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.