omenti

how much is enough? - homer simpson and olympic swimming events

It turns out The Simpsons has little math and science jokes making their way into the show - usually as very short sequences that fans can stop and study.  In a favorite episode Homer scribbles on the blackboard leaving four items.¹    3987¹² + 4365¹² - 4472¹² immediately caught my eye ..  it smelled like they were hinting at Fermat's Last Theorem.

 

There are an infinite number of equations of the form x² + y² = z² ..  like 3² + 4² = 5². Fermat hunted in vain for x³ + y³ = z³ and later claimed xⁿ + yⁿ = zⁿ  has no solution for whole numbers n > 2.  In the margin of a book he stated there are no whole number solutions for the infinite number of equations of that form and then added Cuius rei demonstrationem mirabilem sane detexi, hanc marginis exiguitas non ca- peret - 'I have discovered a truly marvelous proof of this, which this margin is too narrow to contain'.    Nasty.    Frustrated mathematicians toiled for hundreds of years until it was finally proved in a very non-trivial fashion in the mid 1990s .. indeed it wouldn't fit in the margin of a book:)

 

I pulled out my old HP calculator and tapped in the numbers.  Sure enough it worked.  It appeared to be a counterexample .. enough to disprove the proposition.  You suspect a near miss .. something lost as a rounding error.  The next step was to go beyond the ten digit accuracy of the calculator.  I reached for the laptop and fired up Mathematica

the first suggests the first two terms equal the third, but the second shows they aren't in the twelfth digit ..  looking at the first fifteen digits gives a bit more.  For most purposes it isn't necessary to have that many digits of accuracy .. but there are times when the question requires a bit of deeper knowledge - in this case the precision of a calculator.   But swimming pools?

 

There was a bit of a kerfuffle in Rio trying to explain ties in swimming events.  Tightly spaced finishes - often to within a few hundredths of a second are not uncommon and every once and awhile you get two and even three way ties (Phelps, Le Clos, and Cseh in the 100 meter Butterfly).   Identical right down to the hundredth of a second.  That can't be right can it?  Someone must have touched first.  The media mostly butchered the explanation.  Going to thousandths of a second like speed skating and a few other events - seems to be the way to go, but closer examination turns up a flaw.  

 

It turns out swimming pools aren't rigid.  They can't be or they'd break given what they're made of.  The ground around them shifts as does the water.  Even a degree change of water temperature causes a change of size.   In a 50 meter pool the time of the fastest event has a swimmer moving at about 2.39 meters per second.  In a thousandth of a second - a millisecond - the swimmer would travel about 2.4 millimeters.  It turns out the allowable tolerance of the length of a lane is 3 centimeters - something that has to be re-established several times a day during an important event. The pool can vary by more than ten times the distance the swimmer travels in a millisecond ..  timing at that level becomes meaningless.  Records couldn't be compared with each other at that precision and even the lengths of lanes in the pool can't be guaranteed. ² 

 

Which brings up another problem.  How do you have a fair start?  A starting gun goes off and runners or swimmers start out.  The bang of a gun travels at the speed of sound - you probably know the rule of thumb of counting seconds between the flash of lightning and its thunder and multiplying by one thousand feet.  The speed of sound varies with temperature, pressure and humidity, with the standardized value of about 343 meters per second.  Sound travels a about meter in three milliseconds.  Imagine a starting gun at the side of a track.  The path from the gun to a runner can vary by up to fifteen meters in some situations - that's 45 milliseconds .. over four hundredths of a second and a big deal in some of the shorter events.  

 

You might get around that by flashing a light, but the problem is we're wired in such a way that we respond to sound faster than to light - and by a lot ... like 50 milliseconds.  The solution is to put a speaker under each starting block and connect it to an electronic gun.  But another interesting question comes up.  How do you judge a misstart?  That one works out nicely.  Elite runners have a reaction to the starting gun of about 145 milliseconds and remarkably similar to each other.  A misstart is judged at something shorter than what reaction time would allow .. usually 100 milliseconds.   Some runners have very slow starts - reaction time and just getting the muscles going .. Usain Bolt is one of the slowest, so go figure... 

 

And for fun consider a problem I've given to students...  we all know refrigeration is inefficient and that opening the door of a refrigerator is a way to heat a room.  What would it take to detect the effect of a refrigerator in an average house?  What would it take to detect the impact of a single refrigerator on the power grid?  The first case isn't too difficult .. the second case demands some very specialized knowledge.

 

A thread connects these examples.  If you want to make an accurate measurement you need to understand something about what you're measuring and how you're measuring it.   The measuring tool, what you're measuring, its environment and even how they interact over time.  There are limits and when you bump into them it is useful to realize something is amiss so you know if you need to fix it or not - or if the situation needs a complete rethinking.

 

^__________

¹ I suspect this was aimed at physicists as the two physics jokes are a bit deeper than the math jokes.    two math and two physics.  The first has terms you'd see in particle physics - and suggest they predict the mass of the Higgs boson (this was fifteen years before it was confirmed).  The terms may look normal, but the equation don't make sense.  If you plug in numbers it does give about 775 GeV which isn't terribly far from the real figure of 125 GeV. (admission - I did just that immediately)  The third, Ω(t0) > 1,  suggests the universe will implode ..  when things in the room start rushing in on each other Homer replaces the > with a <  which causes a major explosions.  The fourth is a topology joke.  A circle and a triangle are the same, technically homeomorphic, in topology.  Consider a circle made of a perfectly pliable rubber.  You could easily stretch it into a square, pentagon and almost anything else without holes (that's a bit of a stretch, but you get the idea:-)  An object with a hole in it, a doughnut for example, can be stretched into a coffee cup - one hole that goes all the way through is the distinguishing characteristic.  A doughnut is not homeomorphic with a sphere.  Homer suggests that taking a bite out of a doughnut leaves it a doughnut, but you're removed the hole and destroyed its doughtnutness in the process.

For those who love math there are other gems scattered through the shows.

In one a screen comes up in the Springfield stadium:

Tonight's Attendance:

a. 8128

b. 8208

c. 8191

d. no way to tell 

(a perfect number, a narcissistic number and an Mersenne prime)

at one point the spine of one of Lisa's books reads e^iπ + 1 = 0

great stuff... to see this on tv is astonishing.

Futurama has some cultural and writing overlap and tends to leave techie trailings..  

 

² There are other problems with swimming that get a bit complex and pool dependent, but suffice it to say some lanes are better than others and the faster swimmers usually are given the best. 

 

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Recipe Corner

 

Grilled Sweet Potatoes

 

Ingredients

 

° 1 tbl cumin seeds

° about 2 pounds or a kilo of sweet potatoes cut into inchish wedges

° 2 tbl olive oil 

° 1 tbl maple syrup

° salt and pepper

dip

° 1/4 cup tahini

° 1 tbl maple syrup

° 1 tbl lemon juice

° 2 tbl water

° fresh cilantro

 

Technique

 

° get the grill to a good medium high heat

° toast cumin seeds to light brown in a small pan while the grill is heating. Then move them to a chopping board and crush them until fine

° toss the sweet potatoes and everything else up to dip

° make a half dozen aluminum foil packets with two large pieces of foil each (cut it off roughly square) brush the inside with a bit of olive oil

° lay the wedges in a line in each packet and fold tightly

° grill till tender .. about 15 minutes for me, but my grill isn't great.  check in a packet and look for the beginnings of charring.  Flip the packets a couple of times during the process

° remove and carefully open .. let them cool.

° whisk the dip ingredients and drizzle..

° garnish with cilantro 

 

 

 

when swing is a poem of motion

1900 at the Olympics in Paris

François Brandt and Roelof Klein had qualified in the coxed pairs, but had lost a race to a French pair.  Thinking it over with an unspecified amount of alcohol they came to the conclusion they couldn't change their rowing performance, but perhaps Hermanus Brockmann was overweight.  Brockmann was their coxswain.  In the early days the cox was largely responsible for steering and sometimes setting the rhythm of the strokes, but they were frequently seen as dead weight.  Brockmann weighed sixty kilograms.  The pair spotted a young boy in the crowd and ran a trial.  He only weighed thirty-three kilos and they had to attach another five kilos of lead to their shell to make the rudder set properly in the water.

 

The flying Dutchmen leapt to an early lead but couldn't keep their initial pace. Halfway into the race the French started their move, but at the finish the Dutch were still ahead - but by less than a meter. A dramatic race by any measure, but more curious is the fact that the cox disappeared into the crowd never to be awarded his gold medal.

 

That brings me to the trigger for the post.  American Experience in PBS recently aired a piece on the 1936 US men's Olympic rowing team - one of the most amazing bits of magic ever seen at the Olympics.  You can watch it here.

 

How important is that dead weight? In theory a cox aims the boat in the straightest possible line, gives pacing and coaching, and removes the effects of uneven thrust as much as possible. But he or she also adds to the load.  We can easily make an estimate.  

 

In doing very quick calculations to test ideas dimensional analysis is frequently invoked. The kid destined to be a natural scientist would look at the units and note that gallons is a volume or a mass. Assuming you are using the same liquid in your comparisons - at least liquids of similar densities - it doesn't matter ... take your pick. Let's pick volume .. the dimension of volume is a length cubed - L³. Miles are easy - just a length. So miles per gallon is dimensionally L/L³ or 1/L² - an inverse area.  It could be inverse square meters, inverse square feet, or ... hey .. inverse acres has a nice rural feel to it.

 

The question is how much does it cost for a racing skull to carry a cox? Rather than a boat with two rowers let's consider the more general case with N rowers. A four man boat has twice the power of a two man, but its weight is about twice as much. Does having twice the engine and almost twice the weight help or hinder?

 

I've received some complaints about being a tad too technical.  I'll indent and highlight the reasoning behind the estimate.  You can skip it if you like.  It turns out to be about two or three minutes of work at the blackboard and a good example of how physicists play.  You make simple estimates, often little more than dimensional analyses, to build a simple model.  That can give you some insight into the problem and further questions.  It is a very quick method for going through dozens of ideas when you're in conjecture and hypothesis land.  Here it gives a quick and dirty answer to the question.

 

 

The power the rowers deliver mostly is used to overcome the drag on the hull.  It is

p = f_d * v

where p is power, f_d is the force of drag, and v is the boat's velocity relative to the water

The force of drag is reasonably complex and would require some careful measurements that depend on the shape of the hull and some other factors.  The important thing to realize it is proportional to the wetted area - something that goes as a length squared - and the square of the velocity.   So 

f_d ∝ L² * v²

the power needed is 

p_d ∝ L² v² * v or L²v³ 

The power to overcome the drag is supplied by N rowers (assuming all are about equally strong).  A racing shell is pretty minimal, but its volume needs to increase to float with the heavier load.  The volume goes as L³ and also is proportional to N, so L ∝ N^1/3.  Getting tricky we substitute

p_d ∝ L²v³ ∝ N^2/3v³

getting to the end! ....  We know power supplied is proportional to N so 

N ∝ v³ N^2/3

solving for velocity we see

v ∝ N^1/9

 

The speed of the boat only increases slightly with an increasing number of rowers.  

 

For those of you who have rejoined us it only increases at the one ninth power of the rowers or N^0.111.   Try plugging some numbers into your calculator and see!  The time is just to the inverse of the speed over similar distances so t ∝N^-1/9. If you look at world record times over similar distances this holds for one to four person coxless skulls reasonably well and is a good enough model for this purpose. 

 

You can play with some numbers yourself - assuming the cox is just along for the ride the impact of having one decreases with crew number. Since races are tight you don't want to race coxed vs uncoxed boats and you don't want a heavy cox. Hopefully they add more value and perhaps you can get a sense of quantifying that value in real vs expected times.

 

 

The original question was what is the impact of the cox - the little guy or gal with the rudder and the loud voice?  I did a similar analysis increasing the size of the boat, and thus its drag, by going to N + 1 crew members, but assuming only N are contributing to the power.  I won't go through the reasoning as it takes much more time to type than to think about it, but it is very similar.  The predicted time with a cox, t_cox is:

t_cox ∝ (N + 1)^2/9/N^1/3 

To get something that is easy to think about look at the ratio of cox and coxless times and apply a bit of rearranging

t_cox/t ∝ ((N + 1)/N)^2/9

(N + 1) is always bigger than N, so the cox always slows things down ...  not by a lot and the effect decreases with increasing N.  A two man boat should be about nine percent slower if the weight of the cox is similar to the rowers.  But it isn't.  The Dutch used a kid who was about a third the weight of each rower, so a reasonable guess would be about (2.33/2)^2/9 or something over two percent slower - probably several percent faster than an equal muscled crew with a cox that weighed as much as poor Hermanus Brockmann.

 

These days coxswains have much greater value.  They steer the boat and usually run the strategy.  Recently I was thinking about four man (or women) boats - there are coxed and coxless events.  Thinking about the pattern of the rowers led to a slightly veiled surprise. 

 

The pattern of the rowers is called the rig of the the boat.  It looks very symmetric - right, left, right, left for the four and twice that for the eight (note that left, right, left, right is the same by symmetry).  Assuming the crew are evenly matched you would expect the shell to move straight ahead without the intervention of the cox.  If this was true the uncoxed four should be a bit faster with the cox only offering on-bard coaching.  Of course crew members aren't evenly matched so the cox has to work the rudder.  

 

A bit of thinking about the forces on the oarlocks leads to a slightly hidden asymmetry.

 

High school physics tells us we can decompose forces into vector components - when examining the flight of a ball you can break the motion into horizontal and vertical components allowing you to isolate the effect of gravity.  On a shell there is an oscillating force at each of the oarlocks.  When the oar is in the water there is a strong force on the oarlock in the forward direction of the boat's motion.  Assume each crew member is equally strong with the same form and perfect timing.  In a four man boat we get four times this force to move the boat ahead.  Dandy - that's what you usually think about when you calculate the potential speed of a racing shell.  

 

There is another force at the oarlock perpendicular to the forward force.  At the first half of the stroke it is directed towards the shell and in the second half it is directed away.  The force drops to zero at the midway position of the oar - a sinusoidal force.  It is smaller than the forward force and at first glance appears to be balanced by the fact that every oar has an opposing oar on the other side of the shell. Two oscillating forces that are mirror images of each other at any moment.  They should cancel each other so we might be tempted to move on.

 

Here's the nut of the question.  The forces act over a distance.  We're not concerned with a force on the left balancing a force on the right, but rather each oar produces a torque on the rudder.   Now things get interesting.

 

The torque is just a force on a long arm.  Imagine the sideways force is F at any moment in time.  If you are distance L from the rudder, the torque at the rudder is F*L (foot-pounds, newton-meters...).  Let's give our model a distance r  from the rudder - essentially the cox -  to the first oarsman.  Let the oarsmen be separate by the same distance - call it d.  The total torque at any given moment (note I'm just looking at instantaneous torque rather than worrying about a messy integral) for a four man shell with the rig shown would be:

T = rF - (r + d)F + (r +2d)F - (r - 3d)F

 this = -2Fd  certainly not zero over most of the stroke!  This wiggle-waggle torque, remember F is oscillating in magnitude over ⁺/_- its maximum value, either goes into a wasteful motion of the shell or the cox can counteract it.  Both outcomes waste energy.

 

So you wonder about other patterns.  Consider these rigs.  The first is what we just examined and the second is the standard eight which alternates between ⁺/_- 4d as you might expect (check my arithmetic).

 

The third rig is interesting.  Performing a similar calculation I get:

T = rF - (r + d)F - (r + 2d)F + (r + 3d)F

 Now we get zero.  By a clever arrangement we can balance out the torque at every instant.  It turns out this has been tried and successfully, although there isn't any mention they understood what was going on.  

 

Moving to the eight- man gives a trivial wiggle-waggleless rig - 4a is just a couple of the rigs used in example 3.  For fun I found three more and showed that is all that exists. 4b gives zero - you may want to check the arithmetic again.  I found two more in short order and won't give the solution for those of you who find pleasure in figuring things out.  Ping me if you want my solution.  

 

In the real world the forces aren't equal and timing and form are less than perfect - mostly that is.  There is this notion of swing in rowing.  It happens rarely and is the result of near perfect timing and application of force.  Even in world class crews it isn't that common although you may see it in one or two of the races in Rio.  An eight-man boat is producing something over five kilowatts most of the time and can burst much higher. Individual differences on the order of a few watts can destroy swing.  It is a dramatic thing to see and crew members who have been there cite it as something amazing - a very special kind of flow.  You can see it in the motion of the boat and in the expressions of the crew.   Quite literally a poem of motion.

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Recipe Corner

 

From an athlete friend:-)

 

Peanut Butter Smoothies

 

Ingredients

 

° 2 sliced bananas - frozen

° 3/4 cup milk or soy milk

° 3 tbl peanut butter

° 2 tbl unsweetened cocoa powder

° 1/4 tsp vanilla extract

 

Technique

 

° surely you jest - blend until smooth and server