Friday, December 31, 2010

Benford's Law

Given a large set of data (bank accounts, river lengths, populations, etc) what is the probability that the first non-zero digit is a one?  My first thought was that it would be 1/9.  There are nine non-zero numbers to choose from and they should be uniformly distributed, right?

Turns out that for almost all data sets naturally collected, this is not the case.  In most cases, one occurs as the first digit most frequently, then two, then three, etc.  That this seemingly paradoxical result should be the case is the essence of Benford's Law.

Friday, December 24, 2010

Holiday Hidden Message

Evil gun-wielding code-breaking
robo-santa from Futurama
Greetings and happy holidays!  Everyone has gone home for the break, so we will be taking a break from the grossly misnamed "Problem of the Week" for a while.  Instead, here's a "Christmas Code" I made up for a friend.  Figure it out and win the respect of strangers on the Internet!  Largely unhelpful hints after the break.


Saturday, December 18, 2010

A Buffoon's Toothpicks

Figure 1: Two of the thousands of toothpicks on my floor
You're sitting at a bar, bored out of your mind.  You've got an unlimited supply of pretzel rods and a lot of time to kill.  The floor is made of thin wooden planks.  How can you calculate pi?

This is how the problem of Buffon's needle was first presented to me.  Stated more formally the problem is this:  given a needle of length l and a floor of parallel lines separated by a distance d, what is the probability of a randomly dropped needle crossing a line?

Wednesday, December 15, 2010

Problem of the Week #3: Solution

Cold-Blooded Killer
Hello all and welcome to to another roundup of Problem of the Week.  If the time intervals don't seem to be adding up, just remember that "week" is an illusion here at Virtuosi headquarters.  "Problem of the Week," doubly so.  But enough with the excuses, let's see if Mr. Bond lives to Die Another Day.

The situation presented in the problem was one of a glass of water filled all the way to the top with a single ice cube in it.  The goal is to see if any water falls to the floor as the ice cube melts.

Thursday, December 9, 2010

The Law and Large Numbers

Human beings are not equipped for dealing with large numbers. Honestly, 7 thousand, 7 million, 7 billion and 7 trillion all register about the same in my mind, namely 7 big. Unfortunately, there is a world a different between each of these, three whole orders of magnitude, a thousand, the difference between lifting me and a US quarter.

This lack of respect for orders of magnitude has really been rearing its head recently with most of the political discussions surrounding the US budget.

Turns out the US Budget is really large. In 2010 it weighed in at $3.55 trillion. Thats big. Really big. So big that I can't fathom it.

Without getting too political, there has been a site going around recently; the You Cut program, which invites public suggestions for cuts to be made to the budget to try and fix the deficit. Now, personally, I believe we ought to do something about the deficit. To this end, I think it is useful to point out the scales involved. In particular, the link I gave above is to one of the suggested cuts: federal funding of NPR (Disclaimer alert: I love NPR), which weighs in at 7 million dollars.

Seven million dollars is a lot of money. A lot of money, more than I can imagine having personally. But to suggest that a 7 million dollar cut is any sort of progress towards solving a $1.2 trillion dollar deficit is a little amusing. As a fraction, this comes out to

\[ \frac{ 7 \text{ million} }{ 3.55 \text{ trillion} } = 2 \times 10^{-6} \]

Two parts in a million. To give a sense of scale to this, the gravitational influence of the moon on my weight is:

\[ \frac{ \frac{ G M_{\text{moon}} }{ R_{\text{earth-moon}}^2 } }{ 10 \text{ m/s}^2 } = 3 \times 10^{-6} \]
Three parts in a million. So, suggesting that you have made real gains in reducing the US budget by cutting federal funding for NPR is as silly as suggesting that if I want to lose weight, my first concern should be the current tides.

[I want to point out that I don't really mean to get too political, and that I've noticed both parties pulling these kinds of numbers tricks.]

So, wanting to get a little better understanding of the numbers at stake, I collected some data (all from the 2010 budget). My goal is to attempt to represent how the US government spends its money.

Before I begin I need to plug two great tools towards this end: Here the NYTimes graphically represents government spending, helping to give a sense of scale to different categories. Here the NYTimes lets you try and balance the budget, not only for next year but down the line, letting you choose from a wide array of proposed changes.

Monday, November 29, 2010

Your Week in Seminars: Short Thanksgiving edition

Good Monday evening all, and welcome to another edition of Your Week in Seminars. Last week was a half week here in Cornell, but we still managed two talks, the general colloquium and the Wednseday-talk-on-Tuesday.

Tuesday, November 23, 2010

Problem of the Week #2: Solution

Thanks to all who sent in solutions! We are very happy with the vast number of responses, and we will put up a leader board shortly!


Monday, November 22, 2010

Your Week in Seminars: One for Two Edition

Hello everyone and welcome to another week of talks here at the physics department. I was out of Ithaca for a bit this past week, so in this very special edition I'm going to present a full week's worth of seminars (one from last week and two from the previous week) in one post covering two weeks.

Problem of the Week #3

We welcome you to send in solutions, or even any ideas you have about how to solve the problems to with “problem of the week” in the subject line. We will keep track of the top Virtuosi problem solvers.

Welcome to the third installment of
Problem of the Week!  We are very pleased with the number of responses we have gotten so far and we super duper promise to put up some kind of leader board soon.  In fact, I super duper promise to relegate that assignment to Alemi.  We intend to keep this up as long as we can and give out prizes for high scores maybe...? They will be lame internet prizes...?

The solution to the last problem of the week will be up shortly.  Adam is the only one who knows the answer and he was busy all weekend taking a magic ring to Mordor.

One more housekeeping note.  Since we want everyone to have a clean shot at answering the question, we would prefer the solutions to be sent by email instead of posted in the comments.  However, we certainly don't want to stop discussion on the problem, so if you don't want any hints you might want to avoid the comments!

Now for the problem...

Sunday, November 14, 2010

Problem of the Week #2

As always, we welcome you to send in solutions, or even any ideas you have about how to solve the problems to with “problem of the week” in the subject line. We will keep track of the top Virtuosi problem solvers.

Why did the ant cross the rubber band?

A rubber band is held fixed at one end. The other end is pulled at a velocity v. At time t = 0, the rubber band has a length of L, and an ant starts crawling from one end to the other at velocity u.

Does the ant reach the other side? If so, how long does it take to get there? Assume that the rubber band is able to be stretched indefinitely without breaking.

Pet Projects

You're doing it wrong!
Here at the Virtuosi, we have a very specific way of asking a very specific type of question that sounds anything but specific.  These are the "How come [blank]?" questions [1].  These are very simple questions that just about every four year old asks, but likely never get sufficiently answered.  To  get a feel for what I mean by these questions I provide the following translations of problems we have either considered or will consider:

Q:  How come trees?
Translation:  How tall can trees be?

Q:  How come plants?
Translation:  Why are plants green?

These are my very favorite types of questions because they are completely understandable by everyone and promise to have very interesting physics working behind the scenes.  So I've been thrilled to see two such questions considered by scientists lately that have also had a good run  in popular media.  They are:

Q:  How come cats?
Translation:  How do cats drink?


Q:  How come dogs?
Translation:  How do dogs shake?

Monday, November 8, 2010

Your Week in Seminars Dark Edition

Good afternoon everyone, and welcome to another week of seminars here in the physics department. Our theme of the week is dark matter - where does it come from, how do we see it, and why is there so much of it. Along with that we have a little more AdS/CFT, seemingly continuing last week's subjects theme. All in all, it looks like seminars on similar subjects tend to condense here in the department.

Friday, November 5, 2010

Problem of the Week #1

I thought we could spice things up a bit with a more interactive post on The Virtuosi. Starting this week, a new problem of the week will be posted each week. Solutions will be posted the following week. These problems will be a collection of physics and math problems and riddles, and although hopefully challenging enough to be fun and interesting, they should mostly be solvable using concepts from introductory undergraduate physics and math classes.

We welcome you to ponder these problems, and send in solutions, or even any ideas you have about how to solve the problems to with “problem of the week” in the subject line. We will keep track of the top Virtuosi problem solvers.

Here it goes…

Monday, November 1, 2010

Your Week in Seminars: Conformal Edition

Another week has gone by here in Cornell. The last leaves are turning red, a hint of snow passed us on the weekend, and the undergrads have hit the streets and parties in minimal clothing, then did the same again next day wearing a set of cat ears. And in the physics department, we had the usual three talks.

Monday, October 25, 2010

Your Week in Seminars Fermionic Edition

Good evening, and welcome to the second edition of YWiS. Last week I took in the full range of seminars, from colloquium to Friday lunch. I don't know if I can say I took in the full content of these talks as well, but let's see what I learned

Thursday, October 21, 2010

Paradigm Shifts 3: With a Vengence

The last shift I wanted to present is best explained at . There you will find a manifesto (yes, a manifesto) about why we should change from using \[ \pi = \text{180 degrees} \] as the circle constant to \[ \tau = 2 \pi = \text{360 degrees} \] It's quite a convincing argument, and it's a shift that can easily be made. Check the website for more.


Monday, October 18, 2010

Your Week in Seminars Intro Edition

We've done a lot of talking over the past few months here on the Virtuosi, but one important subject has not come up so far. An issue that is central to the day to day life of the average grad student. The subject of free food.

The average graduate student in an American university shops for food 0.7 times per semester, paying a total of $13.22. He eats an average of three vegetables and one fruit, all at home during Thanksgiving. He turns his oven on once per year while trying to ascertain if the power is out or the light bulb in the kitchen needs to be replaced. The rest of his nutrition is made up entirely of free donuts, bagels and pizza.

The place to get all this free food, naturally, is various department talks and seminars. And while we're there, we may as well try to learn some physics.

Saturday, October 16, 2010

Four Fantastic Books (3 of which are free)

Well, we just had our fall break, which means I get a bit of a break, coincidently enough. Somehow I've managed to read three books in the last two days, and each of them were excellent enough that I need to tell people about them.

Street Fighting Mathematics - Sanjoy Mahajan

The art of educated guessing and opportunistic problem solving

Link to MIT Press Site

You know that feeling you get when it's the second half of January and you put on new clothes that have just come out of the dryer? This book is like a cross between that and a kick in the face.

The warm fuzzy-clothes-out-of-the-dryer feeling will come from the realization that you can wield unsurmountable power. The kick in the face will come when you realize you're not doing it yet.

Sunday, October 10, 2010

Caught In The Rain II

I was rather proud of my last post about being caught in the rain.  In that post, I concluded that you were better off running in the rain, but that the net effect wasn't incredibly great.  However, when I told people about it, the question I inevitably got asked was:  What if the rain isn't vertical?  That's what I'd like to look at today, and it turns out to be a much more challenging question.

Tuesday, September 28, 2010

Beards and Pulsars

The bearded half of
A few weeks ago I was on a bus going through Scranton and I read a super-awesome fun fact regarding the Hulse-Taylor binary pulsar in Black Holes, White Dwarfs and Neutron Stars.  Sadly, I have since forgotten it and left the book a few thousand miles away.  So, let's just make up our own!

Sunday, September 19, 2010

Paradigm Shifts 2: Paradigm ShiftER

Last time, I presented reasons why it would be economically infeasible for the US to switch to the metric system. This time, I'd like to talk about a change that could relatively easily be brought about soon. A change that would barely cost a thing, but could improve efficiency dramatically in many jobs and in every day life for many people.

A change of this type would be very handy. Puns aside though, what I'm talking about is this:

(again lots from Wikipedia)

Breaking Intuition

When I walked into my first day of physics class in high school, I carried with me a set of ideas which I learned from simply observing and interacting with the world. In fact everyone builds up what they believe to be intuitive concepts, whether it be in science, math, or any other field. Without any scientific training whatsoever, we begin to build intuition.

If you let go of a ball in the air, what will happen?
If you try to run on the ice of a frozen lake, will it be easier than running on the sidewalk?
If you stand in the sun and on the ground you see a strange dark misshapen copy of yourself imitating your every move... who is following you?

Unfortunately we run into an issue when our intuition disagrees with experimental results or someone else’s intuition. At that point, it is essential to break down and analyze our intuition to find where any problems in our logic may exist. This process of continually breaking down and analyzing intuition is key to progressing in science.

Tuesday, September 14, 2010

Visualizing Quantum Mechanics

Or how I learned to stop worrying and love the computer.

[Note: There's a neat video below the fold. ]

A Confession

I was recently rereading the Feynman Lectures on Physics. If you haven't read them lately, I highly recommend them. Feynman is always a pleasure to read. As usual, I was surprised. This time the surprise came in lecture 9, which the way the course was laid out meant that this was something like the last lecture in the third week that these students had ever received of university level physics.

The lecture is on Newton's laws of dynamics. The start is of course Newton's first (second) law,
\[ F = \frac{d }{dt } (mv ) \]
which, provided the mass is constant takes the more familiar form
\[ F = ma \]

After discussing the meaning of the equation and how in general it can give you a set of equations to solve, he naturally uses an example to illustrate the kinds of problems you can solve.

What system does he choose to use as the first illustration of a dynamical system?

Sunday, September 12, 2010

Microseconds and Miles

The following is an unfinished manuscript found under heaps of rubble and pizza boxes here at Virtuosi headquarters.  It appears to be some sort of screen play, though one would be hard-pressed to figure this out solely from the script.  The true giveaway was the 100 page addendum (not published) full of potential titles and acceptance speeches.  I dare not bore you with these vanity pages in their entirety, but just for completeness and posterity I include some samples.

For possible titles we have: "Dr. Dre, OR: How I Learned to Stop Worrying and Love the Metric," "How to Teach Physics to your Dee Oh Double G (West Coast Edition)," "Bring Da Ruckus: ODEs by ODB" and "Flavor Flav's Flavor Physics...boooyeeeee!" among other even worse and less relevant titles.

Among the acceptance speeches we have one that starts: "I would like to thank the Academy, Scott Bakula and Chuck D.  You know what you did.  Here's a song I wrote...", etc.  It is all very painful.

There is almost no value to this document whatsoever, but it does present a nice fun fact about GPS.  The legible parts of the script are thus presented below.  The illegible parts appear to have been obscured by some caustic mixture of Mountain Dew, pizza sauce and tears.

Wednesday, September 8, 2010

Quantum Chess!

Ever find out when you're playing chess that the Queen you reached for is actually a pawn? Probably not. But most chess games aren't affected by the weirdness of the quantum world. This one is:

Tuesday, September 7, 2010

Paradigm Shifts 1

Hi everybody, I'm Sam, and this will be my first contribution to the blog! (cue applause) It will not be a physical modeling exercise; instead I will be writing a little bit about about paradigm shifts in a series of a few posts. I hope it will provoke some interesting discussion.

"But Sam," you ask, "Isn't 'paradigm shift' just a buzzword that people use to sound important?" Well, maybe, but it's also useful phrase used to describe a substantial change in the way something is done. Consider, for example,


(many details from Wikipedia)

Thursday, September 2, 2010

Remembering two things

One of my professors, Yuval Grossman, was talking about the zoology of particle physics in class the other day. Trying to get us to remember such trivia as the mass of the B meson, he noted that it's easier to remember two things than it is to remember one - and as it happens, the mass of the B meson is about 5280 MeV, which is also the length of a mile in feet (an equally obscure piece of trivia, if you ask me).

This reminded of one of my first calculus classes back home where another professor (Mikhail Sodin) chided us for not knowing the value of e, 2.71828. This is easy to remember, he said because 1828 is the year Lev Tolstoy was born.

Then again, when I came to write this post, I could neither remember e, nor Tolstoy's year of birth - or even that it was Tolstoy, rather than Dostoevsky or some other Russian author. So perhaps two things are not easier to remember than one after all.

Wednesday, September 1, 2010

Caught In The Rain

There's an age old question that mankind has pondered.  I'm sure that noble heads such as Aristotle, Newton, and Einstein have pondered it. I myself have raised it a few times.  The question is: do you get more wet running or walking through the rain?  Now, I know that this question was mythbusted a while back.  So this is one of those situations where I know the result I want to get to with my calculation: according to mythbusters running is better.  Still, I think formulating the question mathematically will be fun, plus if I fail to agree with experiment everyone can mock me mercilessly.

Saturday, August 14, 2010

Ringing A Bridge

Matt and Jared standing on
our experiment
When you strike a bell, it rings at a given frequency.  This frequency is called the resonant frequency and is the natural frequency at which the bell likes to ring.  Just about anything that can shake, rattle, or oscillate will have a resonant frequency.  Things like quartz crystals, wine glasses, and suspension bridges all have a resonant frequency.  The quartz crystals oscillate at frequencies high enough for accurate timekeeping in watches, the wine glasses at audible frequencies to make boring dinners more interesting, and bridges at low enough frequencies that you can feel it when you walk.  It is the resonant frequency of bridges that we decided to measure.

Thursday, August 12, 2010

Terminal Velocity 2: A Theorist's Experimental Experiment

Yesterday we rode down Ithaca's hills in an attempt to estimate the terminal velocity of a bike rider braving the city's potholes. But estimations are easy, and we relied on a number of factors - the drag coefficient and area of the bicyclist, in particular - to get them. To see how well we did, it's time to move on to the experimental portion this exercise. Our tools? My bike (figure 1), and my beloved accelerometer (figure 2), with Google's My Tracks app installed.

Figure 1: Our vehicle

Figure 2: Our instrumentation

Wednesday, August 11, 2010

Teminal Velocity

The impetus for this post lies with three facts. First, I like to bike to work. Second, Cornell sits on a hill. And finally, I'm not very brave.

As a result of all of these, along with Ithaca's less-than-optimal road maintenance, my semi-daily rides home tend to produce a lot of wear on my brakes as I cruise downhill at what appears to me to be very high speeds. I began to ponder just how high this speed really is, and if I could reduce my use of the brakes or if I'm going to end up using them anyway at the bottom of the hill.

Friday, August 6, 2010

The Wrath of Blotto

You may remember when I invited everyone to play my webform version of Colonel Blotto. Well, its still up and has been up for some time, but hasn't seen any action for a while so I thought it might be time to take a look at the results.

Colonel Blotto is an interesting game. It seems to me, that much of this interest derives from the fact that how well your strategy performs is very much a function of which strategies exist in the pool. There is not a clear cut winning strategy, you need to feel out the existing pool and adapt accordingly.

So to stir things up a little bit, in what follows I will share some data from the existing database, refraining myself from commenting too much. Basically, stay tuned for a bunch of pretty pictures which will hopefully get your gears turning. The game is still up, feel free to try to game it now that this information is out. Might be interesting to see what kind of effect releasing the leaderboard will have on the leaderboard.

Tuesday, August 3, 2010

Steak Dinner

Sorry about the blog hiatus. During the summer, without teaching classes, inspiration is harder to come by. But, tonight I cooked a steak. I recently got a new digital meat thermometer. My plan was to slowly cook the steak until the internal temperature got to be about 140 degrees Fahrenheit with the oven at 200 degrees, take it out, wrap in tin foil, crank the oven to 500 degrees, stick it back in, and give it a nice exterior, reaching an internal temperature of about 150 degrees which would put it at about medium.

After I put the steak into the oven though, I started to watch the temperature go up on my digital thermometer and thought, why not take data. And so I did. Here are the results.

Above you see the internal temperature of the steak as a function of time. First some comments about the graph.

Saturday, July 24, 2010

On the Death of Karl Schwarzschild

Every once in a while, in the study of science, one comes across biographical snippets that momentarily breathe life into names that otherwise serve as shorthand for equations and eras.  As an obvious effect of the selection bias involved with including this superfluous information in technical books, they are bound to be pretty interesting.  Such stories range from the hilarious antics of Feynman [1] or Fermi [2], to the heartbreaking stories of Boltzmann and Oppenheimer, and even to the surprisingly scandalous life of Erwin Schrödinger.  But my all time favorite of all these historical "fun facts" is that of the man who provided the first exact solution to the Einstein field equations while fighting in the First World War:  Karl Schwarzschild (pictured left impersonating a surprised walrus [3]).

Tuesday, July 20, 2010

Something Bugging Me

Apparently July is a quiet month here at the Virtuosi.  We're busy with research, travel, vacation, etc.  I, myself, have been busy with only a few of those things, though I've also been studying for my qualifying exam, which is coming up in less than a month.  However, that's not the question before us today.  Today I'd like to think about the density of bugs in the air.  I was walking outside this past weekend, there was a fierce wind blowing, and twice in five minutes a bug hit my ear.  That seemed like a lot.  But for 1 hour of previous walking no bugs hit my ear.  How many bugs would there have to be per cubic meter of air to achieve that rate?

Thursday, July 1, 2010


Here at the Virtuosi, we're concerned. We are concerned that perhaps the world is really not ready for a zombie apocalypse. You know, the kind of zombie apocalypse that you may have seen in such classics as "Night of the Living Dead", "Shaun of the Dead", or perhaps the even more recent "Zombieland" (sweet cameo by the way). The kind of zombpocalypse that could leave major cities void of life and the country plagued with the undead.

Well, Alemi and I were curious as to how likely such a pandemic was to occur and what it would look like in the simplest of models. In typical Virtuosi fashion, we threw some physics at it and this is what we came up with.

The Impossibility of Why

So I think we've all been rather busy here.  Hence the lack of posts.  I'm going to try to keep this one short but sweet.  A lot of people think that physics tells us why things happen.  Why is the sky blue?  Why does the earth orbit the sun?  Why does copper transmit electricity so well?  These all seem like perfectly reasonable questions to ask.  Questions that we, as physicists can answer.  Yet, I entitled this post the impossibility of why.

In general, questions about why are not good questions for physicists   More accurately we answer questions about how.  Or what.  What phenomena causes use to see the sky as blue?  What forces cause the earth to orbit the sun?  How does copper transmit electricity so well?  In general, we can't answer a question of why.

Saturday, June 19, 2010

Life as an Experimenter - Reflections

I've been doing experimental physics for about three years.  I started during my sophomore year in college, went straight to graduate school, and continued here (albeit not quite immediately).  There are people out there who have been doing this for a lot longer than I have, but I've gained a few insights, by working with said people, and through my own experiences in the lab.  I thought I'd try to share a few of these, to help illuminate the past few days I blogged about.

Friday, June 18, 2010

Life as an Experimenter - Day Three

Today marks the third and final day in our beam time at CHESS.  I think the circles under all of our eyes may take away from the glamor a bit, but the right makeup specialist could fix that.  If they ever make a movie about us, which they should, I want to be played by some really awesome british actor.  I think that would be about right.  Someone with a strong jaw.  In case it's not obvious, lack of sleep is getting to me a little bit.  Read on for the final few hours of our experiment.

Life as an Experimenter - Day Two

Today I'm continuing my series on the life of an experimenter.  Today is the longest day, since we have beam time for all 24 hours.  And after the setbacks of yesterday, we feel compelled to use it to the utmost bit.  Read on for more tantalizing glimpses of the grit behind the glamor of the rock-star-like lifestyle of an experimental physicist.

Thursday, June 17, 2010

Life as an Experimenter - Day One

I'm an experimental physicist.  If you think this sounds like a job second in glamour only to rock star you would be right.  Just like being a rock star, you have to deal with the people, the shows, the lights, the groupies . . . okay, maybe I'm lying about the groupies.  Unless you're Brian Greene.  Also similar to a rock star, no one really knows what it is we do behind the scenes (when we're not touring the nation or publishing papers).  I'd like to pull back that curtain a little bit.

Wednesday, June 16, 2010

How Long Can You Balance A (Quantum) Pencil

Sorry for the delay between posts. Here in Virtuosi-land, we've all begun our summer research projects and I think we've just become a little bogged down in the rush that is starting a summer research project. You feel as though you have no idea what the heck is going on, and just try desperately to keep your head up as you hit the ground running, but thats a topic for another post.

In this post I'd like to address a fun physics problem.
How long can you balance a pencil on its tip? I mean in a perfect world, how long?

No really. Think about it a second. Try and come up with an answer before your proceed.

What this question will become by the end of this post is something like the following:
Given that Quantum Mechanics exists, what is the longest time you could conceivably balance a pencil, even in principle?

I will walk you through my approach to answering this question. I think it is a good problem to illustrate how to solve non-trivial physics problems.

I will try and go into some detail about how I arrived at my solution. For most of you this will probably be quite boring, so feel free to skip ahead to the last section for some numbers and plots.

Tuesday, June 8, 2010

My Pepsi* Challenge

The basement of the Physics building has a Pepsi machine.  Over the course of two semesters Alemi and I have deposited roughly the equivalent of the GDP of, say, Monaco to this very same Pepsi machine (see left, with most of Landau and Lifshitz to scale).  It just so happens that Pepsi is now having a contest, called "Caps for Caps," in which it is possible to win a baseball hat.  There are several nice things about this contest.  Firstly, I drink a lot of soda.  Secondly, I like baseball hats.  So far so good.  Lastly (and most important for this post), is that it is fairly straightforward to calculate the statistics of winning (or at least simulate them).

Monday, May 31, 2010

Memorial Day Distractions

Summer is upon us. That means summer research, and online games. In order to help you through this three day weekend and beyond, I thought I'd share some of the more physics inspired games I've been playing lately to pass the time.

I really enjoy physics based games. When done right, I think they can not only be fun and engaging but have the opportunity to teach you something.

Sunday, May 30, 2010


I'd like to start a short series of posts on what I'm doing this summer. Like most all of the first year physics graduate students, I've found a research group at Cornell for the summer, and if everything goes well, I'll continue working with them after the summer is done. This is good for three reasons. First, I'm getting paid, so I can do things like pay rent. I'm not all about the money, but having a place to live is a big deal to me. Second, I'm getting a chance to explore a new area of research, with limited expectation of commitment on my part. Third, if everything works out, I've found the group I'll be working with the for the next 4-6 years. I'd like to spend a few posts here on The Virtuosi discussing first the physics I'm considering, and then what I actually do. Today I'm going to talk about the most exciting sounding piece of my work, cryopreservation.

Wednesday, May 26, 2010

I was born on Wednesday

Probability is a tricky thing. There are a lot of nonsensical answers to be had. I just read an article about the recent Gathering for Gardner meeting that took place. Gathering for Gardner is a unique meeting for mathematicians, magicians and puzzle makers where they get together and talk about interesting things. The meetings were inspired by Martin Gardner, one of the awesomest dudes of our time, who unfortunately just passed away.

The question put to the floor was the following:
"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

Think about that for a moment. Not too hard though. The answer turns out to be surprising. Upon reading the question, I thought about it for a long time and managed to confused myself entirely. Thinking I had gone crazy, I wrote a little python script to test the riddle, which only left me more convinced I had gone insane. I've spent most of the night thinking about it, and after making it half way to crazy, I've come around and am momentarily convinced the puzzle makes perfect sense.

I'm going to attempt to convince you it makes perfect sense, but I plan on doing it in steps so as to reduce the bewilderment.

Tuesday, May 25, 2010

Why is the Grass Green?

I was outside talking with Alemi last week and we were both startled to realize that the frozen white tundras of Ithaca had somehow transformed into fields of green.  Apparently the snow was a temporary fixture that covered real live grass.  Neato, gang!  

The joy at seeing green grass led quickly to surprise, confusion and then anger.  Why the heck is grass green?  Well, things look a color when they reflect back that color.  So grass is green because its pigments (chlorophyll) absorb only a certain range of the visible spectrum, reflecting back the greenish bits.  But if I know anything about approximating the sun as a blackbody, I know that it has a peak output of around 550 nm (i.e. green) light.  So what's going on?  Why are plants blatantly rejecting the most abundant kind of light?

Flying back

Those of us originating on the right side of the atlantic ocean are familiar with a little quirk of international flights: the flights home are shorter. Specifically, going from Tel Aviv to New York takes about one hour longer than going the other way around.

This is an oddity, and the very first explanation that comes to mind the rotation of the Earth. After all, our naive image of a plane going up in the air might be something a little like a rock being thrown up from a moving cart, and we would imagine the plane to pick up some relative speed by not rotating as fast as the Earth. Is this a factor in the plane's movement?

Fishy Calculation Followup & New Contest

So, some you may remember when I attempted to calculate how much the oceans would lower if you took out all of the fish in an earlier post.

Well, the results came in a while ago, but I forgot to mention that I lost the contest. I was about two orders of magnitude off from the winning answer.

In light of my failure, I'm going to try again in the newest contest. This time the question is a bit stranger:

How many buff hamsters would it take to completely power a mansion?

I encourage all of The Virtuosi readers to enter as well, it only takes a minute to come up with some number.

Good luck one and all.

Sunday, May 23, 2010

Esoteric Physics I - The Hall Effect

What we usually do here at the The Virtuosi is take an interesting problem, and work out the physical principles behind what we're seeing.  Or pose a question and try to answer it.  Now, I'm a big fan of this kind of thing, which is why I've done so much of it.  But I worry that it might give a slightly skewed view of physics.  Sure, physics explains things.  That's why we do it.  But not everything in physics is laser guns and solar sails.  There are a lot of interesting physics phenomena that the general public will never hear about, because they're just too, well, esoteric.  What I'm going to do is occasionally talk about such effects, and, for some of them, give you applications for these strange effects you might see on a day-to-day basis.  Today I'm going to examine the Hall effect.

Wednesday, May 19, 2010

Physics of Baseball: Batting

Summer is upon us, and so that means that we here at the Virtuosi have started talking about baseball. In fact, Corky and I did some simple calculations that illuminate just how impressive batting in baseball can be.

We were interested in just how hard it is to hit a pitch with the bat. So we thought we'd model hitting the ball with a rather simple approximation of a robot swinging a cylindrical bat, horizontally with some rotational speed and at a random height. The question then becomes, if the robot chooses a random height and a random time to swing, what are the chances that it gets a hit?

Cell Phone Brain Damage: Part Deux

I thought I'd take another look at cell phone damage, coming at it from a different direction than my colleague. Mostly I just want to consider the energy of the radiation that cell phones produce, and compare that with the other relevant energy scales for molecules.

Cell Phone Energy

So, lets start with cell phones. I looked at my cell phone battery, and it looks like it is rated for 1 A, at 3.5 V. So when it is running at its peak it should put out about 3.5 W of power in electromagnetic waves (assuming it reaches its rating and all of that energy is fully converted into radiation). But what form does this energy take? Well, its electromagnetic radiation, so its in the form of a bunch of photons. In order to determine the energy of each photon, we need to know the frequency of the radiation. Surfing around a bit on wikipedia, I discovered that most cell phones operate in the 33 cm radio band, or somewhere between about 800 - 900 Mhz.

How much energy does each ~ 1 Ghz photon have? We know that the energy of a photon is:
\[ E = h \nu \sim 7 \times 10^{-25} \text{ J} \sim 4 \times 10^{-6} \text{ eV} \]
it will be convenient to know the photon energy in "eV's". 1 eV is the energy of a single electron accelerated through a potential of 1 volt, or
\[ 1 eV = (1 \text{ electron charge} ) * ( 1 \text{ Volt} ) = 1.6 \times 10^{-19} \text{ J} \]

So my cell phone is sending out signals using a bunch of photons, each of which has an energy of about 4 micro eVs.  Lets consider the energy scales involved in most molecular processes and compare those scales with this energy.

Monday, May 17, 2010

Solar Sails III (because two just isn't enough)

One thing that I've wanted to quantify since reading Intelligent Life in the Universe, an outstanding book by Carl Sagan and I.S. Shklovskii, is the idea of exogenesis. Exogenesis is the hypothesis that life formed elsewhere in the universe and was somehow transferred to earth in the form of some small seed or spore. Now since E.T. E. coli presumably do not have little tiny jetpacks or other means of active transport, they would need to traverse the cosmos in some passive way. One such way would be solar sailing.

Way back in Solar Sails I, we derived equations describing the maximum speeds and time-of-travel for various distances for a given solar sail. Each of these equations was a function of the surface mass density of the sail, which is just the amount of mass per unit cross-sectional area. All we need to know is the cross-sectional area and mass of a given object and we can apply these equations to just about anything!

Ask a Physicist: Volume I

We have received our first Ask a Physicist e-query! An entity known only to us as "Hungry" writes:

"We had a dispute at a dinner party about whether blowing on hot food actually makes it cool down faster, or only gives you something to do while you wait for your food to cool."

While it is questionable whether or not I am indeed a physicist (but people do pay me to do physics) and whether I will answer definitively (there'll be some hand-waving), I'll give it my darndest.

Solar Sails II

[NOTE: In my hurry to make up for weeks of non-posts, I managed to almost immediately knock Nic's first post from the top of the page. It's got the LHC, black holes, and about 3 full cups of metric awesome, so make sure you check it out (after reading this one, of course).]

Last time we did some calculations on how fast and far our solar sails can go, but those calculations were just for the sail itself. If you are going to do any science with it, you're going to need a payload. Let's take it a step further and make it an actual spaceship (with people and everything!)

Sunday, May 16, 2010

Solar Sails I

Solar sails are in the news again, and this time not just for blowing up. The Japanese space agency is launching what they hope to be the first successful solar sail tomorrow. In honor of that, we will be discussing the physics of solar sails.

First of all, what the heck are solar sails? Solar sails are a means of propulsion based on the simple observation that "Hey, sails work on boats. Therefore, they should work on interplanetary spacecraft (in space)." Boat sails work when air molecules hit into the sail and bounce back. By conservation of momentum, this gives the boat sail an itty bitty boost in momentum. Summing over the large number of air molecules moving as wind, the boat gets pushed along in the water. A similar process works with solar sails, but instead of air molecules doing the hitting, it's photons. Since each photon of a given wavelength has some momentum, by reflecting that photon the solar sail can gain a tiny bit of momentum. Summing over the large number of photons coming from the sun over a long time frame we can get a considerable boost. So let's see how good solar sails are.

Solar Sails Addendum I

As requested, below is an explicit evaluation of the silly looking integral in Solar Sails I.  If you just want some hints to do the integral, see Solar Sails Addendum II.

Solar Sails Addendum II

This is the schematic version, if you just wanted hints.  The full solution is given in Solar Sails Addendum I.

Why Black Holes from the Large Hadron Collider Won't Destroy the World

Hi everyone. As this is my first post, I thought I'd introduce myself. Like the rest of the Virtuosi, I'm a graduate student in physics at Cornell University. I work in experimental particle physics, in particular on the Compact Muon Solenoid, one of the detectors at the Large Hadron Collider. I'll post more on what I actually do at some point in the future, but I thought I'd start with a post in the spirit of some of the other fun calculations that we've done. My goal is to convince you that black holes created by the LHC cannot possibly destroy the world.

To start with, the main reason no one working on the LHC is too concerned about black holes is because of Hawking radiation. While we usually think of black holes as objects that nothing can escape from, Stephen Hawking predicted that black holes actually do emit some light, losing energy (and mass) in the process. In the case of the little bitty black holes that the LHC could produce, they should just evaporate in a shower of Hawking radiation.

That's great you say, but Hawking radiation has never actually been observed. What if Hawking is wrong and the black holes won't evaporate? Well, the usual next argument is that cosmic rays from space bombard the earth all the time, producing collisions many times more energetic than what we'll be able to produce at the LHC. To me, this is a fairly convincing argument. However, let's pretend we don't know about these cosmic rays and that there's no Hawking radiation. We can calculate what effect black holes produced by the LHC would have on the earth if they do stick around.

Thursday, May 13, 2010

Freezing in Space II - Turn On The Sun!

Yesterday I considered how long it would take a human to freeze in space.  However, I considered only what would happen if you were not absorbing any radiation from nearby sources.  Today we consider what happens if you do have hot objects nearby.  Namely, the sun.  The sun provides a lot of energy, even as far away from it as we are.  It keeps the earth at a comfortable ~20 C, good for us humans, and provides the energy for life on earth, also good for us humans.  That's a lot of energy.  So maybe the sun can keep you alive when you're adrift in space.  Or at least keep you warm.  I still think you'll asphyxiate.

Wednesday, May 12, 2010

Freezing in Space I - Blackest Night

In the last post I made, I discussed the fact that humans radiate energy.  In that post I calculated that we actually radiate quite a lot of power.  This immediately raises a few questions, the most obvious one being: How long would it take you to freeze in space?  This question is multifaceted, and I'm going to split it between two parts.  This first part, 'Blackest Night' is how quickly we'd freeze if we were completely lost in space, nothing anywhere near.  The second part, 'Turn On The Sun!' will address what would happen in near earth orbit.

Sunday, May 9, 2010

Human Radiation

Things are still busy here at the Virutosi.  Hopefully in a week or so we'll be back to normal, and much more active than we've been recently  Anyways, today I'd like to consider human radiation.  It is well known that any object will radiate energy based on its temperature.  Even more interesting, we radiate at all wavelengths, though at the human body temperature our radiation is sharply peaked in the infrared.  Even so, we still put out some x-ray radiation.  As a professor of mine once said, consider that next time you sleep with someone!  Given all this, the question on my mind today is:  how does the energy we radiate daily compare to the energy we consume?  That is, why don't I lose weight sitting here typing on the computer?

Tuesday, May 4, 2010

Letting Air Out of Tires II

In a recent post I calculated how cold air coming out of bike tires should feel.  However, at the end of the post, I did note that there are competing explanations for why the air cools.  There's the approach I took, which is adiabatic cooling, but there's also something called the Joule-Thomson effect.  The Joule-Thomson effect has the interesting property that helium being let out of a bike tire would actually be warmer, which suggests an immediate way to test which effect is dominant.  We pressurize a bike tire with helium, and see if the valve gets cold or hot.  This is exactly what I did.

Monday, May 3, 2010

Physics as Magic?

There's a nice post over at Physics Buzz that I thought I might draw your attention to.  The central quote for me is:

"Speaking strictly about technology - which is often the knowledge attained by physicists put into practical use by engineers - physics has created some pretty amazing things. Cars, planes, iphones, medical treatments, lasers, 3-D movies, and the Large Hadron Collider. We are constantly WOWED by science. Unfortunately, the less someone understands how these things work, the more they begin to believe anything is possible. In other words, if you don't understand the parameters that allow for amazing things (like jets!) you also don't understand the parameters that would prevent other things (like energy generating heart replacements). If you don't understand anything about physics and technology, then it appears to be nothing short of magic, and magic has no bounds..."

Letting Air Out of Tires

Have you ever noticed how when you let air out of a bike tire (or, I suppose, a car tire) it feels rather cold?  Today we're going to explore why that is, and just how cold it is.  Many people consider the air escaping from a tire as a classic example of an adiabatic process.  What is an adiabatic process?  It is a process that happens so quickly there is no time for heat flow to occur.  For our air in the bike tire this means we're letting it out of the tire so quickly that no energy can move into it from the surrounding air.

Friday, April 30, 2010

The Beer Diet

I know it's been pretty quiet over here this week.  The semester is winding down (a week of classes left), and that means that things have been kicked up into another gear.  We've got four or five ideas bouncing around at the moment, so hopefully we'll get some up soon.  Today I'd like to talk about the beer diet.  A while back, there was a rumor going around that if you drank ice cold beer your body would burn more calories heating the beer than the beer contained.  It turns out that this false, and I think the claim relied on a lack of knowledge that the American food Calorie is actually one thousand calories (note the difference in capitalization).  Let's prove this to ourselves.

Tuesday, April 27, 2010

Cell Phone Brain Damage... or not.

For better or worse, cell phones are a part of our lives. I say this because they can be convenient when needed, but there's nothing more annoying than a dropped call or an inconveniently timed ring tone. Since many people carry their phones with them daily, there has been a number of studies which ask what are the long-term health effects of cell phone usage. While the controversy rages among medical researchers, I decided to find my own answers by doing a calculation based on the power output of a simple hand set.

Sunday, April 25, 2010

Some of the Best Advice You'll Ever Receive

I came across what might be the best advice any student (nay any human being) could possibly ever receive reading a book today...

Thursday, April 22, 2010

End of the Earth Physics III- Asteroids!

No day of earth destroying celebration would be complete without that apocalyptic all-time favorite: the asteroid. And to be fair, it deserves to be the favorite. Of all the doomsday predictions out there (nuclear holocaust for the cynics, death by snoo-snoo for the optimists) it is the only one that is just about certain to occur at some point in the geological near future. On top of that, it's one that we could potentially avoid with enough time and some neat ideas .

Let's model an asteroid impact on the earth. We will assume an asteroid starting from rest at infinity and falling to the surface of the earth due to earth's gravity only. Now obviously these assumptions are not exactly correct. We can imagine our asteroid not starting from rest or feeling some force due to the sun. Each of these would certainly change our answer, but should still be within an order of magnitude of our result.

Earth Day - Earth Units

In honor of Earth day, I thought I would take a look at what it would mean to do physics in 'Earth' units. What do I mean by that? Well lets be anti-Copernican here, in fact lets assume the opposite of the
Copernican principle, and state that the Earth is privileged in the universe and define all of our units around the Earth.

So, I will put a little subscript earth on all of the 'earth' units. They are to be read as 'earth meters' or 'earth amps', etc. We will take as our starting point the mass, radius and day of the earth, normalizing all of our standards to that. This gives us our initial conversion factors
\[ 1 g_{\oplus} = M_{\oplus} = 5.9742 \times 10^{34} \text{ kg} \]
\[ 1 m_{\oplus} = R_{\oplus} = 6378.1 \text{ km} \]
\[ 1 s_{\oplus} = T_{\oplus} = 86,400 \text{ s} \]

From this, we can figure out what all of the other 'earth' units would be.

Nobody Really Gets Quantum

Nobody Really Gets Quantum / Etgar Keret

On Yom Kippur eve Quantum walked over to Einstein's house to seek forgiveness. "I'm not home," shouted Einstein from behind a closed door. On the way home people taunted him and somebody even hit him with an empty can of coke. Quantum pretended not to care, but deep inside he was really hurt. Nobody really gets Quantum, everybody hates him. "You parasite" people cry out when he's walking down the street, "why are you dodging the draft?" - "I wanted to enlist," Quantum tries to say, "but they wouldn't take me, because I'm so small." Not that anybody listens to Quantum. Nobody listens to Quantum when he tries to speak up for himself, but when he says something that can be misconstrued, oh, then suddenly everybody's paying attention. Quantum can say something innocent like "wow, what a cat!" and right away the news says he's making provocations and run off to talk to Schrodinger. And anyway, the media hates Quantum most, because once when he was interviewed in Scientific American Quantum said that the observer affects the observed event, and all the journalists thought he was talking about the coverage of the Intifada and claimed he was deliberately inciting the masses. And Quantum can keep talking until tomorrow about how he didn't mean it and he has no political affiliation, nobody believes him anyway. Everybody knows he's friends with Yuval Ne'eman.

End of the Earth II - Blaze of Glory

In honor of earth day today, many bloggers are posting things about how to save the earth, or retrospectives on earth days past. We here at The Virtuosi decided, what better way to celebrate the earth than to figure out how to destroy it? So that is exactly what we intend to do. This post will focus on the destruction of the earth by a laser beam. This is a familiar concept. Whether it is Marvin the Martian or the Death Star, destroying planets with lasers (or threatening to) is a common theme.

We will be considering two questions today.  The first is fairly obvious:  how much power would the death star need to destroy the earth?  The second relates to a topic of continuing interest of mine: how much would the death star recoil upon firing?

Wednesday, April 21, 2010

Laser Launching

Lasers seem to be on my mind recently.  Just yesterday, the class I TA for (E&M for engineers) talked about the momentum carried by E&M waves.  This called to mind a discussion I had with a housemate a few weeks back.  He had heard somewhere that 'they' were thinking of launching satellites with lasers.  No way, I thought to myself.  Satellites are too heavy.  However, his question has been hovering around in my mind, so I've decided to try and answer it: can we use a laser to launch a satellite into orbit?

Laser Gun Recoil: Follow-up

Matt Springer over at Built on Facts has a very nice post following up on my earlier analysis of whether or not a laser gun would recoil.  In my analysis I came to the conclusion that the momentum delivered was much less than that of a conventional gun, but that the impulse, and hence the force delivered, was about the same.  However, that force was delivered over a ~30ns timescale, which left open the question of whether or not the wielder would be able to feel that recoil.

While I had thought to possibly return to this issue at some point, Matt has beat me to it.  This is fortuitous, because I wasn't sure quite where to start with the question, while he approaches it in a very sensible, clear manner.  The gist of his post is that he compares our response to short time scale forces to our ability to sense sound.  I won't go through his calculations, you can check those out for yourself, but he concludes that it is very unlikely that we could feel the recoil of our laser gun.

Case closed.  Though, like any good scientist, we may choose to reopen it later if more evidence comes to light.

Q Factors

When I walk in my door when I get home, I hook my keys, which I keep on a carabiner, onto a binder clip that I've clipped onto my window sill.  Its a great way to never lose your keys.  But one thing I always notice is that when I hook it on, it swings, and every time it swings it makes a click.  This you might expect.  What always surprises me is how long the keys keep swinging.  They seem to swing for a surprisingly long time, minutes.

It always catches me off guard.  In order to explain why, I get to talk about Q Factors

Saturday, April 17, 2010

The End of Earth Physics I

I was reading the Wikipedia page for the Hitchhiker's Guide books the other day and found that it started as a series of radio shows called "The Ends of Earth." At the end of each episode, the Earth would be destroyed. Since I feel like this is the best way to end any TV show/movie/book/news broadcast/Mayan calendar, I will shamelessly steal the idea.

Since this is the first End of the Earth post, we will start small and just consider the boiling off of all the world's oceans. To be precise, we will consider how much energy it would take to turn all the world's water at 0 degrees Celsius to water vapor at 100 degrees Celsius.

Friday, April 16, 2010

Entropy and the Arrow of Time

Somebody I talked to yesterday about what should the topic of my post be suggested that I should write something about cosmology, in particular the Big Bang and the ultimate fate of the universe. Being mostly interested in condensed matter, I am by no means an expert in cosmology (I’ll leave that to Corky). However, I will not abandon the request, but instead, spin it in a more condensed matter direction, by talking about Entropy.

Physics for non-physicists

This is my first (but certainly not the last) post in this blog, so let me introduce myself. My name is Nikolay, and as most of the others contributing to the blog, I am also a first-year graduate student in physics at Cornell.

So, what is my motivation to join the team headed by Mr. Alemi in contributing to the blog? I happen to have many friends that are not in physics. In fact, probably due to the fact that I graduated from a liberal arts college, many of my friends have never taken any physics beyond that one class in high school, which due to both the inherent difficulty of teaching physics at introductory level and the lack of good high school physics teachers was often an unpleasant experience that scared them away from physics for life. Now, imagine what my difficulty is when on a daily basis I have to answer the question: “So what do you study/work on as a graduate student?” I usually try to come up with a sentence or two describing the essence of what I am doing without going into too many details, but even that is a daunting task. There seems to be a disconnect between the world in which a physicist lives and the general public. As Chad Orzel pointed out in the talk that motivated the creation of this blog, this is not the general public’s fault, but our fault as physicists of not really committing enough effort in relating our knowledge to the rest of the world.

In short, my goal is to create a series of posts about physics geared to people with no physics background that would build upon each other and culminate with a post providing an answer to the question of what my research project is beyond the generic words I would often resort to that would rather leave most people confused. I plan to use as little math as possible, which will not be an easy task considering that math is the language of choice for physics. However, mathematics is just a tool, and physics is not about equations and complicated algebra, but about how nature works, which we should be able to formulate in plain English. I know I am embarking on a difficult task, so wish me luck, and please leave your thoughts in the remarks section of the blog, since any feedback would be appreciated.

Thursday, April 15, 2010

Onsager's Tour de Force

In 1943 in a tour de force of mathematical physics, Lars Onsager solved the 2D Ising Model.

His solution has proved crucial in furthering statistical mechanics, allowing theorists to check all of there approximation schemes against analytical results.

I call his effort a tour de force because it was a huge mathematical exercise, his solution spanning 33 pages. I also call it a 'tour de force' because I have seen it referenced as such in no less than 5 different sources, as well as numerous times in speech. This got me wondering, just how many times is Onsager's solution called a tour de force...

Another Reason Why The Core is Stupid

I assume everyone has heard of The Core, the terrible scifi movie from 2003.

If you haven't you're missing out on what appears to be, according to Discover magazine, the worst sci-fi film ever. There are already numerous sites that discuss the bad science in the core (here, or over at Bad Astronomy), but they all seem to ignore another fundamental problem with the plot.

I don't think I'll give too much away if I tell you that the basic premise of the movie is that the earth's core has stopped rotating, and so the earth's magnetic field is collapsing, which they claim will mean that all of the previously deflected microwaves (note: EM radiation is not bent by a magnetic field) will cook us all. Now, a lot of people have focused on the microwaves bit, which while bad science, one could argue that we would still have some bad effects from loosing our magnetic field.

The problem I have is that the Earth's magnetic field cannot change that abruptly.

Wednesday, April 14, 2010

Locating the Sun Photo

This photo has been making the rounds on the internet lately:
Originally from here.

Its a time lapse photo of the sun taken from a pinhole camera from June until December. The real question is: Where was this photo taken?

Tuesday, April 13, 2010

The malleability of theory

There were some undergrads in my office for office hours the other day, asking questions for the midterm. I'm teaching the engineering course on optics and waves (also known as Things that go Sine) and the students were looking at a problem dealing with quarter-wave plates.

A quarter-wave plate, the question goes, is a kind of optical instrument that turns light with linear polarization into circular polarization. Show, it continues, that it also turns light with circular polarization linear.

After a minute or two of discussing how a quarter-wave plate work ("what, does it act differently on different axes?" giggled one, and everybody gasped when I said yes) we went to the math. I wrote down the equations for light with linear and circular polarization,

\[ E_0 \hat{x} \cos\left(kz-\omega t\right) \to \frac{E_0}{\sqrt{2}}\hat{x}\cos\left(kz-\omega t\right) + \frac{E_0}{\sqrt{2}}\hat{y}\sin\left(kz-\omega t\right) \]

with a little arrow in between them to signify that the quarter-wave plate turns one into the other.


Ever wonder why don't you see a standard rainbow when looking at a thin film such as soap stretched across a membrane ready for bubble making? Well, I encountered this problem when I presented my intro physics section with a quiz question today. Properly stated, the question was...

Sunday, April 11, 2010

On the Nature of Many Penny Systems

The other day I was on my way through ye olde internets to the Quantum Leap episode index at Wikipedia, but stumbled over a quote by Feynman by accident:

"There are 10^11 stars in the galaxy. That used to be a huge number. But it's only a hundred billion. It's less than the national deficit! We used to call them astronomical numbers. Now we should call them economical numbers"

So this got me thinking. What if we decided to pay the national pennies? Seems fair enough, we don't really want to pay it and when you don't want to pay something but you have to you pay it in pennies. Hooray!