31 August 2012

Interference ... or not?

My Vanderbilt Commodores helped open the college football season on Thursday night by hosting the #9-ranked South Carolina Gamecocks.  I always root for my alma maters (Vanderbilt and Indiana), but as the football season moves along, it usually becomes clear that the chances of seeing my schools win will improve greatly once basketball season arrives.  Opening this season at home with a top-ten SEC school did not bode well for my desire to see Vandy get off to a good start.

We lost to South Carolina, 17-13, after leading early in the 4th quarter.  Vandy might have looked better than I thought we'd look, and South Carolina might not have looked like a top-ten team.  The two schools appeared more evenly matched than I thought they'd be before the start of the game.

With just under two minutes left in the game and facing a 4th and 7 from our own 38-yard line, Vandy quarterback Jordan Rogers threw a long pass toward the right sideline for Jordan Matthews.  South Carolina safety D. J. Swearinger was defending Matthews.  The pass made it to about the South Carolina 35-yard line.  Check out the two images below (click on each for a larger view).

The above images show Matthews preparing to catch the ball just before it reached him.  Now there is an interesting rule in 2011 and 2012 NCAA Football Rules and Interpretations, specifically Section 3, Article 8, part c on page FR-74, which says, "Defensive pass interference is contact beyond the neutral zone by a Team B player whose intent to impede an eligible opponent is obvious and it could prevent the opponent the opportunity of receiving a catchable forward pass."  Matthews did not make the catch, ending Vandy's shot at a win.  No flag for defensive interference was thrown on the play shown in the above images.  I think grabbing a player's arm and yanking it down suggests that Swearinger did "prevent the opponent the opportunity of receiving a catchable forward pass."  WE WERE ROBBED!

Okay, so I'm a disgruntled fan who hated to see a ref miss a call that would've given my school a chance to beat a top-ten-ranked school in our season opener.  We made plenty of mistakes during the rest of the game and missed calls are just part of the game.  There was no guarantee that we would've scored the winning touchdown had a ref made the proper call.  But it would've been fun to see if we could have done it.

College football is great because year after year, we make an emotional investment in our alma maters' gridiron exploits.  The vicissitudes within each game vex us more than they should -- hey, it's just a game, right?.  After a close loss to a top-ten team, this Vandy alumnus does what comes all too naturally -- chalk up another moral victory.

24 August 2012

Reality and Lance Armstrong

Consider the following list:
  • Alex Zülle (1999)
  • Jan Ullrich (2000)
  • Jan Ullrich (2001)
  • Joseba Beloki (2002)
  • Jan Ullrich (2003)
  • Andreas Klöden (2004)
  • Ivan Basso (2005)
When compiling a list of Tour de France winners from 1999 to 2005, one needs only one name:  Lance Armstrong.  The greatest Tour de France cyclist in history did what nobody at the close of the last century thought possible -- win seven consecutive Tour de France races.  The list above represents those who finished second behind Armstrong.

Lance Armstrong announced that he would no longer fight the United States Anti-Doping Agency's charges against him.  The possibility exists that Armstrong could lose his seven Tour de France titles.  If that happens, then what?  Will there be a repeat of 2006, the year Floyd Landis stood in front of the Arc de Triomphe donning the yellow jersey as the overall champion, only to be disqualified two months later by the International Cycling Union and see Óscar Pereiro Sío be awarded the win?  Will the above list be the new reality?

Of course, the 2006 Tour de France had other doping issues with famous cyclists like Jan Ullrich and Ivan Basso getting excluded before the race began.  The 2007 Tour de France had its share of doping woes, too.  Questions of reality marred those two races.

After the Armstrong news broke, several people asked me, "Do you think he did it?!?"  As if I know.  I've spent a decade modeling the Tour de France and predicting stage-winning times.  Beginning that line of research with Ben Hannas back in 2003, right in the heart of Armstrong's streak, launched my professional career as a sports physicist.  Chapter 4 of my book is devoted to Lance Armstrong and Tour de France modeling.  Does all of that somehow make me any more knowledgeable as to whether or not Lance Armstrong cheated to win all those Tour de France races?  Of course not.  Like many, many people, I was enthralled by his story of beating cancer and dominating his sport like no other.  But I never met Lance Armstrong, and I've no idea what he does when nobody's watching.

So, what is reality?  Surely that's a question pondered by great minds for much of human history.  I watched Ben Johnson run the 100-m sprint in the 1988 Summer Olympics in a time of 9.79 s, faster than any human had ever run 100 m.  Carl Lewis saw Johnson cross the finish line ahead of him, yet Carl Lewis is listed as the gold-medal winner because of Johnson's famous disqualification three days after the race.  The reality for Carl Lewis after that race was that he lost to Johnson.  Lewis never got to experience the feeling of having successfully defended his 100-m sprint gold as he crossed the finish line, a feeling Usain Bolt enjoyed last month.

I love college basketball.  I watched the championship games in 1992 and 1993 as the Fab Five of Michigan lost both games, yet Michigan vacated those Final-Four appearances -- like they were never there.  Imagine what reality shifts would have occurred if Michigan had won one or both of those title games.  I'm pretty sure I watched UMass play in the 1996 Final Four, but that school had to vacate its appearance in New Jersey that year.  I sat in Assembly Hall in Bloomington and watched the 1997 Final Four Minnesota team beat my beloved Hoosiers in overtime.  Oh wait, Minnesota must not have been in Indy that year for the Final Four.  Ohio State joined fellow Big-Ten school Michigan State in the 1999 Final Four, but the Buckeyes weren't really there.  The Florida sun played with people's eyes that year.  Kansas beat Memphis in a thrilling overtime game in the 2008 championship, but, alas, Memphis was really in Tennessee for that game instead of Texas.

Just three days after I turned 37, Alabama beat my Vandy Commodores in football by the score of 24-10.  My alma mater finished 5-7 that year and would've gone to a bowl game if just one of our seven losses had been a win.  Hold on!  In 2009, Alabama had its wins vacated for the 2007 season, as well as wins in a couple of seasons before that year.  Did my 2007 Vandy team deserve to go to a bowl???  The reality that year sure made me feel like we didn't deserve a bowl appearance.

College football was racked by awful scandals this past year.  Penn State topped the list.  Among the penalties Penn State incurred, they had to vacate 112 wins, all but one of which belonged to Joe Paterno.  The 2006 Orange Bowl had Paterno face Bobby Bowden's Florida State team.  Bowden lost a close one, 26-23.  Well, the loss didn't really happen.  In fact, Joe Paterno is not the college football coach with the most wins, as he surely thought he was before he died earlier this year -- Bobby Bowden is.  Bowden got that distinction three years after he retired.  Some reality.

Photoshop makes it possible to look at a picture and not know if it's real.  CGI can turn any YouTube video into pure fiction.  Basketball shots from helicopters and footballs flying into garbage cans from impossible distances are the norm these days.  Well, the images are real, but they don't show reality, or do they?  People pass themselves off as celebrities, wedding guests, etc. and others may never be the wiser.  Was Han Solo real?  He was for me when I was a kid.  Harrison Ford may have been standing in front of a camera dressed as Han Solo, but the idea of Han Solo was quite real in my head.  I knew he wasn't real, didn't I?  Harry Potter is very real for my daughters, especially my older daughter who has read all seven books.

Now everyone in the sports world wants to know what's real.  Did Lance Armstrong really win all those Tour de France races?  Or at least did he win them fairly, whatever than means?  Are all the home runs we see real?  Were they real in 1998?  Were the dingers hit by Barry Bonds real?  Some say they shouldn't count and that Hank Aaron is the true home run king.  Will the Hall of Fame not call Bonds next January, even though all his records are still intact?  What about Roger Clemens?  Will the home-run king (Bonds with 762), the Cy-Young king (Clemens with seven), and the hit king (Pete Rose with 4256) all need to buy a ticket to the museum in Cooperstown, just like everyone else, if they wish to visit?

I want to watch a sporting event and know that the result will stand.  I want athletes to play by the rules.  I want the great things I saw in the London Olympics to stand up for all time.  Queue music for my Pollyanna speech (and, yes, I watched Knots Landing when I was younger -- click here).  Putting aside the atrocious crimes at Penn State, recruiting violations, doping, and so forth, rewriting the past with new sporting results just stinks.  Lance Armstrong helped to set me on a new research path.  I really, really, really want Lance Armstrong to have won all his Tour de France races in an honorable way.  Do I have a guess as to whether he did?  No.  I've no idea what the true reality is.

13 August 2012

Olympic Physics Summary

The 2012 Summer Olympics are over.  As overwhelmed as I felt trying to watch as much as I could, even during holiday with my family this past week, I can't believe how fast the last two-and-a-half weeks have gone.  London and the rest of the UK are right to feel proud of the fantastic effort they put forth.  Great job!

A couple of my colleagues commented to me that they enjoyed reading my blog posts concerning the Olympics, but they found that if I wrote two or three posts on a given day or on consecutive days, they missed some posts at first glance of my blog.  That's my fault for not having a particularly well-organized blog!  To that end, I though I would summarize in this post the two dozen or so posts that I wrote for the Olympics.  I am certainly flattered by anyone who chooses to read what I've written, and I appreciate the kind comments I've received.

Below is a list of the sporting events I wrote about during the Olympics.  I give a link to a post or posts I've written that deal with a particular event in parentheses following the event name.  In some cases, multiple events link to the same post because of discussions of big physics ideas that I applied to more than one sport.
As tired as my fingers got typing all those posts, I look at the above list of events and feel like I hardly scratched the surface of this past Olympics.  I'll try to watch as many replays of the events I missed as I can, and perhaps write a few more blog posts.  Following and studying the Tour de France and Summer Olympics have made this summer a lot of fun!

Just 543 more days until the opening ceremony in Sochi, Russia kicks off the 2014 Winter Olympics!

10 August 2012

Queen of 10 m!

China's Chen Ruolin is simply unbeatable once she steps on a platform that sits 10 m (33 feet) above an Olympic pool.  Be it synchronized diving or individual diving at 10 m, Chen Ruolin has had gold medals around her neck in both the 2008 and 2012 Summer Olympics.  She essentially ended the 10-m platform final yesterday with an 85.50 on her first dive.  The next closest score was 78.00.  With four more dives to perform, Chen's competition faced an uphill climb that was simply too steep.  Chen's last dive was a back 2 1/2 somersault with 1 1/2 twists.  She needed only 30.61 points for gold.  She probably could have belly-flopped and earned gold!  Instead, she earned 86.40 points, which matched her third dive's score.  Not bad for someone who won't be 20 years old until the end of the year!

I wrote an earlier post about how angular momentum conservation plays a crucial role in diving (click here for that post).  Here, I'll consider what happens to Chen after she enters the water.  On her last dive, I calculate that she left the platform with initial speed 3.34 m/s (12.0 km/hr or 7.47 mph) at 78 degrees from the horizontal.  Her time of flight was about 1.8 s.  She hit the water moving at about 14.4 m/s (51.8 km/hr or 32.2 mph).

Now consider how fast she would hit the pool bottom in the pernicious scenario that the pool water magically disappears after she dives.  The pool is about 5 m (16 feet) deep.  Her impact speed on the pool bottom would be 17.5 m/s (62.9 km/hr or 39.1 mph).  I obviously never want to see that dive!  The point here is that a diver is moving quite fast when hitting the water, and would be moving even faster if the water wasn't there.  Even though diving is an "aquatics" event, most of the scoring happens above the water.  Except for the entry and splash, the pool serves no purpose other than to slow the diver down.

I've written about drag before, but mostly in the context of projectiles moving through air.  Water is about 800 times more dense than air.  Consequently, water is capable of much larger drag forces.  Divers sometimes angle themselves in such a way that they curve through the water after entry.  Other times, they may go straight down until they just make contact with the pool bottom.

Besides drag, another important force on a diver while slowing down in the water is the buoyant force.  The density of the human body is roughly 1.062 times that of water.  That means that while completely submerged, Chen feels an upward buoyant force on her that is about 94% of her weight.  With a mass of just 47 kg (corresponding to a weight of 104 pounds), the upward buoyant force on Chen is approximately 434 N (97.5 pounds) while she is completely underwater.

As fast as divers enter the water, it's a good thing there is plenty of upward force from drag and buoyancy to slow them down over a relatively short distance.  Without the pool water, a diver would never make it to the medal stand!  That's surely obvious, but it's nice to think about exactly why and how the pool water functions in the sport of diving.

Will Chen Ruolin make it three in a row when the Summer Olympics hits Rio de Janeiro in four years?  I won't bet against her!

09 August 2012

Second Law of Thermodynamics with Goofy Dives and Pole Vaults

I love thinking about and writing about the physics of the sports world.  Dozens of Olympic events are almost too much stimuli for the nerdy brain in my head that can't stop searching for the physics behind nuances in motions, trajectories, strategies, and so forth.  Sometimes it's fun just to think about what's not there and what's not seen, even in the context of physics.  Allow me to illustrate that idea with a great physics law.

The second law of thermodynamics is one of my favorite topics in physics to teach.  It is not some stale equation or lifeless group of words.  Mathematics may be the language of the universe, but physics is the poetry.  Consider the following quote from C P Snow's famous book, The Two Cultures:

A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists.  Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics.  The response was cold:  it was also negative.  Yet I was asking something which is the scientific equivalent of:  Have you read a work of Shakespeare's?

In other words, know something about the second law of thermodynamics if you wish to consider yourself an educated person possessing scientific literacy.  Why?  Well, for starters, that law drove the industrial revolution, as well as technological developments today.  Imagine holding a lump of coal in the palm of your hand.  That lump of coal contains a certain amount of chemical energy, but we can only extract about a third of that energy for the task of performing work.  Two thirds of that energy is wasted in the process of getting that needed work, often as heat pollution in the sky or in a river.  Energy conservation, which is actually the first law of thermodynamics, doesn't help us understand why so much energy must be wasted, but the second law does.  The second law tells us about the direction in time certain things happen, and that going back to the starting point is impossible.  Once we extract useful work from a lump of coal and lose twice as much heat energy in the process, we can't "fix" the wasted energy and make it useful.

Okay, time for a couple of sports examples to make the second law of thermodynamics a little clearer.  Think back to this past weekend when Wu Minxia of China won gold in the women's 3-m springboard diving event.  She would jump from the diving board, execute a bunch of twists and rolls, and then enter the water with little splash.  Now, what if after entering the pool, you saw Wu Minxia emerge back from the water, do all the twists and rolls in reverse order and backwards, and then land back on the diving board.  Would you believe your eyes?  Or would you think someone hit "rewind" on your television?  The movie doesn't run backwards in real life, right?!?  Of course not!  But, and here's the great thing about it, the law of conservation of energy would not be violated if you saw the dive undo itself!  No energy would be created or destroyed if the energy from the sound wave you heard from the splash, the heat energy of the pool and diver, and the energy of the undulating waves, all recombined back into kinetic energy for the diver and sent her flying back up out of the water.

So why don't we see divers pop back up out of the water and onto the diving board?  The second law of thermodynamics is what provides us with an "arrow of time," as it's often called.  Seeing the "movie" run backwards is so statistically improbable that we say it can't happen.  Disorder is created out of order during a normal dive.  Simply put, there are so many zillions of ways to distribute energy in the disordered post-dive environment that a diver will never return to her more ordered pre-dive environment.  Of course, she can get out of the pool and climb back up to the diving board so that everything "looks" the same, but it's not.  She burned chemical energy and did work to get back to the diving board.  Doing "work" is the key to getting back to the starting point, and that means that some energy is wasted.

Want an even simpler example?  How many possible ways are there for your bedroom to get messy?  How many possible ways are for it to be neat and orderly?  There are many, many more ways for the former to happen than the latter.  Over time, if you don't "work" on it, your bedroom gets messy, not orderly.  That's the second law of thermodynamics in action -- in your very own bedroom!

If the idea is starting to sink in, you'll won't be able to stop thinking of examples.  Did you watch Jennifer Suhr of the US win the women's pole vault event this past Monday?  Her gold-medal-winning vault took her 4.75 m (15.6 feet) over the bar.  Her mass is 64 kg (corresponding to a weight of 141 pounds) and at 1.83 m (6' 0") tall, her center of mass had to raise about 3.75 m (12.3 feet) to clear the bar.  Her gravitational potential energy increase was about 2.35 kJ, but she also needed some kinetic energy to keep moving horizontally over the ball.  That might have accounted for another 5% - 10% more energy.  Where did all that energy come from?

First, Suhr needed to converted stored chemical energy in her body to kinetic energy as she ran faster and faster leading up to her vault.  She lost a little energy along the way to air resistance and track friction.  When she planted her pole on ground and it began to bend, while at the same time elevating her upward, energy was transferred into potential energy in the pole, much like a compressing spring, and gravitational potential energy.  As the pole began to straighten out, like a spring uncoiling, potential energy in the pole got transferred into more gravitational potential energy as she continued to move upwards.  After clearing the bar, gravitational potential energy was converted into kinetic energy as she fell, and that energy was converted into potential energy in the compressed mat, the sound wave you heard when she smacked into the mat, and heat in both her and the mat.

None of the aforementioned energy transfers are 100% efficient, meaning energy is wasted along the way.  She would have needed more than 2.35 kJ just before vaulting.  By the time she jumped off the mat, the kinetic energy Shur had just before her vault was no longer with her.  It was wasted, at least in the physics sense of "waste," meaning it wasn't doing useful work.

So, what if all that wasted energy were to recombine into kinetic energy for Jennifer Suhr and propel her back over the bar?  Energy conservation won't tell us why that can't happen, but the second law of thermodynamics can.

These diving and pole vaulting examples may seem goofy, and they are, but I've always found that the goofier something is, the better it is to remember.  Do you think you'll have trouble remembering something about the second law of thermodynamics after reading my silly examples?!?

08 August 2012

Field Hockey Semifinals

Field hockey in the London Olympics has now reached the semifinal stage for both men and women.  Both of my pre-Olympics picks, Australian men and Argentine women, are still alive.  The women's semifinal is on Wednesday; the men's semifinal is on Thursday.  There is so much interesting physics in the sport of field hockey that I couldn't possibly do justice to the entire sport in a single blog post.  I'll instead focus on one aspect of the stick.

I spent my sabbatical year working with Matt Carré in his Sports Engineering Research Group at the University of Sheffield in England.  One of Matt's research areas concerns field hockey sticks.  Check out the graph below that he sent me before the Olympics started (click on the image for a larger view).
The graph comes from the PhD thesis of one of Matt's students, Mark McHutchon.  The COR on the vertical axis is the coefficient of restitution, which helps measure the efficiency of energy transfer in the collision between ball and stick.  If COR = 1, no energy is lost during the collision, but that's completely unrealistic.  By "lost" I mean that the energy that goes into a sound wave (we hear the collision!), heat generated in the ball and stick, and some wave energy associated with stick oscillation cannot be returned.  Global energy amount remains the same, of course, but the energy associated with the ball and stick goes down after the collision.

The parameter on the horizontal axis is a measure of the stick's stiffness; moving left to right means a stiffer stick.  Note that the data suggest that more efficient transfers of energy from stick to ball are associated with stiffer sticks.  That means that if a player really wishes to smash the ball hard and have it leave the stick with a large speed, he or she should employ a stiff stick.

Ice hockey sticks are not quite so stiff, but of course ice hockey is quite different from field hockey.  The control players feel from their sticks is manifested in different ways in the two types of hockey.  In fact, when taking shots on goal, look for field hockey players to use the edges of their sticks instead of the sticks' faces.  That technique creates an effectively stiffer stick, which means better energy transfer, as the above graph shows.  Improving a stick's "second moment of area," which, loosely put, is connected to a stick's resistance to bending, is one way manufacturer's enhance stick design.

Field hockey can be a tough sport.  Pay close attention to those shots on goal -- they happen fast!  Some of the best players can whop the ball such that it leaves their sticks at around 120 km/hr (33.3 m/s or 74.6 mph).  Now that's fast!

07 August 2012

Strongest Man Alive!

My pre-Olympics pick for the men's +105-kg weightlifting competition was Behdad Salimikordasiabi of Iran, and he did not disappoint me today!  He showed unbelievable power in both the snatch and clean and jerk competitions.  At 1.97 m (6' 6") tall, I estimate that he held his lifted weight at a height of 7' (2.1 m) above the floor.  I'll take his snatch and clean and jerk lifts separately.
  • Snatch:   Salimikordasiabi tied Russia's Ruslan Albegov for greatest lift during the snatch at 208 kg (459 pounds).   Salimikordasiabi took about 3 s to lift the weight.  Simply converting his chemical energy to gravitational potential energy means that he outputted 4.35 kJ.  Assuming an energy efficiency of about 20%, that means that he burned about 5 Calories.  That energy efficiency may be a bit high, but it's not a bad estimate.  Over 3 s,  Salimikordasiabi had a power output of 1.45 kW or about 2 hp.
  • Clean and Jerk:   Salimikordasiabi led all lifters with a clean and jerk of 247 kg (545 pounds). It took him about 8 s to lift the weight.  His energy output was 5.16 kJ.  Using the same 20% energy efficiency, he burned at least 6 Calories.  Given that he had to hold the weight on the top of his chest after the clean phase, he surely burned another Calorie while his muscles were tensed.  One of my book chapters is devoted to energy in the body (in the context of sumo), and I discuss how work is done even though no macroscopic object, like the weight, is displaced.  His power output needed to lift 247 kg in 8 s was 646 W or about 0.9 hp.
Stop and think about those numbers for a moment.  Salimikordasiabi's 2-hp output on the snatch was double what the women's sprint cyclist did over 200 m of biking (click here for my post on women's sprint cycling).  His power output during the snatch was a little greater than a standard microwave oven and essentially the same as the power on each square meter of Earth's area from sunlight!  The roughly 12 Calories he burned during his two gold-medal-winning lifts amounts to three Diet Cokes.

For the 3 s he needed to lift 208 kg, and in a square meter of area where he did his lifting, Salimikordasiabi was as powerful as the sun in that same square meter!

Meares Beats Pendleton!

My pre-Olympics pick for the women's sprint had to settle for the silver medal.  Anna Meares of Australia won the gold-medal final against Victoria Pendleton of Great Britain.  Pendleton actually won the first race in the closest photo finish I've ever seen.  Because she ever-so-slightly came out of her sprint lane, Pendleton was relegated, giving Meares the first race with a time of 11.218 s.  That works out to an average speed of 17.829 m/s (64.183 km/hr or 39.881 mph) over the 200-m (656-ft) sprint.  Meares was originally given 0.001 s off Pendleton's time before Pendleton was relegated.  Multiplying average speed by that time difference gives about 1.78 cm (0.702 inches).  The photo finish looked even closer than that, making me wonder if the 0.001 s time difference was not simply the smallest value the timers could give.

Meares and Pendleton slowed to almost a stop at the top of the track during their second race. Pendleton decided to break out early, but Meares kept pace.  Once the tune was heard to sprint, Meares was close enough to draft and conserve a little energy.  She then moved out of Pendleton's slipstream and sprinted to gold.  It was great strategy!

It's too bad that the first race between Meares and Pendleton ended in a relegation, especially given the amazing photo finish.  It would have been nice to see a third race between those two legends of the velodrome.  Congratulations to Anna Meares for a great ride!

06 August 2012

Tuesday Preview #3: Women's Sprint Cycling

This is the third of three "Tuesday Preview" posts (click here for my first post on men's discus and here for my second post on men's high jump).  This third post, and final of this evening because my fingers are about to fall off from typing, concerns women's sprint cycling, which has its final on Tuesday.

My pre-Olympics pick for women's sprint cycling was Victoria Pendleton of Great Britain.  During the 200-m (656-feet) time trial qualification this past Sunday, Pendleton broke her own Olympic record with a time of 10.724 s.  That works out to an average speed of 18.650 m/s (67.139 km/hr or 41.718 mph).  Using the same physics in my Tour de France model, I estimate that Pendleton had a power output of about 0.77 kW, which is just about 1 hp.  If I further estimate that her body is 25% efficient in its energy conversion, I find that she burns about 8 Calories during her nearly 11-s ride.

What represents 8 Calories?  The chemical energy content in two cans of Diet Coke.  Where have I used Diet Coke before?  When I wrote about archery (click here for that post).  The 72 arrows fired by competitors during the ranking round requires about 4 Calories of internal energy burn, which is one Diet Coke.  Pendleton thus doubles the internal energy burn in about 11 s what archers burn over the time it takes for them to fire 72 arrows.

When you watch a world-class athlete cycle hard across 200 m, know that you are witnessing about 1 hp of output.  Impressive!  I'm sticking with Victoria Pendleton for Tuesday's final, but watch out for Australia's Anna Meares.  She can fly, too!  Meares took silver to Pendleton's gold in the last Olympics in women's sprint cycling.  The bronze medalist in Beijing four years ago?  Guo Shuang of China, who did not win gold on her own soil.  Will Victoria Pendleton have to listen to someone else's national anthem on her home soil, just as Guo Shuang had to do four years ago?  It should be exciting!

Tuesday Preview #2: Men's High Jump

I got enamored by the men's high jump in this year's Olympics because I was asked to work on a story that now appears in the current issue of Outside magazine (click here for a link to the article).  The article concerns Jesse Williams of the US, who is my pick to win Tuesday's high-jump final.

What intrigues me about Williams is that at 1.84 m (6' 1/2") tall, he is a bit shorter than most of his competition.  All jumpers, regardless of height, have to clear a bar that is some distance above the ground.  Taller jumpers have the advantage of possessing centers of gravity that are higher at the start than for a jumper like Williams.  It takes energy to elevate mass vertically upward.  Kinetic energy obtained in the approach and muscle energy released upon jumping are partially converted at the apex of the jump to gravitational potential energy and a little kinetic energy needed to pass over the bar.  All jumpers make use of the Fosbury Flop, which allows their centers of gravity to pass under the bar as they are contorted in a Ç shape over the bar.

How can Williams make up for starting with his center of gravity lower than most of his competition?  Note that as athletes approach the bar and prepare to jump, they do not run in a straight line to achieve sprint speeds.  Instead, they circle into the jump.  Williams, because of shorter limbs, is able to accelerate better than his longer-limbed competitors.  I discussed scaling and notions of top acceleration favoring short people and top speed favoring tall people when I discussed the 100-m sprint (click here for that post).  Because jumpers don't reach top speeds before they leap from the ground, someone like Williams may use his acceleration advantage to help his speed at takeoff.  It turns out that Williams is 3% -- 12% faster on his approach than most of his competition.

Williams also has slightly less lean at the jump compared to others.  His powerful leg muscles and strong core enable him to initiate just the right amount of twist upon leaping.  Why does he need to twist?  He must cross over the bar with his back closest to the bar.  In a sense, he "rolls" over the bar.  That he and others can execute such a move is due to angular momentum conservation, something I've mentioned in other posts (click here for gymnastics and diving, here for long jump, and here for shooting).  Once athletes leap from the ground, their angular momenta are fixed.

The men's world record is 2.45 m (8' 0"), set by Javier Sotomayor of Cuba in 1993.  At 1.95 m (6' 5") tall, Sotomayor combined great height and great technique to set a record that is now 19 years old.  Williams has been clocked at 7.59 m/s (27.3 km/hr or 17.0 mph) on his approach.  To elevate over the bar, Williams needs about 5.2 m/s (19 km/hr or 12 mph) vertical launch speed to match his personal best jump of 2.37 m (7.78 feet).  To match Sotomayor, Williams will need to increase his vertical launch speed by about 3%.

The women's high-jump final is not until Saturday.  My pick there is Blanka Vlašić of Croatia.  At 1.93 m (6' 4") tall and possessor of great technique, she is quite the sight to behold.  As for tomorrow, I'll root for Williams and fellow Americans Erik Kynard and Jamie Nieto.

Tuesday Preview #1: Men's Discus

My pre-Olympics pick to win the men's discus throw was Robert Harting of Germany.  He's in good shape after coming in second in the qualifying round with a throw of 66.22 m (217.3 feet), just behind the 66.39 m (217.8 feet) thrown by Gerd Kanter of Estonia.  The world record for men's discus throw was set in 1986 by Jürgen Schult of Germany at 74.08 m (243.0 feet), which is the longest standing world record for men's outdoor track and field events (click here for the list).  Harting and Kanter will need to increase their throw distances by almost 12% to match Schult's 26-year-old record.

So, what do Harting, Kanter, and the rest of the field need to do to match Schult?  The discus fascinates me enough that I devoted an entire chapter to the event in my book.  My focus in that chapter was on the great Al Oerter, who won discus gold in four straight Olympics (1956, 1960, 1964, and 1968).  Consider the graph below (click on the image for a larger view).
The above graph appears in my discus chapter.  The angle on the horizontal axis is the launch angle, which is measured with respect to the ground.  The angle on the vertical axis is the discus inclination, which is the angle between the long axis of the discus and the ground.  The launch speed for all throws in the above graph is 25 m/s (90 km/hr or 56 mph).  The three "islands" represent all possible values of the two angles that allow the discus to land within 1 m (3 feet) of the range given in the middle of the island.  The top island has a tailwind of 10 m/s (36 km/hr or 22 mph); the middle island has no wind; the bottom island has a headwind of 10 m/s.

Wait, can that be right?  The discus can be thrown farther with a headwind than with a tailwind? It turns out that a tailwind hurts the lift force on the discuss because the air speed is not as great over the discuss as with a headwind.  A headwind allows the discus to act like a sail and stay in the air longer.  Too much headwind, though, increases the drag enough that added lift won't matter.  Note that the world record is sitting in that bottom island, which has the smallest room for error to get the discus to within 1 m of the record.  What Harting and Kanter have done so far in London is essentially in the middle island (assuming they had no wind on their throws).

Harting, Kanter, et al will need a little help from the wind and phenomenal technique if they want to match Schult's record.  Harting has thrown just over 70 m in his career, but reaching Schult will be tough.  I'll still go with Harting for gold on Tuesday.

US women win FANTASTIC football game!

What an incredible semifinal match between the US and Canada!  Canada's Christine Sinclair put her country up 1-0 by halftime.  She would add two more great goals in the second half to complete a hat trick in a stunning performance.  The player who saved the US in the second half, however, was Megan Rapinoe.

In the 54th minute, Rapinoe hit a beautiful corner kick from the left side that had lots of counterclockwise spin (as seen from above).  The Magnus force on the ball pushed it left toward the goal.  The Albert, as the ball is known, sneaked in between a defender's legs and then off the goalkeeper's knee.  Rapinoe got the goal all by herself from a corner kick!

In the 70th minute, Rapinoe got the ball right of center, just outside the penalty area.  She then smashed a "knuckleball" that wobbled its way toward the left goalpost.  It ricocheted off that post and went into the right side of the goal.

In the span of about 16 minutes, my girls' favorite player, Megan Rapinoe, had delivered two equalizers for the first two of Sinclair's goals.  Each of Rapinoe's goals was of a different spin variety that I mentioned in my previous post (click here for that).

Abby Wombach scored a penalty kick that equalized Sinclair's third goal.  After a scoreless first extra-time period, the second extra-time period ended with no goals.  Three minutes of injury time turned out to be just enough as Alex Morgan nailed the perfect header into the upper portion of the goal for the game-winner, just past the outstretched gloved hand of Canadian goalkeeper Erin McLeod.

Canada played a terrific game, and they should be congratulated.  The US moves on to the goal-medal game against Japan on Thursday.  It will be a rematch of last year's World Cup final.  Let's hope this time around, penalty kicks won't be needed!


Some say, "Football!" Some say, "Soccer!"

Each time I check scores on a US website or listen to a US sports commentator, I read or hear about soccer in the Olympics.  Given where the Olympics are being played right now and given that the international governing body is the Fédération Internationale de Football Association (FIFA), football might be the better term to use.  What one calls the beautiful game is, of course, not nearly as important as the action taking place on the pitch (or field, if you prefer).

My girls and I have been following the football action.  They are, no surprise, more interested in the women's tournament than the men's.  We plan to watch the US team take on Canada later today in the semifinals.  The winner plays the winner of the match pitting Japan against France. My girls love the big stars like Amy Wambach and Hope Solo, but they especially enjoy the all-out white-haired enthusiasm expressed in the play of Megan Rapinoe.  Of delight to them as well is The Albert, which is the ball being used in the Olympics.

Adidas created a special Tango 12 series ball for the London Olympics, and it was named The Albert.  It is easy to find online close-up images of the ball (click here for one example).  According to Adidas, the new ball's woven carcass and improved bladder are capable of holding its air better and keeping water out of its interior.  The latter feature is especially important for matches in Great Britain where summer rain is not so uncommon.  Football designers do their best to make balls as spherical as possible.  The balls in the Tango 12 series have 32 thermally-bonded triangles with a grip texture.  As someone who does research on footballs, those details are important to me.  For my young girls, they love the color scheme.

Football trajectories are of special interest to me, and I pay close attention to the ball's flight on corner kicks and shots on goal.  If you see a shot with lots of sidespin, imagine viewing the ball from above.  The Magnus force due to the counterclockwise spin often imparted by a right-footed player causes the ball to deflect to the left.  Clockwise spin is associated with balls getting deflected to the right.  Those "banana" kicks are tricky for goalkeepers.  Even trickier are the low-spin "knuckleball" kicks.

The Magnus force comes about because the boundary layer of air around the ball separates asymmetrically (as defined by the line of the velocity vector) from the back of the ball.  Think of what a boat's rudder does to the water moving around a boat's hull.  There is no sideways deflection if you see a symmetric wake behind a boat.  Turning the rudder, however, causes the wake to deflect to one side, allowing a component of the water's force on the boat to push the boat to one side.  Think about Newton's third law.  If the rudder pushes water in one direction, the water has to push on the rudder in the opposite direction.

The low-spin "knuckleball" kicks present slowly-varying profiles of the ball to the oncoming air.  Because of the places where the panels are joined, there are parts of the ball that are rougher than others.  Rough area usually delay the boundary layer's separation.  Allow the ball to slowly rotate and the the deflection direction changes.  Tough for the goalkeeper!

Air drag is the force that slows the ball down during its flight.  Those rough areas on the ball help reduce drag.  If that seems counterintuitive, check out the graph below (click on the image for a larger view).
That image comes from an invited article I wrote for Physics Today before the 2010 World Cup (click here for the article or here for the Japanese version in the magazine Parity).  The drag coefficient is plotted versus Reynolds number, which is proportional to the ball's speed (to convert to speed, note that the "2" on the horizontal axis corresponds to a ball speed of about 14.1 m/s or 31.5 mph).  What you see is that in the speed range for football, a perfectly smooth ball would have a lot more drag on it.  The two inset images show that as speed increases through the "drag crisis" the boundary layer's separation ultimately moves farther back on the ball, leading to a smaller drag coefficient.

The above is but a flavor of the wonderful physics behind the beautiful game.  My pre-Olympics football picks were US on the women's side and Brazil on the men's side.  Both teams are still alive.  My girls and I will enjoy great semifinal action on the women's side today as we root for our home country!

05 August 2012

Lightning Bolt Strikes Again!

I just watched the men's 100-m sprint final, and it happened exactly as I thought it would (see my previous post by clicking here).  Bolt was in lane 7 with Justin Gatlin to his left in lane 6.  To Gatlin's left in lane 5 was Yohan Blake.  Bolt's reaction time (0.165 s) was faster than both Gatlin's (0.178 s) and Blake's (0.179 s), but Gatlin had the early lead out of the blocks.  The shorter Gatlin at 1.83 m (6' 0") tall had greater acceleration than the taller Bolt, who is 1.95 m (6' 5") tall.  By the time the sprinters were approximately halfway home, they were upright and at top speed.  Gatlin had the acceleration advantage, as scaling arguments tell us, but Bolt had the top speed advantage.  Bolt had enough distance left to ensure that his great top speed would give him the gold.

Usain Bolt broke his own Olympic record with a time of 9.63 s, which is 0.05 s slower than his world-record time.  The 1.80-m (5' 11") tall Blake was able to pass Gatlin at the end for the silver medal with a time of 9.75 s.  Gatlin's bronze-medal time was 9.79 s, just 0.01 s ahead of fellow Americans Tyson Gay and Ryan Bailey.

Scaling laws are powerfully simple approaches to understanding certain aspects of nature.  The acceleration advantage Blake and Gatlin had with shorter legs got them off to a quick start, but the long legs of Bolt helped give him the top speed that ultimately prevailed.  A complete understanding of the race is more complicated, of course, but it's great to see the predictive power of (relatively) simple physics.

Lightning Bolt made sure that Jamaica would sweep the men's and women's 100-m sprints!

Later Today: Fastest Human

The men's 100-m sprint final will be held later today.  The winner is often referred to as the world's fastest human.  Shelly-Ann Fraser-Pryce of Jamaica was crowned world's fastest woman yesterday as she edged my pre-Olympics pick, Carmelita Jeter of the US, by 0.03 s to take gold with a time of 10.75 s.  I was intrigued by Jeter's training after reading a Washington Post article last February that described how science and technology were helping shave tenths of a second off Jeter's 100-m time (click here for the article).  Jeter began training with John Smith about three years ago and made use of Ralph Mann's video analysis work at CompuSport.  Video of Jeter's sprints would be compared against a simulated sprinter racing with optimum technique.  Jeter was able to knock about 0.4 s off her personal best by making using of sports science.  Jeter had the best semifinal time of the London Olympics, and even improved on that by 0.05 s in the final, but Fraser-Pryce beat her own semifinal time by 0.10 s to win gold.

On the men's side, I made a bold prediction before the Olympics.  I claimed that the winner of the men's 100-m sprint would be Jamaican with an A in his first name and an L in his last name.  Okay, with current world-record holder Usain Bolt being joined by fellow countrymen Asafa Powell and Yohan Blake, my prediction was hardly the stuff of the clairvoyant!  What intrigues me about those three men is their height.  Bolt is 1.96 m (6' 5") tall; Powell is 1.91 m (6' 3") tall; Blake is 1.80 m (5' 11") tall.  Scaling arguments tell us that long legs are good for top speed, whereas short legs are good for acceleration.  If Blake makes the final, I'll be interested to see if he is able to get an early lead because of a possible acceleration advantage.  I'll then be interested to see if he can sustain a lead while longer-legged competitors possibly gain on him.

Making predictions for the 100-m sprint is difficult because there are so many factors at play.  One challenging aspect of the race for sprinters is the false start rule.  Two years ago, the IAAF (International Association of Athletics Federations) created the rule that a sprinter who pushes off the starting block less than 0.10 s after the gun goes off is disqualified.  There are no second chances!  Athletes risk disqualification if they try to anticipate the gun.  Starting blocks are electronically linked to the gun, meaning humans don't make disqualification decisions.  Usain Bolt was disqualified in last year's world championships after his false start.

Given the electronic linking of the blocks and guns, it is easy to determine athletes' reaction times.  Those times are typically in the range of 0.15 s - 0.20 s.  For a race that may be decided by hundredths of a second, reaction time could make all the difference between saluting one's flag from the medal stand and going home empty.

Once the race begins, note that sprinters are not completely upright as they accelerate out of the blocks.  They do not reach top speed until they've traveled a distance of 50 m - 60 m.  Being completely upright at top speed, the real skill in the final portion of the sprint is maintaining top speed.  Much of a sprinter's training focuses on the ability to hold off on slowing down for as long as possible.  Powerful muscles help supply energy at a sufficient rate to help top sprinters maintain top speed.  Those guys are not ripped for looks alone!

Being from the US, my rooting interests lie wih Ryan Bailey (height:  1.93 m or 6' 4"), Justin Gatlin (height:  1.83 m or 6' 0"), and Tyson Gay (height:  1.80 m or 5' 11").  When you watch the final of the 100-m sprint, look for the athlete that gets out of his blocks first.  Then, look to see if one of the shorter athletes (perhaps Blake, Gatlin, or Gay) is able to accelerate to the early lead.  Finally, take notice of top speeds and who is best able to sustain his top speed.  Will one of the taller athletes (perhaps Bolt, Powell, or Bailey) sustain the greatest top speed?

I'll write more on scaling arguments and different body types before the men's high jump final on Tuesday.

04 August 2012

Nice shooting, Jamie!

Jamie Gray of the US won the women's 50-m (164-feet) rifle three positions event today.  She was definitely the one I was rooting for today.  My long-shot (pardon the pun!) pick before the Olympics was Sonja Pfeilschifter of Germany, who still holds the finals record that she set in 2006.  If Jamie Gray (age 28) couldn't win, I wanted to see someone my own age (41) win!  After Gray set the Olympic record in the qualifying round with a score of 592, she won gold with a finals score of 691.9.  Pfeilschifter finished 19th in the qualifying round and did not make the finals.

Athletes competing in the 50-m rifle three positions event use a 0.22-caliber (5.6 mm) small-bore, single-loader rifle.  They shoot in prone, kneeling, and standing positions.  Rifle barrels are grooved (called "rifling") so that bullets emerge from the barrel with spin along their longitudinal axis.  Why?  Conservation of angular momentum -- again!  Bullets, like people on moving bicycles, are more stable because of rotation.  Changing a large angular momentum requires a large torque.  That is why it is easier to stay on a moving bicycle than on a stationary one.  It's also how American football quarterbacks reduce air drag on their passes, by spinning the ball along its longitudinal axis, exactly analogous to the fired bullet.

Bullets emerge from the rifles at speeds around 331 m/s (740 mph), which isn't too far from sound speed.  Air drag slows them so that they enter the target with a speed around 302 m/s (676 mph).  I modeled the flight of the bullet with air drag and found that the bullet takes about one-sixth of a second to reach the target.  Gee, where has that time of one-sixth of a second come up in the recent past?  In my table-tennis post (click here)!  A smashed table-tennis ball (initial speed of 20 m/s or 45 mph) with lots of topspin (120 rev/s or 7200 rpm) traverses a horizontal distance equivalent to the length of the table-tennis table (2.74 m or 9 feet) in a time of about one-sixth of a second.  If you want to visualize the bullet's time of flight in "slow motion" on a "reduced scale," watch a table-tennis smash.

So, do you think you could hit the competition target from a distance of 50 m?  Consider the image of the target given below (click on image for larger view).
The outermost circle, which defines the target's size, is 154.4 mm (6.08 in) in diameter.  Just hitting the target means shooting inside an angle of 1/6 degree, which is 10' arc.  Not much room for error, is there?!?  Well, if you want to hit the bullseye, which is defined by the penultimate interior circle, which has a diameter of 10.4 mm (0.409 in), you must fire within an angle of 1/84 degree, which is 43" arc.

In the US, swimmers and gymnasts get all the press during the first week of the Summer Olympics; track and field takes center stage in the second week.  Think about how difficult it is to win gold in a shooting event, and then remember the name Jamie Gray!

Today's Men's Long-Jump Final

I plan to watch the final of the men's long jump later today.  Ever since hearing stories about Bob Beamon's epic jump in Mexico City on 18 October 1968, nearly two years before I was born, I have been enthralled by the long jump.  I even devoted a full chapter to Beamon and the long jump in my book.

As you watch the long-jump final today, keep an eye out for some great physics.  Just before launching off the ground, athletes achieve sprint speed, nearly 10 m/s (36 km/h or 22 mph).  They want to jump with a large velocity component parallel to the ground so as to maximize jump distance, but they need lots of air time, which happens only with a large vertical component of the launch velocity.  Jump too horizontal and the athlete isn't in the air long enough to go far.  Jump too vertical and the athlete might be able to dunk a basketball, but won't win a long-jump medal.

Basic vacuum kinematics tells us that if launch and landing heights are the same, an angle of 45° will maximize a point particle's horizontal range.  That launch angle represents a perfect balance between launching with plenty of horizontal velocity to get good distance and launching with plenty of vertical velocity to get good hang time.  For a long jumper getting ready to leap from terra firma, launching oneself at 45° to the horizontal is nearly impossible because the athlete has so much forward momentum just before leaping.  Athletes typically jump in the neighborhood of 16°28°.  It turns out that it's more important to achieve as great a launch speed as possible than it is to launch at exactly the optimum angle.

You may be tempted to think that if a jumper launches well below 45° that the jump will be mediocre, but hold on!  That wonderful conservation of angular momentum law that I've discussed before helps athletes make up for a lack of vertical launch velocity.  Notice that athletes land in the pit with their centers of gravity at a height below where they started at the launch.  By landing low in the pit, athletes may add approximately 1 m (~3 feet) to their horizontal distance.  How can they do that if their mechanical energies and angular momenta are essentially fixed at takeoff?

Watch what long jumpers do with their arms while in the air.  By thrusting their arms down and backwards, they rotate their upper bodies in such a way that their heads are rotating toward the ground.  Strong cores and well-conditioned abdominal muscles are crucial for powerful in-air rotations.  Because their angular momenta are conserved, long jumpers' legs and lower bodies have to rotate in the opposite direction, meaning their legs rotate up.  Compared to landing standing up, as they were at takeoff, athletes are able to extend their flight times and land at greater distances.

Pay close attention to what long jumpers do while in flight.  They are not merely flailing their arms and legs in random directions.  There is serious technique involved  and lots of great physics!

My girls and I will root for Will Claye and Marquise Goodwin.  Brits will be pulling for Greg Rutherford and Chris Tomlinson.  Australia's Mitchell Watt was my pre-Olympics pick, so I should stick with him as my pick.  Godfrey Khotso Mokoena of South Africa and Brazil's Mauro Vinicius da Silva could also make some noise.  The latter and Goodwin both jumped 8.11 m (26.6 feet) in qualification.  That's still a long way from Beamon's Olympic record of 8.90 m (29.2 feet) and Mike Powell's world record of 8.95 m (29.4 feet), which he set in Tokyo on 30 August 1991.

Will a long jumper distance himself from his competition with a Beamonesque effort today?  To get close to Beamon and Powell, an athlete will need to jump nearly a first down (10 yards or 9.144 m) in American football!

02 August 2012

Shakehand Beats Penhold in Table Tennis!

My pre-Olympics pick for the gold in the men's singles table tennis event was Wang Hao of China.  My pick fell short today as fellow countryman Zhang Jike defeated Wang in the final match, four games to one.  The final point came as Zhang smashed Wang's serve back into Wang, who was to the left side of the table, and Wang miss-hit the ball up into the air.  That the gold-medal point went to Wang's left is significant, as I'll return to in a moment.  Zhang becomes the fourth table-tennis player to win the Olympic gold, World Championships, and World Cup, something Wang would've done had he defeated Zhang today.

Okay, so what's so special about Wang's left side?  Wang uses a penhold grip (we in the US might call this an "upside down" grip) whereas Zhang employs a shakehand grip.  As someone who uses a penhold grip himself, though infinitely less skilled than Wang Hao, I picked Wang for gold.  Some have criticized those who use penhold grips because of weakness on the backhand side.  What intrigued me about Wang's approach is that he uses both sides of his racquet (my racquet has only one playing surface).  Penhold-grip users are capable of imparting enormous amounts of spin with their forehand shots.  Lots of friction between the ball and the racquet's spongy surface generates large torque on the ball, which creates significant angular acceleration to cause the ball to leave the racquet with enormous spin.  The blue forehand side of Wang's racquet is more spongy and tackier than the red-colored backhand side.  Because he is more comfortable hitting with his forehand, Wang uses a backhand surface that gives him more control due to the inability of the surface to create as much spin as the forehand side.  But, alas, today was a day for the shakehand users as Zhang earned his well-deserved gold with a dominating performance.

Table tennis balls have a mass of only 2.7 grams (about 0.095 ounces in weight).  Because they are so light, air forces greatly influence the ball's trajectory.  Consider a forehand smash that imparts topspin to the ball.  Typical values for initial speed and spin are 20 m/s (45 mph) and 120 rev/s (7200 rpm), respectively.  I plot below the aerodynamic forces on just such a smashed ball as it travels a horizontal length of the table, which is 2.74 m (9 feet) long.  Click on the graph below for a larger image (so that everyone's happy with game labels, I label my graph with the term "ping pong" for table tennis).
By the time the ball has moved 2.74 m in horizontal distance, its speed has been reduced to 14 m/s (32 mph), a reduction of 30%.  The terminal speed for a table tennis ball is approximately 8.8 m/s (20 mph), meaning the ball speeds players witness are often well past terminal speed.  Note in the above graph that the initial drag force is just over five times the ball's weight.  Imagine having wind hit you with a force five times your own weight!  The Magnus force on the ball, due to its spin, is initially more than twice the ball's weight.  The Magnus force has a downward component for a ball with topspin, like a curveball in baseball, which means that the air pushes the ball down.  By the way, the buoyant force on the ball due to the ball displacing air is only about 1.5% of the ball's weight.  When the ball reaches the opposite side of the table, the topspin causes the ball to accelerate off the table due to the friction force between ball and table.  Good luck if you ever face a topspin from the likes of Zhang or Wang!  Oh, and don't forget that the ball takes only one-sixth of a second to traverse a horizontal distance equal to the table's length.  How are your reflexes?  It is easy to see why players often stand a good bit back from the table when smashes are going back and forth.

The next time you play table tennis and put lots of spin on the ball, note carefully how the ball moves.  Lots of wonderful physics in that motion!

01 August 2012

Gymnastics, Diving, and Angular Momentum

After watching the US women's team win gold yesterday in my office, I had to keep a straight face as my girls and I watched the NBC replay last night (finished much too late for young kids -- oops, I wasn't supposed to comment on NBC again).  When we saw McKayla Maroney nail her difficult vault, we knew the US women would not be stopped.  It is impossible for me to watch sports like gymnastics and diving, for example, and not think of the beauty of angular momentum conservation.  That powerful law in physics helps us discuss the motion of galaxies, nuclei, and everything in between, including athletes.

Newton's first law tells us that an object moving in motion at constant velocity (could be zero -- doesn't matter) will remain in motion at that constant velocity unless acted upon by a net, external force.  Newton's second law gives us the mathematical machinery we need to calculate trajectories when point masses are subjected to forces.  The rotational analog of Newton's laws usually requires us to think of extended objects instead of point particles.  In such an instance, we think of angular momentum, which, loosely put, is the product of an object's moment of inertia and its angular velocity.  The former quantity gets big when mass is moved away from a rotation axis and small when mass is pulled in toward the rotation axis.  A net, external torque, which, again loosely put, is the product of the net, external force with the lever arm distance to the rotation axis, is required to change an object's angular momentum.

When Maroney left the springboard, she had all the angular momentum she was going to have while in flight.  She was, however, able to twist and turn by altering her moment of inertia.  To keep angular momentum conserved, the product of moment of inertia with angular velocity must remain constant, but each may change as long as the other changes in such a way as to keep the product the same.  By rotating her arms and torso in special ways, McKayla Maroney was able to dazzle the world with her sensational vault.

When Viktoria Komova of the silver-medal-winning Russian team, for example, earned one of the top scores on the uneven bars, angular momentum conservation was on full display.  She began her dismount motion with large swinging circles on the top bar with her body completely elongated, giving her a large moment of inertia.  Once she let go, her angular momentum was set.  She twisted and turned in the air before landing.  Watch the replay and you'll see a slight twist as she lets go, then her arms rotate in such a way as to initiate more twisting motion.  Analogous to a speed skater's finale, when Komova pulled her knees in close to her torso, her rotational speed increased.  The US team lost only the uneven bars, coming in third behind second-place Russia and first-place China.  If they could all fly like Viktoria Komova, Russia would have won uneven bars in a landslide.  Unfortunately for all other countries, McKayla Maroney, Kyla Ross, Jordyn Wieber, Gabrielle Douglas, and Alexandra Raisman were simply too dominant to be denied the gold.

My girls and I also saw the final of the women's synchronized 10-m platform diving event.  Like their male counterparts, the Chinese women's team was methodical and spectacular in earning gold.  The great Chen Ruolin and her partner, Wang Hao, put on a memorable show.  If you are new to diving, and not familiar with Chen Ruolin, keep an eye on her.  She is as good as it gets.  From a height of 10 m (33 feet), divers hit the water in just under two seconds with speeds around 14 m/s (50 km/hr or 31 mph).  For divers Ruolin and Hao to be in sync so well for such a short time and at great speeds is remarkable.  To be so in sync with their difficult dives is worthy of gold.

There is a rule in diving that you may not know.  Here is Fédération Internationale de Natation (FINA) rule D 8.4.6:

In dives with twist, the twisting shall not be manifestly done from the springboard or platform. If the twisting is manifestly done from the springboard or platform, each judge shall deduct ½ to 2 points, according to his opinion.

Now that's some rule!  Divers can't get help from the platform when they want to twist and turn.  They rely on physics!  Watch an expert like Chen Ruolin and you'll see her throwing her arms and bending at the waist in an effort to get her body to spin.  Once she leaves the platform, her angular momentum is fixed while in the air.  She may throw one arm over her head and the other might be thrown outward.  That gets a twist going.  Twists have the smallest moments of inertia, followed by tucks, pikes, and layouts (to keep this from getting too technical, I'm glossing over the rotation axes about which those moments of inertia are measured).  For fixed angular momentum, the previous list is the order of rotational speed from largest to smallest.  Divers want a layout position when entering the water because it has the smallest rotational speed, and hence gives them the most control.  Layouts also help minimize splash, that is if the timing is just right to get the diver into the water with her body perpendicular to the water's surface.  The next time you watch diving, look for the motion of the divers' arms and torsos.  Watch the accompanying rotations.

Angular momentum conservation truly is a thing of beauty, and you can bet that I'll mention it again when writing about other sports.