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Archive for September, 2013

Move over, McDonalds!  There’s a new worst slogan in the world.

Budweiser (a “beer” company) has a new ad campaign about sports superstitions.  In a nutshell: sports superstitions (like sitting in your “lucky” chair) are funny, charming, and gosh darn it, might even be real!  Budweiser’s tagline: “It’s only weird if it doesn’t work”.

I disagree.  It’s weird, period.

What’s more, it’s ignorant, embarrassing, and frankly makes me a little pessimistic about humanity.  Do you really think that wearing that unwashed jersey will help your team win?  If yes, then please, please unfriend me on Facebook.  I don’t want to have anything to do with you.

aztecs40

This is only weird if it doesn’t work!

Superstitions have always been a force for evil in the world.  Yes, evil.  Superstitions caused Aztecs to pull the beating hearts out of innocent people.  Superstitions caused intelligent women to be burned at the stake as witches.  Superstitions caused Okonkwo to kill his son Ikemefuna to appease the village elders.  Superstitions put Galileo under house arrest, and drove Alan Turing to commit suicide, and prevent a sizeable number of otherwise educated adults from believing in the plain fact of man-made global warming.

Superstitions even keep a huge number of South Koreans from having fans in their bedrooms.

[Cue double-take]

I’m not making this up.  For some strange reason, many South Koreans think that a simple oscillating fan can kill you in your sleep.  This, despite the fact that fan death has never happened in human history.  And despite the fact that the rest of the entire world uses fans in their bedrooms to no ill effect.

Fan+Death

But wait! you might say, in Korean I presume.  People have been found dead with fans running nearby!  The fans must have killed them!  Case closed!

I’ll leave it to the reader to punch holes in that kind of “logic”.

You may have heard of the famous experiment in which B. F. Skinner discovered “superstition” in pigeons:

“Skinner placed a series of hungry pigeons in a cage attached to an automatic mechanism that delivered food to the pigeon ‘at regular intervals with no reference whatsoever to the bird’s behavior.’ He discovered that the pigeons associated the delivery of the food with whatever chance actions they had been performing as it was delivered, and that they subsequently continued to perform these same actions.” [http://en.wikipedia.org/wiki/B._F._Skinner#Superstitious_Pigeons]

pigeon

A typical Budweiser drinker.

Your team wins while you’re wearing that lucky shirt?  The shirt must have done it!  Of course, you should be ashamed of yourself.  You’re not any smarter than a pigeon.

Carl Sagan wrote a book called “The Demon Haunted World: Science as a Candle in the Dark”.  The idea is that science, and only science, illuminates; there is no other way to learn anything about the world.  The next time you’re around a “person” who exhibits superstitious nonsense around you, cough into your hand and say “Pigeon!”  Don’t worry; they won’t know what you’re talking about.  Like Giordano Bruno’s torturers, or the chicken-eater Wade Boggs, or the people who stoned Tessie Hutchinson, they have no idea what science is, or logic, or common sense.  They won’t have heard of B. F. Skinner or Carl Sagan or Alan Turing or Giordano Bruno.

They will, however, be familiar with Budweiser “beer”.

And they’ll be enjoying it, pathetically, in the dark.

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Formula snobs

I am a formula snob.

We all know about grammar snobs: the ones who complain bitterly about people using who instead of whom.  Many people know how to use whom correctly; only grammar snobs care about it.  I gave up the whom fight long ago (let’s just let whom die) but I am a grammar snob when it comes to certain words.  For example, ‘til is not a word, as I have discussed before.

However, I am almost always a formula snob.

Consider this formula from the text I’m currently using in freshman physics:

x = v0 t + ½ a t2.

Beetlejuice-square

Robin Thicke, c. 2012

To me, looking at this equation is like watching Miley Cyrus twerk with Beetlejuice.  I would much, much rather the equation looked like this:

Δx = v0 Δt + ½ a Δt2.

The difference between these two formulas is profound.  To understand the difference, we need to talk about positions, clock readings, and intervals.

A position is just a number associated with some “distance” reference point.  We use the variable x to denote positions.  For example, I can place a meter stick in front of me, and an ant crawling in front of the meter stick can be at the position x=5 cm, x=17 cm, and so on.

A clock reading is just a number associated with some “time” reference point.  We use the variable t to denote clock readings.  For example, I can start my stopwatch, and events can happen at clock readings t=0 s, t=15 s, and so on.

Here’s the thing: physics doesn’t care about positions and clock readings.  Positions and clock readings are, basically, arbitrary.  A football run from the 10 yard line to the 15 yard line is a 5 yard run; going from the 25 to the 30 is also a 5 yard run.  The physics is the same…I’ve just shifted the coordinate axes.  If I watch a movie from 8pm to 10pm (say, a Matt Damon movie) then I’ve used up 2 hours; the same thing goes for a movie from 9:30pm to 11:30pm.  Because a position x and a clock reading t ultimately depend on a choice for coordinate axes, the actual values of x and t are of little (physical) importance.

Suppose someone asks me how far I can throw a football.  My reply is “I threw a football and it landed on the 40 yard line!”  That’s obviously not very helpful.  A single x value is about as useful as Kim Kardashian at a barn raising.

nebraska_amish_barn_raising

Can you pass that hammer, Kim?

Or suppose someone asks, “How long was that movie?” and my response is “it started at 8pm.”  Again, this doesn’t say much.  Physics, like honey badger, doesn’t care about clock readings.

Most physical problems require two positions, or two clock readings, to say anything useful about the world.  This is where the concept of interval comes in.  Let’s suppose we have a variable Ω.  This variable can stand for anything: space, time, energy, momentum, or the ratio of the number of bad Keanu Reeves movies to the number of good (in this last case, Ω is precisely 18.)  We define an interval this way:

ΔΩ = Ωf – Ωi

So defined, ΔΩ represents the change in quantity Ω.  It is the difference between two numbers.  So Δx = xf – xi is the displacement (how far an object has traveled) and Δt = tf – ti  is the duration (how long something takes to happen).

honey badger

Honey badger doesn’t care.

When evaluating how good a football rush was, you need to know where the player started and where he stopped.  You need two positions.  You need Δx.  Similarly, to evaluate how long a movie is, you need the starting and the stopping times.  You need two clock readings.  You need Δt.

I’ll say it again: most kinematics problems are concerned with Δx and Δt, not x and t.  So it’s natural for a physicist to prefer formulas in terms of intervals (Δx = v0 Δt + ½ a Δt2) instead of positions/clock readings (x = v0 t + ½ a t2).

But, you may ask, is the latter formula wrong?

Technically, no.  But the author of the textbook has made a choice of coordinate systems without telling the reader.  To see this, consider my (preferred) formula again:

Δx = v0 Δt + ½ a Δt2.

The formula says, in English, that if you want to calculate how far something travels Δx, you need to know the object’s initial speed v0, its acceleration a, and the duration of its travel Δt.

From the definition of an interval, this can be rewritten as

xf  – xi = v0 (t– ti) + ½ a (t– ti) 2.

This formula explicitly shows that two positions and two clock readings are required.

At this point, you can simplify the formula if you make two arbitrary choices: let xi = 0, and let ti = 0.  Then, of course, you get the (horrid) expression

x = v0 t + ½ a t2.

I find this horrid because (1) it hides the fact that a particular choice of coordinate system was made; (2) it over-emphasizes the importance of positions/clock readings and undervalues intervals, and (3) it ignores common sense.  Not every run in football starts at the end-zone (i.e. x = 0).  Not every movie starts at noon (i.e. t = 0).  The world is messier than that, and we should strive to have formulas that are as general as possible.  My formula is always true (as long as a is constant).  The horrid formula is only true some of the time.  That is enough of a reason, in my mind, to be a formula snob.

critique-film-the-informant-l-infiltre-matt-damon-steven-soderbergh-texte

A formula snob?

Bonus exercise: show that the product

ΩKeanu Reeves  x  ΩMatt Damon  ≈  3.0

has stayed roughly constant for the past 15 years.

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Prescribed_burn_in_a_Pinus_nigra_stand_in_Portugal

Run for your lives!

A few days ago I heard a story on NPR about wildfires in Yosemite.  It turns out that something like 360 square miles of forest have burned.  Being a math geek, I immediately took the (approximate) square root of 360 in my head:

360 ≈ 19 x 19

I did this without really even thinking about it; I did it in order to be able to visualize the size of the Yosemite blaze.  I now had a picture in my head of a square, 19 miles by 19 miles.  A burning square.  That’s how big the conflagration was.  And the mental math was important because I have no intuition at all about square units.

[Disclaimer for my readers not in the USA: I use the S.I. units (m/kg/s) in my physics research, but in American culture units like miles, inches, gallons, etc. are still endemic.  Sorry about that.]

Quick: how many square feet is a baseball diamond?  If you’re like me, absolutely nothing comes to mind.

I do know that a baseball diamond is 90 ft. x 90 ft. square.  So that’s the answer: 8100 sq. ft.  (752.5 m2)  The problem is that, somehow, psychologically, 90 ft. x 90 ft. seems much smaller than 8100 sq. ft., even though they are the same.

The county I live in, Jackson County, NC, is 494 sq. mi. (1,279 km2).  Somehow, this seems big to me.  But in order to better visualize this area, take a square root: the county is like a 22 mile x 22 mile square (36 km x 36 km).  In those terms, the county seems puny (although it is still bigger than Andorra).  The area of Jackson county is less than 1% the total area of the state of North Carolina.

What about the Yosemite fire?  360/494 = 73%.  So that fire is about three-fourths the size of the puny county that I live in.  A big fire, sure, but not apocalyptic.

The problem that all of this illustrates is one of scaling.  Most of my students know that 1 m = 100 cm.  However, very few know (initially) that 1 m2 ≠ 100 cm2.  Instead, 1 m2 = 10,000 cm2.  That’s because a square meter is a 100 cm x 100 cm square.

This fact leads people’s intuitions wildly astray.  Suppose I double the length and width of an American football field.  The area goes up by a factor of 4.  What was approximately 1 acre has become 4 acres.  Suppose I switch from a 10-inch pizza, which feeds 2, to a 20-inch pizza.  That pizza feeds 8.

It gets even stranger if you imagine the switch from length to volume.  Michelangelo’s David is almost 17 ft. tall.  Assume David was 5’8’’ (68 inches).  Then the statue represents a scaling factor of x3 in terms of length (3 x 68 = 204 in. = 17 ft.)  Imagine a real-life David, 17 ft. tall.  How much would he weigh?  If the life-size David is 160 pounds, the 17 ft. David would be 160 x 33 = 160 x 27 = 4,320 pounds.  To most people, this seems very strange.

David

He weighs 4320 pounds. If he weren’t made of stone, that is.

But back to my original idea: I had mentioned that I had no intuition about square units.  I don’t think many people do.  What intuition I do have is based on experience, and comparing unknowns to knowns.  500 sq. miles is about the size of the county I live in.  An acre is about a football field.  1000 sq. ft. is about the area of a small house.  500 sq. in. is about the area of a modest flat screen TV.  100 fm2 (a barn) is about the cross-sectional area of a Uranium nucleus.  A hectare is about 2.5 football fields stuck together.  And so on.  I’m sure you have your own internal mnemonics to help you gauge area, or volume.

If not, just remember: you can also do the square root in your head.  So if that guy on NPR says there’s a fire that’s 100,000 sq. miles in area, you can visualize

100,000 ≈ 316 x 316

and since this is very similar to the size of Colorado (380 miles x 280 miles) you can start kissing your love ones and planning for the apocalypse.

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