I am currently teaching BIO 385 Sport and Exercise Biomechanics at National University in Carlsbad California. It’s an online course and we have a weekly discussion question. Last week it was on angular momentum. Here’s the question:
The total angular momentum of a diver changes from zero to some high value during the takeoff phase. Miller and Munro (1985) reported that Greg Louganis had an angular momentum at takeoff of 70 kg.m2/s in the forward three-and-a-half somersault dive performed in the pike position, an angular momentum of -54 kg.m2/s in the reverse two-and-a-half somersault dive in the pike position, and an angular momentum of 17 kg.m2/s in a straight forward dive. Note that the two multi-somersault dives are; the forward three-and-a-half and the reverse two-and-a-half.
1. What does a negative and positive angular momentum indicate?
2. Why does the angular momentum differ for these three dives?
3. Which of the two “multi-somersault” dives will be spinning the fastest? Explain why in terms of moment of inertia, angular momentum and angular velocity.
The quick answers are:
1. In this case negative means reverse spin and positive means forward spin, so the reverse 2 ½ has a negative angular momentum, the forward 3 ½ and the forward dive both have positive angular momentum.
2. The angular momentum differs because more angular momentum is required to complete more somersaults and even though the forward dive is in the straight position it still only needs a relatively low angular momentum because only ½ a somersault is completed. So we have 70 for the 3 ½, 54 for the 2 ½ and only 17 for the forward dive (½ a somersault).
3. The simplest answer is that the 3 ½ will be spinning faster than the 2 ½ because there are more somersaults to do in about the same amount of time. But looking at the angular momentum numbers we can also figure it out. 70 is higher than 54 and the diver is in the pike position in both of these dives, that means that his Moment of Inertia (MOI) is the same for each. Moment of Inertia is the resistance to rotation, a larger radius has generally has a larger MOI, so the layout has a larger MOI than the pike; but the 2 ½ and 3 ½ are both in the pike position so there MOI’s should be about the same (depending on if the both pikes were as tight as each other). Now angular momentum is MOI times angular velocity or:
H = I x w
H is angular momentum
I is moment of inertia, and
w is angular velocity
So if I is the same for both dives then:
H = k x w where k is constant
So in this case if H goes up w must also go up, so the 3 ½ at 70 will have a higher angular velocity than the 2 ½ at 54 since I is the same in both cases.
It was very interesting to discuss this idea and the idea of “Conservation of Angular Momentum” with my students. This concept is very important to any twisting and flipping skills such as dismounts in gymnastics, tumbling, trampoline, diving, ice skating, half pipe and aerials (skiing), to mention just a few.
The “Conservation of Angular Momentum Principle” basically states that once you are in the air and are no longer subject to external rotational forces (torques) then your total angular momentum will be conserved until you hit the ground or the water. But how do you generate the angular momentum in the first place? Well, you generate it off the ground, the board or the apparatus depending on the sport. A torque is created while in contact with some object. In the case of the diver the torque is created by the rebound force of the board pushing against the diver’s feet and the diver’s center of gravity will be either forward or backward of the point of application of the force creating a forward or backward torque. Now the torque acts for a period of time while the diver is in contact with the board and due to the impulse/ momentum equation; torque times time equals the change in angular momentum; the angular momentum of the diver will increase. This simply means that the diver gets all the angular momentum from the board during the take off; once in the air that is all he has. If the takeoff was poor then he hasn’t got much angular momentum (and probably not much height either) to work with. If the takeoff is good then he probably has all the angular momentum he needs.
Once in the air he cannot change the angular momentum but he can change his moment of inertia. He can hence manipulate his angular velocity (rate of spin). If he wants to spin fast he tucks tight (small MOI); if he wants to spin slow he stretches as tall as possible (large MOI).
So here’s a summary of the forward 3 ½ pike:
Good approach; high hurdle; strong board depression; slight forward lean to create a forward torque, while waiting for the board to propel him as high as possible; (note that there is a tradeoff between height and spin, too much height will reduce the spin and too much spin will reduce the height, but that’s a topic for another discussion); then once off the board he pulls into as tight a pike as he can (chest and legs compressed). Being in the pike speeds up his rotation because his moment of inertia is at the lowest it is going to be during this dive.
He stays in the pike as long as is needed to compete 3 ½ somersaults; opening out at the correct time to reach for entry. Coming quickly out of the pike to the stretched position increases his moment of inertia to its maximum value and slows the rotation speed down to its lowest value allowing the entry to look smooth and as if the rotation has almost stopped.
In summary, the trick to fast rotation is high angular momentum at takeoff. This angular momentum can be manipulated to speed up the rotation speed by tucking or piking or slow down the rotation speed by stretching into a layout position.
An interesting side note and a great topic for another discussion is that this angular momentum can also be transferred to the long axis of the body and hence converted into twist; so twisting can occur at any stage during the flipping, and without any influence from the ground or board. Maybe I’ll talk about that next time.