Wednesday, February 29, 2012

Measuring the Height of a Building with a Barometer

Here's a video made by my winter term students Sophia Chen, Kathryn Hasz, David Morris, and Jenny Schloss.  The music is by the lovely and supremely talented Anna Lackaff.


Tuesday, February 28, 2012

Lessons from Stick Figures: Quantum Mechanics

Sometimes I make cheesy science cartoons.  Here's one featuring the vocal talents of Matthew Grace and Stacey Smith.


video

Learning Chinese

I love my dad.  Sometimes I forget how cool he can be.  Like all parents, there are times he embarrasses his kids, but every once in awhile he does something amazing.

About 12 years ago, my dad came up to visit me in college.  Like many college age students, I didn't always enjoy introducing my parents to friends because of the fear they might reveal some embarrassing childhood story or shame me in some other way.  For this reason, I told my dad to meet me downstairs in the dorm lobby, thinking this would minimize the potential for an embarrassing moment.  Imagine my horror, when, after arriving and meeting the cute Asian student sitting behind the front desk, my dad started speaking what appeared to be mock-Chinese gibberish at her.  The feeling was something akin to what I imagine ESPN's editors experienced after reading the racist Jeremy Lin headline earlier this month.  Much to my surprise, the student in question was not at all offended and instead returned the remark with a laugh and a response in Mandarin.  My dad, perhaps seeing the look of shock and unease on my face, smiled, put his hand on my shoulder, and continued conversing with her.  It was only then that I remembered my dad used to be a Chinese translator in the United States Air Force.

To this day, I don't know exactly what my dad said, but he seemed to make a good impression.  Now every time I get a fortune cookie at a restaurant, I look at the Chinese word on the back and wonder if I'll ever be able to speak with him in Chinese.  How much would I have to spend on Chinese food to learn every word?

Like any language, specifying the exact number of words is tricky.  Words are constantly being invented (and sometimes removed) from usage.  However, judging by the size of the Chinese-to-English dictionaries I've seen in book stores, I would assume the number of Chinese words is roughly equivalent to the number of English words, which would put the exact number somewhere in the vicinity of 100,000 words.  Assuming Chinese words are uniformly represented in fortune cookies, I would need to buy at least 100,000 meals.  At about $10 per meal, this would mean spending at least $1 million on Chinese food, and that's assuming I never got the same word twice (a very unlikely proposition.)  If I ate Chinese food every day for breakfast, lunch, and dinner, this would still take 91 years.  Needless to say, it may take awhile before I can speak to my dad in any language but English.

Wednesday, February 1, 2012

Skewered Contest!!!

Yay!  We have a new contest.  The rules are a little different from previous DON contests, but you can still win a free copy of How Many Licks?  Here's how it works. I’m posting a numbers question below. To enter, come up with an answer and send it to “aaron at aaronsantos period com.” If I think your answer is the best, I'll mail you a free signed copy of How Many Licks? Submit your entry on or before March 15, 2012 for consideration. Don't worry…I won't spam you or share your email with any third parties. Here’s the question:

I was reading an Asimov book the other day and came across something called Skewes' number.  At one time, Skewes' number was the largest number ever to appear in a mathematical proof.1  It would be impossible to write Skewes' number out in the conventional way, but using exponents we can write it as

10101034.

This is an exceedingly large number.  It is vastly bigger than the number of grains of sand in the Earth.  In fact, it's vastly bigger than the number of grains of sand in the solar system.  In fact, it's vastly bigger than the number of atoms in the solar system.  In fact, all of these physical quantities are extreme underestimates of Skewes' number.  What physical quantity comes closest to Skewes' number?

 Note: It must be something physical like a molecule, atom, or grain of sand (i.e. not something abstract like a number appearing in a mathematical proof.)

[1] It has since been replaced by other large numbers.  For more info, see Graham's number and Moser's number.

Alton Brown Wants to Burn Your House Down


I'm a big fan of Food Network's resident gastrophysicist, Alton Brown.  Mr. Brown excels at explaining food science to lay people in fun and entertaining ways.  With this in mind, I'm kind of sad to call him out on an egregious error in one of his recipes.  The recipe, "Plain Brown Popper", gives the following instructions:

"Toss the popcorn with the olive oil, salt, and jalapeno seasoning mix in the paper bag. Fold the top of the bag over and staple the bag twice to close. Place the bag in the microwave and microwave on high for 2 minutes to 3 minutes..."

Come again?  Staple?!  As in a metal staple?  In a microwave?! Perhaps staples are too small to do any damage.  How bad is it to put a metal staple in a microwave?

Microwaves work by generating an oscillating electric field that vibrates the electrically polarized water molecules inside.  Humans observe this enhanced molecular movement as an increase in the temperature of the food.  However, water molecules aren't the only things that are getting pushed by the electric field.  Any mobile electric charges will vibrate as well, including the electrons in any conductive metal that found its way into the microwave.  Charges in the metal can move around in  a way that will make the electric field even bigger.  If the field is large enough (~3×106 V/m), sparking will occur, which can start a fire.

If there's any electric field inside a metal, it will push the electrons from one side of the metal to the other.  In fact, the field will keep pushing electrons from one side to the other until the electric field created by these electrons exactly cancels out the original field.  This happens very quickly because electrons move very fast.  Electrons cancel out the electric field in metals so quickly that we can often assume the electric field inside a conductor is always zero.  How many electrons would it take to cancel out the 2×103 V/m field created by many microwaves?  Well, if you take a plate and put charge on the opposite faces, you'll find the charge on a surface is given by

total charge = (permittivity of space) · (electric field) · (area),

where the "permittivity of space" is a constant equal to 8.854×10-12 F/m.  Plugging in 0.5 mm by 1 cm for the area, you get a total charge

total charge = ( 8.854×10-12 F/m) · (2×103 V/m) · (0.5 mm × 1 cm)
= 1×10-14 C.

That's a small charge, and it doesn't create much of an electric field when spread out over broad side of a staple.  However, when the same charge aggregates on the sharp end of the staple, the electric fields can get quite large because the area is so small.  If sharpened, the points might have and area of 0.05 mm by 0.05 mm,

electric field = (total charge) / [ (permittivity of space) · (area) ],
= (1×10-14 C) / [ ( 8.854×10-12 F/m) · (0.05 mm × 0.05 mm) ],
= 4×106 V/m

That's more than large enough to generate a spark. The electric field just outside the staple will depend greatly on how sharp the surface is, with sharper objects being more likely to spark.  With the possible exception of pets and small children, a staple might be the worst of all possible things to put in a microwave.1,2,3


[1] Being a scientist, I had to test whether or not putting staples in a microwave was actually harmful.  Mr. Brown, you owe me one microwave.
[2] Special thanks to Melinda Keller, who directed me to a very nice article in The Physics Teacher titled "Microwave Mischief and Madness", by Heather Hosack, Nathan Marler, and Dan MacIsaac.
[3] Immediately after posting this, I was struck by the realization that the quasi-static approximation I've made might not be legit.  I think I'm right on this (my microwave was incinerated after all), but feel free to call me out on it if I'm making a mistake.