11 March 2013

When one second feels like forever ...

My beloved Indiana Hoosiers basketball team faced a tough challenge yesterday.  We were to play a tenacious Michigan team on its home court with the Big Ten conference title on the line.  A loss, and we would tie with three other teams for the league crown, including Michigan.  A win, and we would have the title all to ourselves, something we've not had in 20 years.

As a myopic fan, my blood pressure rises and falls with each play.  Successes and failures are magnified to, admittedly, insane heights.  Any die-hard sports fan reading this knows exactly what I mean.  You know all the implications of winning and losing, and you know them well in advance of any game you watch.  If your team wins, you show up for work the next day with a big smile on your face, and you are dying to talk about your team's victory.  If your team loses, you get to work and feel like eyes are upon you.  Are there fans of the team that beat you who are anxious to "ask" you about the game?  Of course you want to talk about the loss, as if conversation can help you understand what happened.  This is all lunacy, but isn't is nice to have things in your life about which you feel so much passion?  After my family and my work, I'm thrilled being passionate about college basketball; I'm a true fanatic.

The game went back and forth.  We got up seven early; Michigan was up 11 with about five minutes left in the first half.  We cut that lead to three by halftime.  Both teams had bursts of excellence and seemingly longer stretches of futility.  The second half was a whirlwind of missed shots and mistakes by both teams.  There were also plenty of fantastic plays as each team kept claiming the lead.

Michigan was up five points with about 45 seconds left and I thought we had lost.  We had missed some shots and turned the ball over twice in the past minute and a half to get ourselves in a hole.  In that final 45 seconds, Michigan would not score again and Indiana's Cody Zeller would account for all six of IU's points.  We won by a single a point.  My kids and I were going bananas.

What made the game especially gut-wrenching for fans of both teams was the final possession by Michigan.  Click here to see our final basket and Michigan's attempt to win it.  Michigan's great point guard, Trey Burke, missed a tough shot near the basket.  Jordan Morgan was in perfect position for a tip-in, but the ball rolled on the rim for what seemed like an eternity, only to fall out of the hoop.

To understand what happened on the final play, I turned on the physics side of my head.  The image below shows the ball precariously perched on the rim after Morgan's tip-back (click on the image for a larger view).
"The Rock," as a basketball is sometimes called, is seen with the Michigan logo on it.  According to my timing, the ball rolled around the front of rim for a time of about one second.  A basketball has a diameter at least 9.39 in (23.85 cm) and at most 9.55 in (24.26 cm).  The black channels you see on the ball cannot be more than 1/4 in (0.635 cm) deep.  The metal ring that comprises the hoop is 18 in (45.74 cm) in diameter; the ring itself is about 5/8 in (1.59 cm) in diameter.

What forces did the "The Rock" feel while the IU/Michigan game was in doubt?  The ball's weight is between 20 ounces (5.56 N) and 22 ounces (6.12 N).  Taking an average weight and an average ball diameter, the buoyant force on the ball is about 1.5% of the ball's weight.  If the ball rolled on about 30% of the rim, the ball moved along 17 in (43 cm) of rim in roughly one second.  That results in an average speed of around 1.4 ft/s or 0.96 mph (1.6 km/hr).  Air resistance on the Michigan shot would have been small while the ball was on the rim, and it would have caused the ball to slow a little while rolling.  The normal force from the rim would have been comparable to the ball's weight.  Friction between ball and rim would have reduced both the rolling rate and the transnational speed of the ball, arresting the motion slightly to allow a gravitational torque to send the ball into the hoop.

What Michigan needed was a little more lever arm on the gravitational torque on the ball.  We are unstable if our center of gravity is outside our feet.  Had the line of the ball's weight vector been more in the hoop than through the rim or just outside it, Michigan would have won.  Of course, the torque created has to arrest motion that takes the ball out of the hoop.  Imagine a ball travelling fast over the hoop -- an air ball.  The ball's weight vector will pass through he hoop, but the ball won't go in because it would have been moving too fast over the hoop.

As passionate as I am about Indiana basketball, I never want to pour salt in an opponent's gaping wound created by a tough loss.  I've got good friends and even some family members who are fans of Michigan.  They are as sickened by the loss as I would be if the ball had gone in.  Agony and ecstasy came down to mere millimeters.  My team is lucky to have won.

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