omenti

a look at the movements of a goat

a pre-Olympic minipost

 

About a year ago I taught a course on the physics of sport. Recently a high frame rate video of Simone Biles was posted. Making a few frame by frame measurements (helped by another view and knowing her size and some simple home-brew software) it’s possible to do a quick back of the envelope analysis.

 

First the this slow motion video from a meet on the 4^th of June -watch it even if you’re not interested in the analysis. Her motions are beautiful.

 

Some background information .. she’s 142 cm tall - a shade under 4’8” and her weight is currently listed at 52 kg (about 114 pounds) Her top sprint speed is 8.0 m/s (18 mph), although she doesn’t reach it here (not enough room)

 

It’s important to note that a couple of innovations around 1980 revolutionized gymnastics. The floor is one of them .. notice how springy it is. It’s a thin carpet bonded to a layer of hard foam that’s glued to plywood sheets. All of this is mounted on a large array of heavy springs which are attached to the main floor. Think of it was a stiff trampoline that allows the gymnast to trade some of her kinetic energy into a higher jump. A pole vaulter uses the pole to do the same thing.

 

She launches at about 7 meters per second (about 15.7 mph) at an angle of about 70°. Her center of mass reaches a height of about 2.9 meters (9.5 feet) and she lands about 2 meters (6.5 feet) from her launch point. She lands on a piece of foam to slow the landing a bit. Even with that and the springy floor she’s experiencing a deceleration of over five times the force of gravity.

 

What goes on in flight is the remarkable part. Once launched, Simone is a projectile that is only acted on by the force of gravity (ok - there’s a small correction for wind resistance, but it can be ignored). She has three axes of rotation to play with. All three go through her center of mass, which is near her waist. Call the one that runs from head to toe her “twisting” axis. Call the second her tumbling axis ‘’ this one's at a right angle to her twist axis and is described by the motion about her waist when she falls forward. The third is at right angles to the other two and is the rotation to the left or right. She doesn’t use that one here, so we’ll ignore it.

 

Back to her sprint. She wants to go as fast as possible so she has the largest possible time in the air to accomplish her motions. Launch angle and speed together determine how high she’ll go and how much “hang time” she has. She also adds a bit of a twisting motion when she launches. She can speed it up by pulling her arms in like a skater. Likewise the tumbling motion can be speeded up by pulling her arms and legs in to lower the moment of inertia about that axis. The high speed of her flips - each is about two thirds of a second long and the smoothness of control is testament to her athleticism and precise control of her moments of inertia.

 

I did the calculation of a launch speed required to get another flip - another two thirds of a second - in… it would have to be about 10 meters per second (22.4 mph). Even with a very long runway we’re not going to see four full flips. A faster sprinter (world record for the 100m women’s sprint peaks around 12 meters per second) wouldn’t have the space and would be too tall and would lack the core body strength to do flips that fast anyway. We’re probably at the limit given the floor design and size along with human capabilities. Of course there are other innovations that can appear in future performances.

 

This is a great example of a sport that dictates body size and type - at least for the elite level.  That's an interesting subject on its own...

 

beyond the peloton

a minipost

 

A few days ago I forwarded the first of what promises to be a series of pieces of a cultural study of Peloton users by Anne Helen Petersen. It’s interesting - she’s certainly an enthusiast and she’s making interesting observations.  There was some interesting feedback- mostly on how big it might get, where Apple fits in and so on. I have some educated guesses where it sits in the home exercise stack and where Apple is as well as a very different direction they may take, but I think there’s something more interesting.

 

Around the world many found themselves isolated for a good deal of the past 15 months.  Exercise equipment became an item for those with the money and time.  Now that people in the US are coming out of isolation a good deal of that equipment, perhaps kit not as engaging as the Peloton,  is turning up at yard sales.¹  There are reports of more engagement in outdoor activities - amateur sports.

 

Over the years I’ve come across several social science papers on the sociology of exercise and amateur sport. Robert Putnam (the author of Bowling Alone way back when) pointed out that communal exercise - particularly amateur sports - creates a meeting ground where people of different beliefs and ideologies come together regularly.   While they may not become close friends, many become familiar faces you look forward to seeing and chatting with.  They're part of the 50 to 150 Robin Dunbar identifies as friendly familiars.  More recently Putnam and others have talked about the service groups, amateur sports, etc that became popular during the Depression and persisted through about 1965.. a period of high social cohesion in the US (Putnam documents this more recently in his perhaps overly-optimistic book Upswing). Apparently it’s still true that social cohesion is high in these communal events - running, cycling, softball leagues, etc. etc.  It’s certainly in evidence in pro-sports fandom worldwide - BBC's Why Factor had an excellent piece on the subject:  Why do we care so much about games.

 

Norway has a unique approach to youth and adult sport and participation at the amateur level is much higher than any other country. The budgetary justification is two-fold: better health and better social cohesion.   There's much more going on there and in the Nordics in general that leads to social cohesion, but in many parts of the world it seems to be decaying.  Perhaps getting involved with others you don't know well at first is healthier for society than doing things by yourself or in your bubble.

___________

¹ The need of social connection is strong.  Peloton provides a bit, but it isn't rich and it's mostly tied to a rather narrow form of exercise - spinning.  Isolated and semi-isolated workouts can be important, but I doubt it has a huge amount of growth potential unless it's very cheap.  I think we're in a Peloton bubble that doesn't have much more life given an improving health outlook. 

 

tapping your inner duck

a pre-Olympics minipost

 

To swim faster you want to decrease the amount of drag caused by moving through the water and increase the amount of propulsive force you can deliver.   Focusing on propulsion it would be nice if you had more hand area - perhaps webbing between your fingers.  It turns out fluid dynamics offers an easy to implement trick.

 

I've done a fair amount of thinking about the motion of balls through the air.  Balls can be made to do something interesting: the dimples in a golf ball letting it travel much farther, the sudden drop or rise in the trajectory of a spinning ball, the wobble of a perfectly executed knuckleball, and so on.  These are all artifacts of a layer of air that is dragged along the surface of the ball -  the boundary layer. 

 

Boundary layers of one form or another are present whenever an object moves through a fluid:  a container ship through the ocean, a mosquito wing beating in the air, a meteor entering the atmosphere, a hand moving through the water.  The study of boundary layers can be mathematically challenging  so wind tunnels and towing tanks found heavy use until the last few decades.  These days good approximations to many interesting problems can be found using computers.  Competitive swimming - particularly the Olympics - is an application area for computational fluid dynamics.

 

It turns out a finger-sized cylinder moving through the water at swimming speeds is accompanied by a large boundary layer.  Spread your fingers a bit and this layer effective fills in the gap between your fingers with an effective web made of water rather than skin.   For most people the average distance is a bit less than half a finger width. Competitive swimmers will train with spacers that let them feel the proper distance until it's natural.  It turns out to be good for as much as a six percent increase in force and a two percent increase in speed, so it's absolutely worth it. An amateur swimmer can get a good effect without trying for perfection.

 

 

 

repairability

Growing up our family always had Fords.  For most of grade school through junior high that meant a blue Ford Falcon.  My Dad's reasoning was simple - a neighbor had good mechanical experience with Fords, particularly the straight-6 engine in the Falcon.  He also had a nice set of tools. Much of it was simple enough that I was involved in part of the routine maintenance even before I had my driver's license.  Simple things like changing the oil every 1,500 miles, gapping and replacing the spark plugs, making sure the battery levels were ok, draining the radiator, testing and replacing the antifreeze, and taking off the air cleaner to spray ether in when cold-starting it when it was below -20°F.   

 

By the time I was driving legally  (I drove a pickup on a wheat farm during harvest the year before I had my license. I suspect that was common in Montana.  Learning how to drive a manual transmission amounted to me trying to not stall the pickup while my boss watched laughing for an hour.)  By high school I graduated to more sophisticated work.  Replacing the head gasket on my own and even replacing the rings under the watchful eye of my Dad's friend.  Everything was exposed and simple - most of it was just sorting out fuel, air and spark.   Sometimes, driving to King's Hill on a hot Summer day, the engine would quit.  That meant pouring some cold water on the fuel line near the engine and waiting a few minutes to break the vapor lock.  Each car had a set of peculiar tricks, but makes had personalities.  Changing to a Chevy or Dodge would have a learning curve.

 

While the Falcon was simple and to work on, almost everything was poorly made.  Perhaps not to the eye, but tolerances in the engine and transmission were sloppy.  A careful break-in period was necessary if you wanted an engine and transmission to last.  It was inefficient even though it didn't weigh much by today's standards.  Although it was a compact, it only got 18 miles per gallon on the road and 13 or 14 in town.  Engine and transmission oil had to be replaced regularly.  Tires went about 10,000 miles if you were lucky, brakes needed adjustment every few thousand miles, the chassis needed lubrication regularly and so on.  The body and chassis designs were primitive and non-crashworthy by today's standards.  Cars often remained fairly intact even in fatal collisions.  

 

The seventies brought more efficient designs along with increasing safety regulation.  More sophisticated tooling that produced dramatically improved engine and transmission tolerances that allowed more efficient designs.   They lasted longer, but it was becoming difficult to work on them.   Repair was moving towards replace.  The failure of a wiring harness on the mechanism for an electric window might be a fifty cent part, but the replacement module could cost hundreds and require several hours of work to replace.

 

Home electronics followed a similar path.  Our first TV was a 19" black and white Magnavox from about 1961.  It had a few dozen tubes, a lot of point to point wiring and hand-soldering, and mechanical switches.  If something went wrong the first step was to take off the back of the set and see if any of the tubes weren't glowing.  You'd pull the suspect out and test it at the local drugstore - most had a tube testing kiosk with a stock of replacement tubes.  That happened a few times before we got our first color TV in 1969 - a Motorola "Quasar" which only had two tubes including the picture tube.  It's main selling point was reliability. By the 80s integrated circuits began to invade home electronics and home repairs became next to impossible.  It didn't matter as we perceived our devices were improving faster than their failure rate.  Prices dropped so rapidly that throwing them away became popular.

 

This replace rather than fix mindset applies to many consumer goods, and many aren't engineered for a long life.  During the pandemic our eight year old refrigerator failed.  I was able to get it working again in about a day, but the repair was anything but easy.  The dishwasher also failed.  A gear on the motor, but the replacement module was $260 and special tools are required. It was eight years old so the leap to a new unit was an easy choice.  I probably wouldn't have tried to repair these things if I didn't have the background of repairing things that broke at home.

 

Ten years ago a mechanical engineer friend thought there might be a market for simple good design and quality manufactured items - mostly home appliances.  They would cost more, but last twenty or thirty years and be easily repairable.   He built a few prototypes, but was laughed at.   It turns out we're mostly ok with disposable.

 

The design integration in most smartphones and watches makes repairs difficult.  If compact, rugged and waterproof are goals, replaceable batteries are an issue even though you may need to replace them every few years.  A cynic might say it gives the user a push to get a new device.  

 

More important is the level of software integration across the system.  Apple's walled garden is problematic for many, but the value of more security and privacy (assuming you don't live in China), and mostly well-behaved apps is worth it to many.  Trading price and flexibility for stability and privacy.   

 

This raises the question of what should we be using?  Cars are much safer and efficient, although more expensive to operate, consumer electronics are mostly better and much cheaper.  Thinking of different ways to do things probably makes sense.  Getting rid of a second car and adding an electric cargo bike if your roads are safe enough is an option.  Three very sophisticated friends have traded their smartphones for simple "dumb" phones.  Well-made and designed clothing that lasts a decade.  Not everyone has multiple choices, but the pandemic seems to have people questioning many things. 

 

I'll end with a recommendation that applies almost universally..  staying away from IoT (Internet of Things) devices makes a lot of sense unless you know exactly what the security and privacy issues are in each case.  And a fundamental flaw in many - they have software flaws and the only way to update is to throw them away and buy something new.  And while it's beyond the main message, some of these IoT devices send information back to organizations that market it.  They turn you and your behavior into a product.

 

 

bananas, bagels, pizza and gauss

minipost

I watched by the sidelines as a few people managed got in a verbal tangle about what makes proper pizza.  As the religious wars raged I remembered pizza illustrates a deep feature of mathematics - Gaussian curvature and the Theorema Egregium (remarkable theorem). It turns out you can get a good feel of it with a ball, banana, bagel and a slice of pizza.  

The always enthusiastic Cliff Stoll demonstrates:

 

Math, properly taught, is beautiful and not that hard ...  And go out and get some pizza.  I won't judge.

 

a congressman, the sun and the moon

Someone sent this rather bizarre segment of a Congressional hearing involving Rep. Louie Gohmert of Texas.  

 

Rather than being clueless about science (he may be that too), I suspect he's echoing some baseless claims used to claim humans have no impact on global warming.  While the orbits of the Earth and Moon are dynamic, they have no impact on current global warming.   That said, the Moon's orbit is changing at a measurable rate and that's worth a short post.

 

The Moon and the Earth orbit a common center of mass called a barycenter that's about 1,700 kilometers beneath the Earth's surface.  Seen from the Earth, the Moon's orbit is elliptical with a slight wobble. The gravitational interaction between the two bodies raises tides on the Earth, making our days longer and pushing the Moon out a bit less than four centimeters per year - roughly at the rate your fingernails grow. 

 

So the Moon's orbit is changing, but slowly.  Unchecked this process would continue for nearly fifty billion years when the Earth's day would have slowed to nearly fifty of our current days and the Moon would take the same amount of time to orbit the Earth.  At that point we'd be permanently locked with the Moon.  

 

Fifty billion years is a long time and something else will happen long before then.  About five billion years from now the Sun will become a red giant.  There are a few scenarios - here's my favorite.  

 

The Sun's atmosphere, as it heats and expands,  will create drag on the Moon's orbit bringing it closer to Earth.  There's a limit called the Roche limit to how close a gravitationally formed body can hold together before tidal forces rip it apart.  For the Earth-Moon system that's a bit under 19,000 kilometers (say 12,000 miles).  The Moon is ripped apart and scattersinto a Saturn-like ring that might stay around for a few hundred thousand years.

 

If the Sun keeps growing the Earth will eventually be in or near enough to its photosphere that the Earth would spiral into the Sun and be no more.   But that's a long time away.  In the meantime we need to worry about the here and now of global warming - perhaps not as dramatic as the ending five billion years out, but deadly nonetheless. 

 

are athletes getting better?

Olympic minipost

One of Simone Biles' floor routines as seen through a high speed camera. (go full screen)  Many sports have undergone dramatic changes in the past fifty years.  The introduction of springy floors and beams has allowed more dramatic routines that revolutionized gymnastics.  It's a sport where powerful muscles and a short stature are required. Simone's incredible acceleration combined with the strength to jump explosively and change her moment of inertia in mid-air give her claim to the great athlete of our time title.  At times she's experiencing g-forces usually found in aerobatic aircraft.  Here's a short video from Quartz (it won't embed) before the Rio Olympics  explaining a bit of what's going on as well as giving a sense of how far the sport has come. 

 

Three major innovations have changed elite sports - equipment, extreme specialization of body type to match the demands of specific sports, and a much larger gene pool to selection from.  David Epstein gave a nice talk about seven years ago touching on all three:

 

Many sports - particularly Olympic sports - lack the financial rewards seen in the top professional sports, but at the elite level it's clear the expanded gene pool has become important.  Of course none of this applies at the amateur level -  people find a sport or two they like and "just do it."  There's a lot of evidence continuing these activities has a huge positive impact on health - particularly as one ages.  It may be elite athletes encourage many of us to move.  Perhaps the endeavor is cost effective at the societal level - it's both desirable and part of the public health policy in Norway.

 

playing with dimensions

Many occupations develop tricks to get "good enough" answers question.  These range from rules of thumb to shortcuts that are based on some insight.  In physics dimensional analysis - checking to see the technique or answer has the right dimensions - is important. After all - you don't want to measure someone's height and come out with an answer in pounds.   

 

The common shorthand is to use square brackets to denote the dimension of a physical quantity.   For a distance r [r] = L, velocity [v]  = L/T, g (the acceleration of gravity) [g] = L/T².  Generally M is the dimension for mass, L for length, and T for time. 

 

With a bit of additional information you can make quick estimates and even gain some insight.  Consider two athletes of similar build and physicality, but different heights.  One is six feet tall, the other five feet.  If you want to compare the power to weight ratio all you need to know is the power a muscle generates is proportional to its cross sectional area.  Their weights scale as the cube of their height, so  [power/weight] = L²/L³ or 1/L.  The shorter athlete has a twenty percent power to weight advantage.   You can use this type of analysis and freshman level physics to sort out why different sports dictate very specific body types at the elite levels of sport.  

 

Techniques like these are used during the early and very playful part of asking questions.  They can tell you if your approach is plausible or if it's going to be wrong.  Plausible doesn't mean right, but it's very useful to recognize what can't work so you can try something else.  It's a lovely way to learn by failing gently.  With time and experience you develop an intuition for what might work. 

 

One of these quick calculations is legendary.  A series of photos of the Trinity after the end of WWII.   Four stages of the explosion with a length and time indicators.  Details like yield were classified.  A British physicist didn't realize calculations were classified and did his own back of the envelope calculation in a few minutes.  His estimate was published and caused a kerfuffle worrying that secret information had been leaked.  

 

There wasn't a leak. The truth is many of the physicists involved made quick estimates that were good enough - enough to tell them the device's yield was about what they thought it would be.  They just didn't publish for obvious reasons. 

 

Here's my ten minute estimate using dimensional analysis and a bit of physical intuition. I'm avoiding a very difficult set of differential equations with some very messy boundary conditions - not to mention some quantities I have no way of knowing as an outsider.  To make it as simple as possible consider a spherical blast. The ground will have a minimal on the radius of the blast in the atmosphere, so ignore it.  The blast creates an expanding bubble that has to push against air outside the bubble. The most important terms in any expression will include the radius of the blast, the density of the air outside the blast, and the energy of the device.  There are many other things going on - this is just an attempt to see if a very simple high level relation of the relevant variables can be found in a few minutes.

 

Basically rearrange these terms dimensionally to see if you can find radius as a function of time after the device was set off.   Remarkably a simple relationship emerges.  So why not plug some numbers in?   The time and radius can be found in the photos and the density of air is well-known.    My radius measurement is crude - I know the device was on a tower and found a point that looked like the center.. draw some lines on a piece of paper, put it up to the picture and estimate 90 meters... perhaps a bit less.  A bit of arithmetic and a yield of equivalent to fifty thousand tons of TNT falls out.   

 

(pardon the handwriting, but it's much quicker and more natural than trying to typeset it)

 

 

The estimate isn't that far off - it gives you a good sense how the radius scales as the one fifth power of the blast's energy and to the two fifths power of time.  Those relationships are more illuminating than the estimated yield.  Since there are four images you could do something a bit more elaborate measuring all of the photos and making a plot of the growth of the fireball.