The Bill O'Reilly Problem

 “Tide goes in, tide goes out. Never a miscommunication. You can’t explain that. You can’t explain why the tide goes in.”

—Bill O'Reilly

 
Political pundit Bill O'Reilly elevated himself to meme status this past month because of his assertion that you can't explain the rise and fall of the tides.  Most people probably don't know the exact mechanism, but even in our Are You Smarter Than a 5th Grader? society, people should at least recall that there was some explanation learned in middle school science.  After all, it was over 300 years ago that Newton figured out the tides were caused by the Moon's gravitational field.  How does the Moon produce the tides and can we use physics to estimate just how big they should be?

We live in a gravitational field.  At the surface of the Earth, the field causes all masses to accelerate to the ground at a rate of 9.8 m/s².  But the Earth is not the only object that has a gravitational field: all masses are attracted to each other by gravity.  Right now, the mass of your body is exerting a very slight pull on the pyramids of Egypt due to gravity.  While your gravitational field is very weak, massive things like the Moon can create very large gravitational fields.  We can show the gravitational fields due to the Moon and the Earth schematically like this:

Gravitational field lines around the Earth (blue) and Moon (orange).

At each point in space, an arrow tells you what direction an object placed there would be pulled.  As you can see, the more massive (i.e attractive) Earth has more arrows pointing towards it.  Only arrows very close to the Moon point towards it.  The reason is that gravitational fields get weaker as you get further away, so if you're very close to the Moon you'll be pulled more towards it than the Earth.  This decrease in strength as you get further away is the reason there are two tides per day.  The Moon pulls very hard on the part of the Earth that is facing it.  The water there will shift a little closer to the Moon creating high tide.  Why is there a second tide 12 hours later?  In addition to pulling on the water, the Moon also pulls on the center of the Earth.  It pulls a little harder on the center of the Earth than it does on the water at the opposite end of the Earth because the center is closer.  This effect elongates the surface of the water giving it an oval shape as seen below:

The gravitational pull of the Moon effectively stretches the Earth.  (Not to scale.)

This is a nice qualitative description, but is there a way we can compute the height of the tides?  Yes!  It turns out that fields are not the only useful way to describe gravity.  There's also something called "gravitational potential".  Masses tend to move from areas of higher potential to areas of lower potential.  The potential V at some point due to a mass M is given by

V(r) = -G M / r,

where r is the distance between the point and the center of the mass and G = 6.67×10^-11 N·m²/kg² is the fundamental gravitational constant of the universe. If water in the oceans can flow to a lower potential, it will.  For this reason, we know that the gravitational potential should be the same everywhere on the surface of the ocean, because if it wasn't water would flow until it was.  We can write the potential due to the Earth and the Moon as

V = -G ( M_Earth/ r_Earth + M_Moon/ r_Moon),

where M_Earth/Moon and r_Earth/Moon are the mass and distance from the Earth/Moon, respectively.  We can plot lines of equal potential much like you can plot lines of equal height on a topographic map: 

Contour plot showing lines of equal gravitational potential.  The white circle on the left represents the Earth while the dark speck in the center represents the Moon. 

Notice the the potential lines are slightly warped by the Moon giving the characteristic oval shape one observes in the tides.  We can look up the mass of the Earth (M_Earth=6.0×10²⁴ kg) and the mass of the Moon (M_Moon=7.4×10²² kg).  When the Moon is directly overhead, r_Moon is roughly 3.7×10⁸ m and r_Earth=6.4×10⁶ m.  From this, we can calculate the gravitational potential at the surface of ocean to be  V=-6.25445×10⁷J/kg.  Halfway around the Earth, the height of the water will be lower but the potential must be the same.  If you calculate this height using the values listed and a little geometry, you find the difference between high tide and low tide is about 22 m.  In the Bay of Fundy, which has the largest tides in the world, the water rises and drops about 16 m, so our estimate is very good considering we didn't consider land masses or other complicating factors.

The Fruit Hunter

My in-laws got me a copy of Adam Leith Gollner's The Fruit Hunters for Christmas.  I'm just a few chapters in, but there's already a few good knowledge nuggets,

"The sweetness issue actually went to the United States supreme court in 1893. They ruled that tomatoes are vegetables because they aren't sweet," 

and an estimation,

"There are an estimated 240,000 to 500,000 different plant species that bear fruit.  Perhaps 70,000 to 80,000 of these species are edible; most of our food comes from only 20 crops."

 How long would it take to try every fruit?

Most people eat three meals per day or 1095 per year.  If you ate one new species of fruit at every meal, it would take about 70 years to try every edible fruit.  Even if I start now and live to be 100, there is literally no way I will every try every fruit.

Beginning a New Semester...

It's been a busy beginning to the new semester.  Fortunately, I've had a couple very nice people send me interesting estimations links.  Enjoy!

Courtesy of Joey:

"If we were to take all that information and store it in books, we could cover the entire area of the US or China in 13 layers of books..."

 
Courtesy of Sean "The Cool Physics Guy":

The amount of snow in Boston is equivalent to about 0.49 Shaqs or 0.60 Nate Robinsons.

Louis CK Is Dead Wrong

"Out of all the people that ever were, almost all of them are dead."

—Louis CK

Louis CK (very funny guy) does a nice stand-up routine, but I couldn't help noticing that one line in his opening.  As I calculated in Skeptically Speaking...(part II), there's an unexpectedly large percentage of people that are still among the living:

"Surprisingly, I computed about 30% of people who have ever lived are still living. I was a little skeptical of this, so I decided to look it up and found that other people have done this problem as well and they estimated 5.8%. Since these numbers are easily within an order of magnitude of each other, they're likely close to the actual result."

No matter which result you take, it's clear that not "almost all" of us are dead.  This counter-intuitive result stems from the fact that population growth can happen exponentially.  Put another way, there simply weren't that many people back in the day, but they had a lot of sex so there are a lot more of us now.  Admittedly, it's probably a bad sign that I'm more irked by this mathematical inconsistency than I am by the whole having sex with a dead kid thing.

But Was It...MURDER???? [1]

"Meanwhile, Dr. Johnson’s cursory examination revealed the body was not quite cold; he concluded that death had occurred three to four hours earlier."

—E. J. Wagner's "A Murder in Salem" from the Nov. 2010 issue of Smithsonian

While stuck in Philadelphia International Airport after missing a connecting flight, I came across E. J. Wagner's "A Murder in Salem" in the Smithsonian magazine.  I was struck by the simple physical analysis implicit in Dr. Johnson's examination of the murdered man's body.  It seemed Sherlock Holmes-ian.  How much would a dead body cool in four hours?

In principle, a dead body loses most of its heat through the skin.²  There is a well known physics equation describing the amount of heat flowing through a material from a hot temperature to a cold temperature.  In symbols, this equation is written as

dQ / dt = (k · A / d) · ( T_hot -T_cold )

Here, dQ is the heat energy that passes through the surface in some small amount of time dt.  The variables k, A, d, T_hot, and T_cold represent the thermal conductivity, the surface area of the material, the thickness of the material, and the temperatures of the hot side and cold side, respectively.  Immediately before death, a healthy human body has a temperature of 37 °C (~98.6 °F).  The average April temperature in Salem, MA is 8.7 °C (~47.6 °F), but room temperature is 25 °C (~72 °F).³   I'll assume the temperature difference between a healthy body and air in the room is about 12 °C.  A large percentage of the energy our bodies use gets converted to heat.  We consume about 2000 Calories per day or about 100 W, so we can estimate the heat energy as

dQ / dt = 100 W.

Dividing dQ/dt by the temperature difference, we can estimate the ratio of k·A/d

k · A / d = ( dQ / dt ) / ( T_hot -T_cold )
= (100 W) / (12 °C) 
= 8.3 W/°C

The thermal conductivity equation is helpful, but we still need to know how much heat energy is stored in the dead man's body.  An object's capacity for storing heat is called the "heat capacity".  This heat capacity can be described by the equation

dQ = m · C_p · dT

where dQ is the heat energy lost when temperature decreases, m is the mass of the body that stores the heat, C_p is the heat capacity, and dT is the change in temperature.  The heat capacity of a human body is about 3470 J/kg·°C and I'll assume the deceased man weighed 60 kg.  Now you might think we could just solve these equations to obtain the time it takes for all the heat to flow out of the body, but there's one problem.  Since the body's cooling, the temperature difference between the body and the air is always changing.  To deal with rates of change, we generally use calculus.  To do this, we combine the two equations and integrate to get

T_final =  T_air + (T_{healthy body} - T_air) exp(- C t),

where,

C =k · A / [ m · C_p · d ]

= ( 8.3 W/°C)  / [ (60 kg) · (3470 J/kg·°C) ]

= 0.00004 Hz.

Plugging in, we can plot the body temperature as a function of time:

We get a final body temperature of 32°C.  This is still above room temperature but not yet cool, consistent with Dr. Johnson's observations.

[1] Given that Captain Smith had his head bashed in by a blunt custom-made club, I suspect the answer is yes.
[2] According to at least one reference, skin has a thermal conductivity of about 0.3 W/m·°C.  I tried using this number directly and estimating human skin's area and thickness, but I kept getting screwy answers.  Presumably, other tissues (fat, muscle, etc.) and clothing also effect the thermal conductivity which might explain this.  As you'll see, there's a nice estimation trick you can do using the fact that we consume about 2000 Calories per day to get around this problem.  
[3] They didn't have the best heating systems back then, so the air may have been colder.  Fortunately, the qualitative results don't depend on the exact number.

Punxsutawney Phil Stats (Courtesy of My Sister Laurie)

Here's a good Facebook discussion by my sister and her friend on the statistical accuracy of Puxatony Phil:

Sister: Just spent the last few minutes calculating Puxatony Phil's statistical accuracy. Of 114 years of forecasts, he has only been 39% accurate. Unfortunately, this means his shadow is not at chance, but instead is actually statistically *inaccurate* at p < 0.02 two-tailed. Based on his prognosis today, this makes me sad.

Sister's Friend: I'm not sure your statistical approach is valid, Laurie. I don't think long and short winters are equally likely, and Phil seems to have a bias towards seeing his shadow. We need to do a signal detection analysis.

Sister: Here are the actual stats in case anyone would like to run their own analyses: Sees Shadow- Phil was right 37 of 99 times. No Shadow- Phil was right 7 of 15 times

Sister's Friend: So early springs are more likely than long winters (69/114), and Phil tends to see his shadow more often than not (99/114). Since a shadow is supposed to predict a long winter, this accounts for the negative relationship. But that doesn't make Phil a useful anti-guide: d' = -0.4, which is pretty near chance.

Sister: Great work, [Friend]. Also nice to see that early springs are slightly though not statistically more likely than late winters.

For reference: http://www.lifeslittlemysteries.com/punxsutawney-phil-weather-prediction-accuracy-1292/

The Curse of "Aaron"

"I feel your pain, Aaron Rodgers."

There are many perks associated with being named "Aaron" (e.g. you're almost always first in line in kindergarten), but there's two major drawbacks that come with the name:

• people constantly ask you how your name's spelled even though it's a very common name because people are stupid and can't quite figure out that a word can begin with two A's.
• pocket dialing.

Being the first name in everyone's phone means being the first person pocket dialed.  But pocket dialing is not nearly as bad as its evil bastard cousin: pocket texting.  It wouldn't be so bad if it was only the occasional blank text, but having your cell phone company charge you 10¢ for every button pushed by an errant buttock starts to add up.  How much money do Aarons spend each year for butt dialed texts?

I like to consider myself a fairly average Aaron.  I get about 1 errant text per month, or about 12 errant texts per year.  Some Aarons have unlimited texting and others don't have cell phones, so I'll assume only 10% of Aarons pay for texts.  Judging by my Facebook friend list, about 1 in 200 or roughly 1.5 million Americans are named "Aaron".  This means the total money spent by Aarons for pocket texting is

(10%) · ( 1.5×10⁶ Aarons ) · (12 texts per year per Aaron) · (10¢ per text)

= $180,000.

That's $180,000—four times what the average high school teacher makes—given to cell phone companies by Aarons because of pocket dialing.

Skee Ball Game Theory

As a kid, I always wanted to win the stereo at the Dream Machine arcade in the North Dartmouth Mall.  The stereo was worth 10,000 tickets.¹  How long would it take to win that many tickets and how much would it cost?

Anyone who's spent time in an arcade knows that the best way to get tickets is through skee ball.  In addition to giving out lots of tickets, you could steal−er−acquire even more by pulling the tickets out of the the slot very quickly.  Usually one skee ball game would net you about 5 tickets.  Back in the day, it cost 25¢ to play and you might get $2 allowance each week from mom. At this rate, it would take 

(10,000 tickets) / [ (5 tickets per 25¢) · ($2 per week)]
= 5 years,

and a total cost of

(10,000 tickets) / (5 tickets per 25¢)
= $500.

Given that it was at most a $100 stereo, I was probably better off spending the money on baseball cards.

[1] I was about 10 at the time, so my memory could be flawed on the exact number.