member.<br />
Also we want to make an observation that not only the above is true, but as team grows larger it becomes more sluggish and bureaucracy bound, now the only question is what are the relations, so , obviously when a lot of people work on the same project with no proper guidance (i.e. tyranny or otherwise simple dictatorship) the average work tend to scatter in all directions equally thus yielding zero work.On the other hand it is a well known fact that given a strong and small team, work is done in greater efficiency.<br />
so I propose the next relations:<br />
\[eff \propto Ne^{-\frac{N}{N_0}}\]<br />
Where N is the number of people working on the same project, and \(N_0\) is there for normalization purposes, and \(eff\) denotes the efficiency.<br />
<br />
also it is a well known fact that you need a team of people to fix one persons stupid mistake, and if I am VERY lenient I will only take a quadratic decrease in efficiency in respect to the amount of money, thus I propose a refinement:<br />
<br />
\[eff\propto \frac{B}{\beta B_s}\frac{N}{$^2}e^{-\frac{N}{N_0}} \]<br />
<br />
Where \(B\) is the average brains of the team, \(B_s\) is the brains possessed by the dumbest person on team, \(\beta\) is the coefficient that represents how many other team members it takes to fix one dumb mistake...<br />
<br />
Keeping in mind that idiocy is the reciprocal of efficiency we get the ambient idiocy of a team or organization to be:<br />
<br />
\[I_{diocy}\propto \frac{\beta B_s}{B}\frac{$^2}{N}e^{\frac{N}{N_0}}\]<br />
<br />
This could very well account for the coastguard behavior in that instance, albeit in military organizations this dynamic is really not enough to account for all the stupid s@#% one encounters there, and I have rich experience of such encounters... <br />
<br />
Also - basically whichever organization it was that hired me, probably raised it's idiocy bar by several orders of magnitude... :p<br />
<br />
That's all for now folks, part II of this discussion will be very interesting, with an explanation of why, for instance,it's highly unreasonable that the Klingon ship captained by Kurn son of mogh, with Worf on board out maneuvered two Klingon birds of prey by flying into a star's corona, and while Kurn's ship was able to escape and break orbit, the pursuing ships fell into the star's sea of lava and consequently exploded...Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com3tag:blogger.com,1999:blog-4827059151570253428.post-63646100718169881512011-12-20T13:02:00.000+02:002013-07-09T18:26:03.737+03:00Physics of marriage and hypoxia<script type="text/javascript">
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<div style="text-align: center;">
<span style="font-size: large;"><u>Abstract:</u></span></div>
<div style="text-align: center;">
<span style="font-size: large;">Got married, took a hike,</span></div>
<div style="text-align: center;">
<span style="font-size: large;">got high, got dehydrated and almost suffocated.</span></div>
<div style="text-align: center;">
<span style="font-size: large;"><span style="font-size: small;">oh and a gerbil.</span> </span></div>
<br />
<br />
I lied.<br />
I said the next post would deal with how being fat can save your life, and there are actually several ways that can happen. but that will have to wait for another instance.<br />
<br />
A couple of years ago I got married.<br />
Yes. I, like loads of better men than I, donned the world's smallest handcuffs, and walked the "green mile" as it were of single men, to happy cheers of "dead man marrying".<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0HqvE3eoX1F-sMMqfMVlex8mqYnryvLLYDewYFs2oU48duDcWD2mz5Wozi8g6HTdJ8Nl3pNmFYOFl4Ev4U-Bd-BwyZmQtHU1Ww1oTGYStIsELeIte58HqK_JKLORhO3kBhPv3YafofxjL/s1600/demotivation.us_Wedding-rings-The-smallest-handcuffs-in-the-world_130799732844.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="his:+5 ring against common sense" border="0" height="319" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0HqvE3eoX1F-sMMqfMVlex8mqYnryvLLYDewYFs2oU48duDcWD2mz5Wozi8g6HTdJ8Nl3pNmFYOFl4Ev4U-Bd-BwyZmQtHU1Ww1oTGYStIsELeIte58HqK_JKLORhO3kBhPv3YafofxjL/s320/demotivation.us_Wedding-rings-The-smallest-handcuffs-in-the-world_130799732844.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">his: +5 ring vs. reason&accountability.<br />
hers: one ring to rule them all and in the darkness bind them.</td><td class="tr-caption" style="text-align: center;"><br /></td></tr>
</tbody></table>
Well, actually I count myself lucky. most men get fucked on the deal, but as for myself, I got myself a ninja for a wife and a Jedi knight to boot. You should see her performing Jedi mind tricks on me or using The Voice on people - damn she's good!<br />
<br />
Anyway after about a year of marriage we decided it's time to go for our much overdue honeymoon.<br />
so somehow we came up with the obscene amount of money it now costs to fly to the USA, and we went to visit grandma, the family and take a hike. no really. we traveled California's national parks and hiked some...<br />
at some point we even had oxygen shortage due to height sickness.<br />
<br />
What had happened was, we trekked from Camp White Wolf to Ten lakes in Yosemite national park, and back, in one day.<br />
Now Camp White Wolf is at \(\approx 2,400 m\), and the mountain overlooking Ten lakes is at \(\approx 3000 m \). so all in all it doesn't seem like a distinctive height difference right?<br />
<u>BUT</u> (and it's a big butt for sure!) being overzealous and not so experienced hikers we spent the night near Briceburg which is at \(\approx 350 m\).<br />
<br />
Now for a bit of physics:<br />
Humans <u>usually</u> breath air which is comprised among other things of roughly 21% \(O_2\).<br />
Assuming air is an ideal gas we get :<br />
\[PV=NRT\]<br />
Which means in a cubicle meter at sea level pressure, at \(298^\circ K\), we get \(\approx 40.9 {} moles \) of air which translates roughly to \(8.6\text{ moles of }O_2\).<br />
<br />
Now we will consider a thin spherical layer \(\Delta r\) of gas:<br />
\[ A(r)\cdot P(r)-P(r+\Delta r) = V\cdot\rho (r) \frac{GM_{\oplus}}{r^2} \]<br />
\[\Rightarrow -4\pi r^2 \Delta r \frac{\partial P}{\partial r}=4\pi r^2 \Delta r \rho (r) \frac{GM_{\oplus}}{r^2} \Rightarrow -\frac{\partial P}{\partial r}= \rho (r) \frac{GM_{\oplus}}{r^2}\]<br />
<br />
For all intents and purposes the gravitational acceleration on the earth's surface and at 3 km differs by a factor of \(1 \pm 10^{-4}\) so really there's no need to take into account the gravitational change in that small a difference in distance.<br />
Thus we will treat the gravitational potential as \(\Phi_N=gr\), and thus the force per kg is directed downwards and is given by \(F_g=g\hat{r}\).<br />
\[\rho = \frac{N\cdot m}{V} \Rightarrow \rho=\frac{Pm}{RT}\]<br />
and so we quickly get:<br />
\[\frac{dP}{P}= -\frac{gm}{RT}dr\]<br />
Which assuming a constant temperature (which is a WRONG assumption as we will see immediately) we get:<br />
\[ P=P_0\,e^{-\frac{gmr}{RT}} \]<br />
Where \(m\) denotes the molar mass of air, \(R\) the ideal gas constant, and \(T\) the temperature. <br />
<br />
Now using this formula we get that the air pressure at 3000 m above sea level is \(\approx 52595\) pa. which means the amount of moles in a cubic meter in 3000 meters of height is 21.2 moles. keeping in mind that \(O_2\) is only slightly heavier the \(N_2\) we allow ourselves the assumption that the \(N_2\sim O_2\) ratio in 3000 meters is similar to that of about ground level we get that the air up there has only ~11% oxygen in the same volume.<br />
Just for the sake of good form - at 15-19% you get decreased ability to work strenuously. impaired coordination may occur and there is a chance of induced early symptoms with individuals that have coronary, pulmonary, or circulatory problems. at 12-15% respiration and pulse increases; impaired coordination, perception, and judgment occurs. , at 10-12% respiration further increases in rate and depth; poor judgment and bluish lips occur. At 8-10% symptoms include mental failure, fainting, unconsciousness, an ash-colored-face, blue lips, nausea, and vomiting.<br />
<br />
you get the picture right?<br />
<br />
So 11% is quite close to being knee-deep in shit. <br />
<br />
Let's have a more realistic estimate though, because we know the temperature decreases with height (at least in the atmosphere).<br />
<br />
So basically the temperature is a physical function and we <u>like</u> to think of those as analytical so the first approximation of \(T\) around \(T_0\) is given generally by:<br />
\[T(r)\approx T_0 - ar\]<br />
The minus sign owing to our understanding of temperature decreasing with height.<br />
and so the above differential equation becomes:<br />
\[\frac{dP}{P}= -\frac{a\,gm\,dr}{a\,RT_0\left(1-\frac{ar}{T_0}\right)} \Rightarrow P=P_0\left(1-\frac{ar}{T_0}\right)^{\frac{gm}{aR}}\]<br />
And so using the corrected formula with \(a\) denoting the linear-approximate rate of temperature decline in the atmosphere which is given by \(a=6.5 \cdot 10^{-3}\) we get the pressure up there was about 70970 pa which is <u>much</u> likelier, whereas the effective air percentage there is almost 15%.<br />
<br />
Meaning that we we're only in an ankle-high depository.<br />
<br />
Just as an additional reference, in normal circumstances the oxygen percentage in our exhaled breath is about 16%, and in some circumstances could go as low as 14%.<br />
So imagine yourselves in a hot, non-ventilated room, stuffed with cheap foreign workforce, and imagine the wicked headache you'll get. Now multiply it, as we were climbing up at a quick pace thus taking even shallower and quicker breaths.<br />
<br />
At some point we actually sat there at the top of the mountain, breaking for lunch, not realizing we are only making matters worse...<br />
<br />
The payoff was worth it though - it's such a beautiful view up there - it's AMAZING!<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_GMO09i3MkmZLbDCUgb0iU0CI7BlX1BZ9-InZyVVYn9FNgsvQjLRqhZnxGogegXN5ySzDMw9PHnq-O5tIk0KQzti1_MNcS6Ybc6c86q9GXWTpljLLW-nijbmjgjv9pgdUV6cSCADxT2yB/s1600/tenlakes.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_GMO09i3MkmZLbDCUgb0iU0CI7BlX1BZ9-InZyVVYn9FNgsvQjLRqhZnxGogegXN5ySzDMw9PHnq-O5tIk0KQzti1_MNcS6Ybc6c86q9GXWTpljLLW-nijbmjgjv9pgdUV6cSCADxT2yB/s320/tenlakes.JPG" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Ten Lakes view, headache and dehydration included.</td></tr>
</tbody></table>
<br />
Anyway, on our way back we also came to the undeniable conclusion we were idiots, as we ran out of water, we had a pounding headache, and we had to virtually run all the way back to Camp White Wolf before dark, since we were really afraid to walk in the woods at night, with no light, no fire, no reception, and loads of wolves howling in the distance.<br />
<br />
We got there in the nick of time, the sun was already setting as we hit the marked trail back to camp, about 20 minutes later, in full dark, we got there.<br />
Sore-foot, dead-tired and otherwise completely wretched, we decided that we are NOT making camp to sleep on cold bare ground. I drove us to a nearby motel outside the national park, and we took a shower, and slept the deepest sleep achievable by mankind...<br />
<br />
<br />
oh, and a gerbil:<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6V6Hr4DCIfooITxNlMw02IuvFboCrwJM5T1ISuhdTTYWAnckftO-TuE-RVGRAw3RgUxx2YLnIJE5LMmhwI31cbPF6P-dp5Yq9I-pw0CF7TzSCEQGBdseNMt3RdAaniQpCzcZuVoQz8-Ed/s1600/Gerbil.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6V6Hr4DCIfooITxNlMw02IuvFboCrwJM5T1ISuhdTTYWAnckftO-TuE-RVGRAw3RgUxx2YLnIJE5LMmhwI31cbPF6P-dp5Yq9I-pw0CF7TzSCEQGBdseNMt3RdAaniQpCzcZuVoQz8-Ed/s320/Gerbil.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Gerbil</td></tr>
</tbody></table>
<br />
Next time: Lenses and idiots.Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com0tag:blogger.com,1999:blog-4827059151570253428.post-55596772390394211272011-12-14T18:04:00.000+02:002011-12-18T14:04:39.230+02:00Physics of falling from a second floor window<script type="text/javascript">
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<div style="text-align: center;"><u><span style="font-size: large;">Abstract:</span></u></div><div style="text-align: center;"><span style="font-size: large;">Fell from second floor, </span></div><div style="text-align: center;"><span style="font-size: large;">and the first thing I thought of was physics!</span></div><div style="text-align: center;"><span style="font-size: large;"><span style="font-size: small;">oh - I also crashed into a parking semi-trailer.</span> </span></div><br />
<br />
<br />
A long long time ago and in a galaxy not so far away, though it does seem like a whole different life to me now, I lived in Jerusalem, in the then new student dorms situated in mt. Scopus.<br />
<br />
One fateful Sunday, after returning from a weekend at my parents house, I found myself locked out of my second-floor apartment. a quick search through my pockets yielded little but lint, and pretty soon the reality of forgetting my apartment keys in my parents house dawned on me.<br />
<br />
Now ordinary people might contact the maintenance person, get a spare key, or - I don't know, call their roommates or something creative like that.<br />
<br />
But it seems that either prolonged army service fucks your brains up, or maybe I was dropped on my head one time too many when I was a kid...<br />
<br />
Instead of the logical path, I took the "highway" literally. I climbed up to my second floor using nothing but my hands. I got up on a nearby trashcan, and jumped to hang on a ledge by my bear hands, proceeding to pull myself up to our outside porch. from there it was a pretty easy business to climb the rest of the way using the stonework to the second floor porch window, opening it (it was a sliding window and those are REALLY easy to open if you know the trick), and getting in my warm and fuzzy apartment.<br />
<br />
Now keep in mind, I am a <u>heavy</u> guy so this is no simple feat.<br />
<br />
Anyway, the next day, as if it was even possible, I was even dumber!!! instead of asking one of my roommates for an extra key, I slept in with the immediate result of getting myself locked in...<br />
<br />
Being a smart bugger, I tried to climb down the way I clambered up... and to my surprise, climbing down proved to be A LOT EASIER than climbing up... I just fell down.<br />
<br />
Now again to illustrate two points:<br />
1. I'm <u>heavy</u> and quite big - this will play a significant role later in this post.<br />
2. The mind works in mysterious ways, specifically, my mind is a little bit warped.<br />
<br />
Now the first point is probably responsible for the fact that I am alive today, more on this later.<br />
The second point is directly responsible for this post! as the following physics consideration and their astonishing result is the FIRST thing that went through my mind as I was hitting the ground. <br />
<br />
So lets start shall we?<br />
<br />
some data first:<br />
<ul><li>the standard floor-to-floor height in western countries is about 3 meters.</li>
<li>my weight at the time was about 100 kg. </li>
</ul><br />
and so from Energy conservation consideration:<br />
\[E_{pot}=E_{kin}\Rightarrow gh=\frac{1}{2}v^{2}\Rightarrow v_{impact}\approx\sqrt{120}\,_{\frac{m}{sec}}\approx 40\,_{\frac{km}{hr}}\]<br />
So basically falling from the second floor means you hit the ground with a velocity of about 40 Kph. And the momentum at which I arrived to that fateful event is given by:<br />
\[ mv_{impact}=100 kg \times \sqrt{120}\, \frac{m}{sec}\approx 1100 \, \frac{kg\cdot m}{sec}\]<br />
And by Newton's second law: <br />
\[F=\dot{P}=\frac{\partial P}{\partial t}\approx \frac{P}{\Delta t}=2200\,N\]<br />
with \(\Delta t\) set to say half a second right? we all remember slow-mo pictures of people get hit in the face - that's about a 1000 frames per second and so half a second sounds about right...<br />
for the sake of proper reference an m-16 bullet has a barrel velocity of about 905 \(\frac{m}{sec}\) where the bullet's mass is about 4.1 grams, and we end up with a measly 37.105 \(\frac{kg\cdot m}{sec}\), and given the event of a hitting bullet takes about 0.01 seconds we get a force of about \(3700\, N\), which is comparable to the previous result.<br />
<br />
So now one would ask why the hell didn't I die right there and then?<br />
well the answer has actually two parts:<br />
<ol><li>it's not just the force that counts. it's the pressure that will determine if my skin raptures or not and a bullet has a very low area of interaction with a body, thus the pressure at the tip is immense, and the bullet is able to enter the body, where the body's consistency versus the bullet's energy determines if and when the bullets go through, or stays in (for the night) and discharges all it's energy thus making a lot of damage. - hmm this actually gives me an idea for a future post on bulletproof vests...</li>
<li>I fell straight on my back thus enlarging my area of interaction. had I fallen head first, or in such a way as to present a thinner profile, or simply falling in a position to hurt essential organs, I would not be writing this post now...</li>
</ol>Yet another nice comparison to make is that of a car-crash:<br />
Consider a car going at about 100 kph, on a wet highway, near Keisarya. <br />
Suddenly the driver in front of our car brakes, and our car brakes and veers right towards the shoulder, only to find there's a parked semi-trailer there.<br />
Our car then proceeds to crash head-on with the truck's aft, at about 80 kph (remember, the car braked, so it slowed a little).<br />
Some of the impact is absorbed by the car's front end, and the whole event takes about 1 second.<br />
thus, the driver's body, that weighed around 100 kg just then has a momentum of about <br />
\[P=100kg \times \frac{80}{3.6}\approx 2200 \frac{kg\cdot m}{sec}\]<br />
And since the event takes about a second the total force is given by \(F\approx 2200 N \)<br />
<br />
Notice the similarity in both cases, where the force on my body was comparable and the total result almost the same:<br />
in the first case it took my body about a day to recover from the first adrenalin rush, it took me to that day's end to feel the pain, and then about 2-3 days of recovery after which I ached for about a month.<br />
in the second case, the first day was spent in emergency ward checking that everything is in it's place and the pain came the morning after, incarcerating me in bed for 3-4 days with a month's worth of inability to practice due to - you guessed right! - after effect aches...<br />
<br />
next time: how being fat can save your life! <br />
<ol></ol>Anonymoushttp://www.blogger.com/profile/10800774578693954901noreply@blogger.com1