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## The Franck-Hertz ice cream truck

Imagine that there’s an ice cream truck parked near a school.  It’s 3pm; class is dismissed and a steady stream of kids approach the truck.  For simplicity, let’s stipulate the following rules:

1. If a kid has enough money, he or she will buy an ice cream cone;

2. A kid will only buy one ice cream cone;

3. A kid will buy the most expensive ice cream cone he or she can.

After the kids buy ice cream (or not) they continue on their way, having whatever leftover money they might have in their pocket.

My question is this: by seeing how much money kids have left, what can you determine about the prices of ice cream cones?  (Astute observers will note that this toy model has many similarities to the Franck-Hertz experiment of 1914, but we’re not there yet.)

Let’s look first at some hypothetical data.  Here we have plotted L (money left over in a kid’s pocket) as a function of M (how much money a kid started with) for a school with 11 kids:

There doesn’t seem to be a pattern.  Maybe we don’t have enough data?  Let’s increase the number of kids to 42:

The sawtooth form of the function is characteristic of problems of this type, and can be understood intuitively.  First of all, notice that the function is linear to begin with, with a slope of exactly one; this means that below a certain threshold value of M, a kid can’t afford any ice cream, so he/she ends up with the same amount of money he/she started with.  Eventually, if M ≥ \$1, a kid can buy a cone; L then plummets because the kid has spent a dollar.

It seems obvious from the graph, then, that ice cream cones are priced at \$1, \$2, and \$3; beyond that we don’t have any data so can’t draw more conclusions.

Now, if we relax the stipulation that kids only buy one cone, then our data is ambiguous.  Maybe the cones are priced \$1/\$2/\$3, or maybe cones are always just \$1, and kids are buying more than one cone if they can.  We can’t distinguish between these cases; that’s why I put the original stipulation there to begin with.

Let’s try a more challenging graph:

This graph is much harder to interpret.  The first peak is much bigger than the others; there is also a very tiny peak on the right-hand side.  It helps if you know where the jumps are: L jumps down to zero whenever M is equal to \$0.67, \$1.02, \$1.34, \$1.69, \$2.01, and \$2.04.  If you want to work out the ice cream cone prices for yourself, feel free.  I’ll wait.

The solution?  A little trial and error will give you two different prices for ice cream cones: A= \$0.67 and B=\$1.02.  Then the jumps occur at A, B, 2A, A+B, 3A, and 2B.  If we were doing physics, we’d make a prediction: we would expect the next jump to occur at \$2.36, which is B+2A.  Observing this would support our hypothesis.  But if the next jump occurred at \$2.22, say, then we’d have to revise our theory and posit a new ice cream C priced at \$2.22.

What does any of this have to do with physics?

I use this example when I introduce the Franck-Hertz experiment to my students.  This experiment was first performed in 1914 (an auspicious year!) and provided support for Bohr’s idea that atoms have specific (quantized) energy levels.  Electrons are accelerated and shot towards mercury atoms (in a vapor).  The electrons may then give energy to the atoms (exciting them) or they may not, bouncing off without loss of kinetic energy.  We look at how much energy the electrons have to start with, and how much they end up with, and thereby deduce the energy levels of the mercury atoms.

How is that possible?  In terms of the analogy, make the following transformations:

kids –> electrons

ice cream truck –> Hg atoms

kids buying ice cream at certain prices –> electrons giving certain amount of energy to Hg atoms

prices of ice cream cones –> energy levels of Hg

M (initial amount of money) –> V (proportional to initial kinetic energy of electron)

L (leftover money) –> I (proportional to final kinetic energy of electron)

If energy levels of an atom are truly quantized, then we would expect a graph of I vs. V to look like our graphs above, with an increasing sawtooth pattern.  Where the drops occur will then tell us the specific energy levels of the Hg atom.  (Incidentally, why do we use V and I?  Well, in the actual experiment the easiest way to measure initial kinetic energy is by measuring the voltage used to accelerate the electrons; the easiest way to measure final kinetic energy is to measure the current of the electrons after they have passed through the Hg vapor.)

How did the experiment go?  Here are the results:

It should be obvious that atomic energy levels have specific “prices”, and that there’s a minimum amount of energy that an electron must have in order to “buy” an atomic transition (exciting the atom at the expense of the electron’s kinetic energy).  It remains for the experimenter to do some elementary unit conversions, to translate I and V into final and initial kinetic energy.

[Note for advanced students and/or physicists: it is interesting to ponder the following questions, which highlight the fact that the ice cream analogy is not perfect: (1) why is the I vs.V graph not linear at low voltage? (2) After “spending” kinetic energy to cause the first excited state, why does the electron’s energy not drop all the way down to zero?]

## Asteroid/meteor as a 1 in 100,000,000 coincidence? Not really…

CNN recently posted a story about how the Feb. 15 asteroid/meteor event was very, very unlikely: a 1 in 100,000,000 coincidence.  I disagreed.  I was all ready to blog about how CNN, yet again, got a non-scientist to write about science…and my indignation was already half out of the bottle.

Then I saw who wrote the article: Meg Urry, a highly respected Yale astrophysicist.

So, I sat on my hands for a second and re-evaluated the article.  It does not contain any errors as far as I can tell.  But I still contend that the article is misleading: saying that the asteroid/meteor event was a 1 in 100,000,000 coincidence is the wrong way to look at it.

I agree that if you multiply 1 in 3,650 days times 1 in 36,500 days you get something close to 1 in 100,000,000.  But all you’ve proven is that for any given random day, there is only a 1 in 100,000,000 chance of such a coincidence occurring.

However, we now live in a post-Nate Silver, post Bayesian controversy world, right?  We’ve known about asteroid DA14 for exactly a year (as of today).  So the right question to ask, before it flew by last week, was: what is the chance that a human-injuring meteor will fly by on the same day?  Well, given that an asteroid will already pass that day, the chance of a once-in-a-decade meteor flying by that same day is just 1 in 3,650 (that is, once in a decade).

I have the utmost respect for Dr. Urry.  I suspect that the hyperbole-filled title of her CNN post was written by a CNN webmaster, not her.  I still agree that the coincidence was unlikely, but given that DA14 was already expected to fly by, the Chelyabinsk meteor hitting on the same day does not sink into the realm of unbelievability.

[Trivia note: Chelyabinsk is the birthplace of Evgeny Sveshnikov, the chess grandmaster for whom the Sveshnikov variation of the Sicilian defense is named.  And I do know that, as much as I like the Sveshnikov defense, I tend to go down in flames like a meteor whenever I play it.]

## The top 10 foods I would not have eaten as a kid

The following is a list of foods I love today, at the age of 44.  I can’t imagine the 10-year-old me even trying any of these foods, much less liking them.  But we grow up; our tastes change, and hardly a day goes by without me craving Marmite.  I mean, come on?  Isn’t that weird?  Please tell me I’m not crazy!

10. Fresh spinach

I didn’t really learn to enjoy salads all that much until I discovered that fresh spinach is infinitely tastier than lettuce.  As a kid, I think I tried spinach, in a can, like Popeye; the problem is, canned spinach is barely edible.  When you eat a fresh spinach salad, maybe with apples and a splash of bacon vinaigrette, you can only wonder: is this really the same plant as found in canned spinach?  Hard to believe.

9. Snails

I’ll be honest; I’ve only had snails (at most) 10 times in my life, and always in a fancy restaurant.  But they’re tasty.  They taste even better when you call them snails, since calling them escargot leaves a bit of a snooty aftertaste.

8.Soft-shelled crab

I first tried this in a Thai curry dish.  I still find the texture a bit strange, but overall I find the taste delectable.

7. Mushrooms

I’ve liked mushrooms for a long time, but can’t recall whether I tried them as a kid or not.  I think most Americans are first exposed to mushrooms as a topping on pizza, but I don’t really like mushrooms that way.  They invariably go straight from a can to the pizza.  Fresh mushrooms are better, and stuffed mushrooms may be the best of all.  As a bonus, there are many, many varieties, and they all taste different.

To my mind, mortadella is the king of sandwich meats.  It’s a bit like deli bologna, but more flavorful: it is marbled with pork fat, and often has pistachios and olives mixed in.  I might have liked this as a kid, but I never tried it.  I had never even heard of it.  I’d have to drive over a hour from where I live to find this today.

5. Runny eggs

Yolk freaks kids out.  But not me.  Over-easy is the way to go.  And when the Hollandaise in an Eggs Benedict runs together with the yolks…

4. Coffee

I guess I haven’t grown up entirely.  I don’t drink coffee, unless it’s cold with plenty of milk and sugar.  Sort of like a melted coffee ice cream.  My favorite way to have coffee is the way they serve it in the Vietnamese restaurants: over ice, with sweetened condensed milk.

3. Pomegranate juice

So bitter.  Yet so good somehow.  I don’t really know why I like it.

2. Stilton

The king of cheeses.  Radically strong flavor; almost hallucinogenic.  Not just for Wallace and Gromit anymore.

1. Marmite

What can I say?  Marmite, objectively, doesn’t taste all that great: it’s almost pure umami, like chewing on a bouillon cube.  And yet it is alluring for some unfathomable reason.  On bread, with butter, it is divine; mixed with honey and corn syrup, it is the ideal pizza sauce.  I can get Marmite (or its complex conjugate, Vegemite) where I live, but I don’t eat it all that often.  I may go 6 months without having any.  Then, on a random day…the 19th of February, say…I will start to crave it, and even start to blog about it.

## What color is the bear? A new class of solutions

In the 1985 film Young Sherlock Holmes, Holmes tells Watson the following riddle:  “You’re sitting in a room with an all-southern view.  Suddenly, a bear walks by the window. What colour is the bear?”  (He says “colour” not “color” because he’s from the UK.)  As I recall it takes Watson most of the movie to give the obvious answer—white—because the bear must be a polar bear.  All southern view…North Pole…polar bear…you get the idea.

Over the years I’ve hear other versions of this riddle.  The most common seems to go something like this: “A hunter travels a mile south, a mile east, shoots a bear, then travels a mile north to her starting point.  What color was the bear?”  People assume that this riddle is isomorphic to the previous one, because (supposedly) there is only one place on Earth you can travel the same distance south, then east, then north, and return to the beginning.  But this is wrong.  There are an infinite number of places on Earth you can travel in a loop that is 1 mile south, then 1 mile east, then 1 mile north.

Of course, starting at the North Pole is one solution.  But there are also many, many more solutions close to the South Pole.  Imagine, for example, a latitude roughly 1/(2π) miles north of the South Pole; in such a case the parallel along that latitude is the circumference of a circle, given by

C = 2πr = 2π [1/(2π)] = 1 mile.

(I said roughly because we’re on the surface of a sphere, so the circumference of a parallel is not exactlyr—because r is an arc length, not a straight line—but 1 mile is so much smaller than the radius of the Earth that we can assume a locally flat geometry.)  If a hunter started his journey 1 mile north of this latitude, then of course he could go 1 mile south, then 1 mile east (circumnavigating the South Pole!) and then 1 mile north, and return to his starting point; I presume there would be no bears.

There are actually infinitely many solutions that work.  In each case, after going a mile south, the hunter would have to reach a latitude in which the circumference C was an integer fraction of 1 mile.  That is, it must be true that

C = 1/n miles,

where n is an integer.  This means that the parallel would have to be

r = C/(2π) = 1/(2πn)

miles from the South Pole (again, this is a flat approximation to a spherical problem).  So the most general South Pole solution is that the hunter should begin 1 + 1/(2πn) miles north of the South Pole.  For example, take n = 5.  If a hunter starts 1.0318 miles from the South Pole, she can go south 1 mile, east 1 mile (circumnavigating the South Pole exactly 5 times) then north 1 mile, and relax in her hot tub.  No bears will be harmed, unless some evil genius has released them in Antarctica.

The original version of the riddle, as given by the young Sherlock Holmes, is superior, since it has only one solution.  We can conclude that Sherlock Holmes was good at math.

I mean, maths.

## The phantom crane fly

One day in the summer of 2011, while mowing the lawn, I saw a strange creature flying through the air.

Actually, “flying” is too generous a term.  The creature was lilting through the air.  Lurching.  It appeared to have ten legs, and was about the size of a silver dollar.  I was puzzled, to say the least, but that lawn wasn’t going to mow itself so I went back to work.

The mystery bug…

I live in a rural area in the mountains of North Carolina, only 30 minutes away from the entrance to the Great Smoky Mountains National Park.  There’s a lot of wildlife here: I’ve seen bear, elk, deer, raccoons, opossums, groundhogs, voles, and squirrels; our bird feeders are always full of cardinals, chickadees, towhees, finches, and titmice; and I once came home to a 4-foot black snake inside my house.  As for arthropods, I’m very familiar with flies, no-see-ums (family Ceratopogonidae), moths, wasps, honeybees, crickets, ants, beetles, and spiders of all kinds.  I’ve had close encounters with black widow spiders no less than 3 times in my life.

But this thing?  With 10 legs?  Lurching through the air like a drunken hang glider?  Incomprehensible.

Over the course of that summer, I saw such creatures on numerous occasions.  I gradually came to realize that they were insects, since subsequent sightings showed 6 legs, not 10.  My working hypothesis was that what I saw that first day was a mating pair: two of these things stuck together.  But I still had no idea what the confounded creatures were.

In appearance, the insects were bizarre to say the least.  They were striped, like zebras, and their legs appeared  to have at least 3 joints each, so that the legs took on a zigzag character.  They didn’t appear to use their wings, which I guessed were vestigial; rather, picture a 6-legged starfish up on one end, clawing and grasping its way forward.  As a physicist, it looked very much like the creatures were literally swimming through the air.  And so I resolved, with the help of the internet, to positively identify them.

Rutherford said that “all science is either physics or stamp collecting.”  A lot of people take this to be a disparaging comment about sciences other than physics, but I don’t.  I kind of like stamp collecting.  I like being meticulous, and being detailed.  That’s why I like pastimes such as putting together 1000-piece puzzles.

But my search for the identity of the “mystery bug” took stamp collecting to a whole new level.  It literally took me a month of sleuthing to identify the things.  I tried the obvious first: I googled things like “strange zebra striped bug” and “bug that swims in the air” but had no luck.  I posted a question on an entomology bulletin board.  I looked at websites dedicated to “insects of the Appalachians.”

Finally, I had a breakthrough: I saw one of the bugs hitting up against a window in our house.  For the first time, I could see the creature close up and for more than just a second or two.  I verified that the creature did have six legs; I verified that it did have wings, although they seemed useless.  I realized that my mystery bug was a crane fly.  Here’s a more typical, run-of-the-mill crane fly:

A typical crane fly

Regular crane flies are common where I live; kids often mistake them for gigantic mosquitoes (which they are not).

Once I realized that the mystery bug was a type of crane fly, my task was eased enormously.  And eventually I found this assortment of photographs.  Eureka!  I had done it!  They were phantom crane flies, of the family Ptychopteridae.  Specifically, they were the species Bittacomorpha clavipeswhich, according to this Wikipedia article, are “known for the odd habit of spreading out [their] legs while flying, using expanded, trachea-rich tarsi to waft along on air currents.”

It turns out that the phantom crane fly is one of the very, very few creatures on Earth that fly without using their wings.  They are literally swimming, somehow taking advantage of a high Reynolds number (let’s say, 265?) to sludge through the atmosphere without those wings that evolution gave them.  Consequently they look more like seed pods drifting on the wind than they do insects.

What is my point?  I don’t have one.  I just think these bugs are cool, and you should try to find them if you ever visit the Eastern United States.  They hang out in marshy areas in late summer.  Oh, and if you’re a physicist or an entomologist, think about studying these little guys.  The field’s wide open as far as I can tell.  Somebody needs to video the flight of the phantom crane fly, so get on it!  [Note added later: I did find this video which shows the weird flight, are there more?]