PseudoRandomJekyll2017-01-13T22:50:56+05:30http://arkyaC.github.io/Arkya Chatterjeehttp://arkyaC.github.io/arkya.chatterjee@iitb.ac.in<![CDATA[The Pole-Barn Paradox Resolved]]>
http://arkyaC.github.io/personal/pole-barn-solved
http://arkyaC.github.io/personal/pole-barn-solved2016-07-17T00:00:00+05:302016-07-17T00:00:00+05:30Arkya Chatterjeehttp://arkyaC.github.ioarkya.chatterjee@iitb.ac.in<p>Last evening, I suddenly realised that I hadn’t yet written down the solution to the pole-barn paradox that I wrote about a while ago. Gawd, it’s been more than a month!<br />
Just to put the problem in perspective, let me refresh our memories with this image.</p>
<p><img src="http://arkyaC.github.io/images/barn.jpg" alt="The pole-barn paradox" /></p>
<p>Like most other paradoxes of relativity, this one too stems from the fact that simultaneity has no meaning at speeds <em>relativistic enough</em>. Now what do I mean by that? As Lorentz’s transformation goes,<br />
<script type="math/tex">\begin{aligned}
\Delta x' = \Delta x \cosh \theta + \Delta t \sinh \theta\\
\Delta t' = \Delta t \cosh \theta - \Delta x \sinh \theta\\
\end{aligned}</script><br />
theta being the rapidity, the primed and the unprimed coordinates representing those in the rocket frame and laboratory frame respectively.<br />
So suppose there are two events which happen simultaneously in one frame, but have a spatial separation, <em>i.e.</em> if <script type="math/tex">\Delta t'=0</script>, then <script type="math/tex">\Delta t</script> will not be <script type="math/tex">0</script>, thus those two events are not simultaneous in the second frame.<br />
Coming back to the problem at hand, in our case there are two events, the pole head merging with the barn back door, and the pole tail merging with the barn front door. In the barn frame, it seems that the two events are simultaneous. And hence, someone standing by the barn would say that the pole was (momentarily) fully contained inside the barn. Now, to you (in the pole’s frame), these two events happen at different instants of time, which basically means that to you, the pole is never completely contained inside the barn.<br />
The next logical question would be, what is <strong>really</strong> happening? How could two things so different physically, <em>i.e.</em> being or not being contained by the barn, be happening in the two different frames? There must be only one of these that is <strong>actually</strong> happening, right?<br />
Well, no.<br />
What you’re implicitly assuming when you say this is that there must be a grand <em>universal</em> frame of reference which shows the <em>actual</em> truth. However, one Mr Einstein showed us how every frame, no matter how big or grand it may be, has an equal footing in terms of the validity of its observations. So to the observer standing by the barn, the pole is contained fully by the barn (momentarily), while, to you, carrying the pole, it seems that the pole is <em>never</em> completely contained within the barn.<br />
If you’re keen enough on taking this further, you might say, <em>what if the observer in the barn frame closes its back door at exactly the moment he sees the pole head merging with the door?</em><br />
Well, in that case the pole obviously won’t be able to move any further ahead after its head reaches the back door. So in the pole frame, it seems that the pole should crash with part of it’s tail sticking out of the front door of the barn. All we’ll be left with then is a pole at rest inside a barn with part of it sticking out of the front door. <strong>But</strong>, nothing can stop instantaneously, not even this pole. So what that amounts to is the fact that once the pole’s head is stopped, a shockwave is sent out along its length, carrying the information of the crash. Think of it as a kind of a vaporising effect being carried along the length of the pole, so that as soon as the wave crosses a point, it gets vaporised.<br />
However, the speed of that shockwave is also limited by the speed of light. In the barn’s frame, the pole was already inside, so when the shockwave reaches the tail of the pole, it’ll be well inside the barn. It turns out, even in the pole frame, the shockwave can never reach the tail of the pole before it’s well inside the barn’s front door. A more graphical explanation of this last statement can be found <a href="https://youtu.be/0TU1tKTOIj4">here</a>.<br />
As always, if you’re reading this, I’d love to know your opinion, and suggestions on the write-up. Thanks for visiting. :simple_smile:</p>
<p><a href="http://arkyaC.github.io/personal/pole-barn-solved">The Pole-Barn Paradox Resolved</a> was originally published by Arkya Chatterjee at <a href="http://arkyaC.github.io">PseudoRandom</a> on July 17, 2016.</p><![CDATA[Treat for the eyes (and the brain!)]]>
http://arkyaC.github.io/personal/heat-equation
http://arkyaC.github.io/personal/heat-equation2016-07-08T00:00:00+05:302016-07-08T00:00:00+05:30Arkya Chatterjeehttp://arkyaC.github.ioarkya.chatterjee@iitb.ac.in<p>It’s been a really long time since the last post; well, in all fairness, I’ve been busy with my ITSP - an autonomous foosball foosball table. It sounds really impressive, but our team could not accomplish much. We were only able to get our goalie running.<br />
Anyways, I’ve been studying some numerical techniques to solve differential equations in the meantime. There’s this nice course on Coursera on Scientific Computing (they’ve removed it recently after they shifted to the new platform), which gives an introduction to the basics of the numerical/computational approach to various problems. The course especially includes the numerical treatment of differential equations of diverse systems. The material is really interesting for a beginner like myself. In this post I’m going to talk about a really cool problem I learned to solve. <br />
One of the most famous partial differential equations is the diffusion equation, or the heat equation. What it basically looks like (ignoring the coefficients and other trivial constants), in its two dimensional form, is this:<br />
<script type="math/tex">\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}</script><br />
Here <script type="math/tex">t</script> and <script type="math/tex">\left x,y \right</script> represent the time and spatial co-ordinates respectively while u represents temperature (in context of the heat equation). Thus this partial differential equation conveys information about the transfer of heat over space and time. Solving this analytically is probably not that trivial. So, we try to attack the problem from a numerical perspective, which brings us to the finite differences method. We chop up the domain into a fixed number of grid points, and basically use the first principle of derivatives, albeit without the <script type="math/tex">\lim_{\delta x \to 0}</script>. The final product is really a visual treat.<br />
<img src="http://arkyaC.github.io/images/HeatEqn/dissipation.gif" alt="Dissipation of heat, visualised" /><br />
It’s obtained from a <a href="https://github.com/arkyaC/CodeForCourse/blob/master/heatEqn_Lec14.m">piece of code</a> I wrote on MATLAB, though the idea behind it is taken from the course material. What’s astonishing is how beautifully clear the partial differential equation becomes once the solution is visualised in such a manner. MATLAB is just awesome! That’s all I have to say.<br />
P.S.: It’s a gif, and in case it’s not working for you, check it out <a href="http://imgur.com/pXPhKim">here</a></p>
<p><a href="http://arkyaC.github.io/personal/heat-equation">Treat for the eyes (and the brain!)</a> was originally published by Arkya Chatterjee at <a href="http://arkyaC.github.io">PseudoRandom</a> on July 08, 2016.</p><![CDATA[Resonance Detection]]>
http://arkyaC.github.io/summers2k16/SURP-going-good
http://arkyaC.github.io/summers2k16/SURP-going-good2016-06-18T00:00:00+05:302016-06-18T00:00:00+05:30Arkya Chatterjeehttp://arkyaC.github.ioarkya.chatterjee@iitb.ac.in<p>Our next module for our summer research project under Prof Nandi is now complete.<br />
We generated pseudo-data for two body decay processes and processed it in another independent code, which looked for resonance in the available set of data. This is a very common method for looking for particles in the humongous heap of data that the LHC produces, though we’re doing it at a much smaller scale, obviously (<em>duh</em>).<br />
Going into the details, what we basically did was take pairs of Kaons, <script type="math/tex">K^{+}</script> and <script type="math/tex">K^{-}</script>, from the available data set (generated by a separate code running Monte-Carlo simulations that we wrote in the previous module), and then calculated their invariant masses. The ones that came from the same <script type="math/tex">\varphi</script> meson, would show a strong correlation, in terms of their invariant masses all adding up to the same value, <em>i.e.</em> around 1020 MeV (mass of the <script type="math/tex">\varphi</script> meson). The other pairs would only contribute to a random background. Hence plotting a historam of counts versus invariant mass for all possible Kaon pairs would give us a peak around 1020 MeV. This is what we call the <strong>resonance</strong> peak.<br />
This is the plot for invariant mass that we obtained: <img src="/images/resonance/inv_mass.png" alt="Invariant Mass of Kaon pairs produced from Monte Carlo decays of $$\varphi$$ mesons" /> As always, you can check out our code in our Github <a href="https://github.com/arkyaC/en_Root">repo</a>, and leave comments.</p>
<p><a href="http://arkyaC.github.io/summers2k16/SURP-going-good">Resonance Detection</a> was originally published by Arkya Chatterjee at <a href="http://arkyaC.github.io">PseudoRandom</a> on June 18, 2016.</p><![CDATA[Two Body Decay]]>
http://arkyaC.github.io/summers2k16/SURP-first-run
http://arkyaC.github.io/summers2k16/SURP-first-run2016-06-08T00:00:00+05:302016-06-08T00:00:00+05:30Arkya Chatterjeehttp://arkyaC.github.ioarkya.chatterjee@iitb.ac.in<p>It’s a great feeling when you finally get to compile your code. Now that we have successfully run the first set of Monte Carlo simulations of a two body decay, we’re going to move on to more complex problems. Oh, by the way, this is for my Summer Undergraduate Research Project under Prof B K Nandi, which I’m doing in collaboration with Mandar and Sagar. Anyways, it might seem pretty noob-like, but looking at the graphs we generated gives <em>major feels</em>. <img src="/images/TBD/energy_profiles.PNG" alt="Energy Profiles of Mother and Daughter particles" /><br />
We studied the decay of the <script type="math/tex">\varphi</script> meson into <script type="math/tex">K^{+}</script> and <script type="math/tex">K^{-}</script> mesons.<br />
<script type="math/tex">\varphi \rightarrow K^{+}K^{-}</script><br />
If you want to look at the code we used, you can reach our Github repo <a href="https://github.com/arkyaC/en_Root">here</a>.</p>
<p><a href="http://arkyaC.github.io/summers2k16/SURP-first-run">Two Body Decay</a> was originally published by Arkya Chatterjee at <a href="http://arkyaC.github.io">PseudoRandom</a> on June 08, 2016.</p><![CDATA[The Pole-Barn Paradox]]>
http://arkyaC.github.io/personal/an-interesting-paradox
http://arkyaC.github.io/personal/an-interesting-paradox2016-05-31T00:00:00+05:302016-05-31T00:00:00+05:30Arkya Chatterjeehttp://arkyaC.github.ioarkya.chatterjee@iitb.ac.in<p>I came across this really interesting, and famous, paradox the other day while reading Wheeler’s Spacetime Physics; it’s called the pole-barn paradox. What it basically says is this: You have a 20 metre long (<em>true length</em>) pole in your hand, running into a 10 metre long (<em>true length</em>) barn, at such a speed that your friend standing just outside the barn sees the pole contracted to 10 metres. So it seems to him that you, along with the pole, can be entirely contained within the barn. <strong>But</strong>, to you, the barn seems contracted to 5 metres, and so, there would be no way you, along with the pole, could be contained within the barn. Puzzling, eh?</p>
<figure>
<a href="http://arkyaC.github.io/images/barn.jpg"><img src="http://arkyaC.github.io/images/barn.jpg" /></a>
<figcaption>This is roughly how it looks like.</figcaption>
</figure>
<p>To make the question more clear, let us look at it this way: If someone standing outside the barn closes the back door (the one on the right), just at the moment the pole’s front end merges with the back door, the entire pole seems to be contained within the barn momentarily (before you crash! :P). Our question is, how is it that to you (running with the pole) it seems completely impossible?</p>
<p>I’ll keep you guys thinking about it ;)</p>
<p>Will be back with a detailed discussion about the resolution of this intriguing paradox… till then, <em>Ta-Ta</em>!</p>
<p><a href="http://arkyaC.github.io/personal/an-interesting-paradox">The Pole-Barn Paradox</a> was originally published by Arkya Chatterjee at <a href="http://arkyaC.github.io">PseudoRandom</a> on May 31, 2016.</p><![CDATA[The Year That Was]]>
https://stab-iitb.org/electronics-club/reviews/arkya/
http://arkyaC.github.io/insti/elec-club-article2016-05-24T00:00:00+05:302016-05-24T00:00:00+05:30Arkya Chatterjeehttp://arkyaC.github.ioarkya.chatterjee@iitb.ac.in<p>This is the link to a review I wrote for the Electronics Club of IIT Bombay for my freshman year. Would like to know your feedback, so fire away in the comments section.<br />
Signing off, like they say at Elec Club, swag <strong>amplified</strong>! ;)</p>
<p><a href="http://arkyaC.github.io/insti/elec-club-article">The Year That Was</a> was originally published by Arkya Chatterjee at <a href="http://arkyaC.github.io">PseudoRandom</a> on May 24, 2016.</p><![CDATA[First Blog Post]]>
http://arkyaC.github.io/personal/first-commit
http://arkyaC.github.io/personal/first-commit2016-05-22T00:00:00+05:302016-05-22T00:00:00+05:30Arkya Chatterjeehttp://arkyaC.github.ioarkya.chatterjee@iitb.ac.in<blockquote>
<p>"The beginning is the most important part of the work."
<small><cite title="Plato">Plato</cite></small></p>
</blockquote>
<p>It’s 4 in the morning, and I’m finally done with setting up my blog. Took me almost a full day to understand how to use Jekyll. Nevertheless, it’s been worth the wait.<br />
It’s vacation time, and I’m working on a couple of projects. I’m also reading up a little computational physics, which is not exactly orthogonal to the project I’m doing under Prof Nandi, of our department, for the summer. Maybe this interest will grow into some interesting work, who knows. The other project I’m doing is a robotics project. We’re making an autonomous foosball table - it’s like a robot against which you can play foosball, when your friends aren’t around, or maybe just for the challenge of playing against AI! How cool is that! :D<br />
Stay tuned, for more updates… Hoping that the summers are gonna be fun!</p>
<p><a href="http://arkyaC.github.io/personal/first-commit">First Blog Post</a> was originally published by Arkya Chatterjee at <a href="http://arkyaC.github.io">PseudoRandom</a> on May 22, 2016.</p>