PARTITION OF ENERGY IN DIVING Francis STROUP Northern Illinois University U.S.A. Although I realize that time is at a premium, I would be negligent indeed if I did not take a ~oment to express my gratitude for the opportunity to participate in this seminar and my delight in exchanging ideas with students of international eminence on subjects that have been of interest to me for nearly half a century. And of the subjects to be discussed here, the one that, through the years, has presented the greatest personal challenge to me - the one that has aroused my abiding curiosity - has been the cause of rotation around the frontal axis of the body in a diving performance. In fact, my concern with this subject has been so consistent for so long that my presentation today may sound more like an autobiography than a research report. As judged by the available literature, the conventional wisdom of the 1920's seemed to be that the way a diver ne1d his head on leaving the board or the way he flexed his joints after leaving the board were the sole causes of such rotation. As a persistent-- if not spectacular-- competitor during that decade, I was not convinced tnat the identification of tnese bodily actions was an adequate explanation of the cause of rotation. Newton's thiro law of motion seemed to demand that any rotation which required energy of such magnitude must have its origin in some relationship between the diver and the base from which he extended himself. Observation suggested that angular acceleration for a diver seemed to be related to the number of degrees from the vertical he was leaning as he left the board. But further reflection indicated that the degrees of lean could not be considered the sole determiner of angular acceleration because a long shallow dive involving little rotation can be effected with an extremely large angle of lean. The search for other contributing variables indicated that the direction of force which projected the diver into the air was independent of the direction of lean at the time of projection. The direction of lean and the direction of force converging at the feet of the diver seemed to form an angle that was positively related to the angular acceleration generated for the dive. For the purpose of our discussion, this angle is identified as angle theta. (9). The hypothesis relating angle theta to rotation could be explored by use of several homely illustrations. One of these illustrations was to balance a wand on a finger, allow it to lean from the vertical and then exert a force that would propel it through the air. The size of angle theta could be varied by exerting the force toward the center of gravity of the wand, slightly away from the center of gravity, or a greater distance from the center of gravity. With an apparent constant force, the degrees of revolution se~~d positively related to the magnitude of theta. 227. DIRECTION OF LEAN AND DIRECTIONS OF FORCE TO GIVE CLOCKWISE AND COUNTER-CLOCKWISE ROTATION. B c - Figure 1.- Direction of Lean and Direction of Force To Give Clockwise and Counterclockwise Rotation. 228. The wand demonstration proved to be a useful teaching device and any neophyte diver who sought my advice during the 1930's was likely to get the full wand treatment with illustrations of big thetas and little thetas and the corresponaing degrees of rotation. However, one limitation to the wand-on-the-finger illustration was that the student had no visual cue as to the direction of the force applied to the wand. The use of a string at the base to~ the wand gave visual evidence of the direction of force but precluded any demonstration of the effects of a theta of zero degrees. In order to produce a visible illustration of a force that could be directed through the center of gravity or to either side of it, a hanging ball struck by a wooden cue as in billiards, was devised. Such an arrangement provided tangible evidence of the direction of force and allowed variations in theta that would produce rotation of the ball in either direction or no rotation at all. The illustrations provided convincing evidence that rotation around the frontal axis of the body of a diver could be explained in terms of angle theta and I began to draw pictures in attempts to place the problem in geometric perspective. The pictures usually looked like Figure 1. In this Figure, the diver is at the free end of the board and leaning slightly forward. His feet contact the board at A and his center of gravity is at 0. Line AC through 0 indicates his direction of lean. If the force which projects him from the board is also in the direction AC, he will receive no angular acceleration from the force. However, if the force which projects him is in the direction AB, it will impart cloc~ise rotation ; and if the force is in the direction AJ, it will impart counterclockwise rotation. From this kind of picture it was possible to identify angle theta as the acute angle in right triangle AFO. With the constant hypothenuse AO, OF was proportional to the sine of angle theta and suggested that the rotation could be described quantitatively in terms of the sine of angle theta. The corresponding relationships prevailed in right triangle AEO with the distance EO proportional to the sine of a theta producin~ counterclochlise rotation. Due mainly to demands of employment and the fact that a war was in progress, my attention was largely diverted from the problems of diving from 1940 to 1950. But during this decade, two studies were reported in the United States which attempted to explain rotation. Fred Lanoue studied moving pictures of divers and hypothesized that the turning movement of divers is causes by two factors : 1. The amount and speed of bending at the waist in the direction of the desired turn, and 2. The angular speed witn which the center of gravity of the body is rotating around the toes at the instant of leaving the board. Calculations from these two factors allowed Lanoue to predict 12.1 per cent less than his observed degrees of rotation for tne back one-andone-half somersault and 4 per cent more than his observed degrees of ro229. 8= 20°- DIRECTION OF 9= 75° 8= 20:..01RECTION OF 9= 10° Figure 2.- Thetas with the same Magnitude but Different Directions. 230. ration for the half gainer. layout position. Both observations were of dives done in the Also from the study of motion pictures, Wi11iam Groves observed a positive relationship between the angle of lean from the vertical and the degrees of rotation in divers performing forward spinning dives. To eliminate the variables in the human jump, Groves projected a wooden plank 6' x 1' x 2" from a diving board in a manner similar to a diver's body leaving the boarc. and concluded that "with the same amount of initial force with which the body leaves the board, a greater lean will insure a greater angular velocity". When I had an opportunity to return to school in the 1950's and to review the reports of Lanoue and Groves, it seemed to me that they failed to adequately describe the cause of rotation in diving. Neither of them examined all classes of dives. Lanoue cited only backward spinning dives and Groves studied only dives spinning away from the board. Inclusion of all classes of dives might have altered their conclusions. Also, Lanoue's placement of the center of gravity is questionable and his assumption that the rate of rotation of a body around its feet on the board is the same as its rate of rotation when it becomes a free-falling body is contrary to physical principles. The conclusion reached by Groves ignores the possibility of performing reverse and inward dives. In an attempt to get some quantitative results with various combinations of directions uf force and lean I divised a model diving board that could be rotated to different angles and a wooden model diver that could be leaned at different angles guaged by a protractor in the background. The board could be depressed against the tension of a spring and when releasee tossed the model in much the same manner as a diver's body is projected from a diving board. Degrees of rotation were estimated for each trial as the diving model completed its trajectory and are shown in Table 1. Lean of boay 0 0 4 0 8 0 12 ° 16 c 20 ° 24 ° • Degrees of rotation for various body leans when force is applied with board at different angles. Board Board Board depressed 10° depressed 5° horizontal -80 -5 0 • 60 110 165 15 • 90 115 130 190 170 240 195 270 205 28 ° 285 force nearest perpendicular to direction of lean. -5 4. 90 120 135 165 180 231. PARTITION OF TOTAL ENERGY BETWEEN TRANSLATION AND ROTATION AS & INCREASES FROM 0 TO 90 DEGREES. 100 90 ~ ' 80 > ~ w z ~ '\ "~"v / /' 70 w 60 ..J ~50 ~ 11. 40 iw 30 0 ~ w 20 a. 10 0 V / Sin2 ee ~ ?eo I<~ V / J ~~ ?-0 / V ~ ' "' ?--94~ ..... 'S(. 4 1'to~y ~ € "" ---k ~ m -a. l 10 20 30 40 50 60 70 DEGREES Figure 3.- Partition of Total Energy Between Translation and Rotation as S Increases from 0 to 90 Degrees. 232. ! 80 90 With the board in a horizontal position, zero degrees of lean brought zero degrees of rotation ; and as the lean increased, the rotation increased. This supported Groves' conclusion that ... "a greater lean will insure a greater angular velocity". But as the board was rotated and thus altering the direction of the force, a reverse rotation could be induced as long as the angle of lean for the diver was less than the angle of rotation for the board. While the instrument lacked refinement, the readings supplied evidence supporting the idea that both the direction of lean and the direction of force were relevent to rotation and that both clockwise and counterclockwise rotation could be explained in terms of angle theta. Perhaps it is wise to emphasize that the magnitude of theta is related only to the angular momentum of the diver and that other variables such as the time spent in the air and the radius of the body exert influence on the number of degrees of rotation. So angle theta has not only a magnitude but also a direction which we will define as the direction of the b1sector of the angle. Figure 2 emphasizes the influence of the direction of theta on the execution of a dive. The lower drawing shows a theta of 20 degrees pointed in the direction of 10 degrees from the vertical. This is somewhat typical of what might be seen in a springboard dive. The thrust is upward with sufficient angular momentum to bring the desired rotation. The upper drawing shows the same theta of 20 degrees but pointed in a direction of 75 degrees from the vertical. This theta will provide the same angular momentum but less degrees of rotation because less time is spent in the air. My study of theta has always been handicapped by a limited background in mathematics and physics. I tried to console myself in the knowledge that even Sir Francis Galton had limitations and had to rely on Kar1 Pearson for technical assistance. In the 1960's I found my Pearson in the person of Dr. David L. Bushnel1 of the Physics Department of Northern Illinois University. Dr. Bushnell was willing to listen to my problem with theta which by then I was describing as the need to express the proportion of force contributing to rotation and the proportion contributing to projection as theta increased from zero to 90 degrees. Dr. Bushne11 first told me we should think in terms of energy rather than force and then began to derive an appropriate formula. In order to focus on the effects of a varying theta, we assumed a number of constants such as a constant weight for the diver, a constant depression of the board, a constant magnitude anri direction of the sum of a11 forces acting on the body during a dive. But even with these assumptions it was necessary to quantify a parameter k which was defined for any diver as the ratio between the moment of inertia about his center of gravity and the product of his mass and the square of the distance from his feet to his center of gravity. From fundamental physical laws, it was then possible to evolve the ratios shown in 233. Table 2. - Ratios of Rotational Energy and Translational Energy to Total Energy Expressed as a Function of Theta. Rotation Trans 1a ti on k It should be noted that the sum of these ratios is unity as the problem requires. In order to quantify the proportions for a single diver we proposed a "typical" diver who would provide a k of 0,2. With angle theta increasing from zero to 90 degrees, the partition of energy between translation and rotation is show~ in Figure 3. Degrees of theta from zero to 90 are shown along the bottom. Percent of total energy is shown at the left. Energy related to translation is shown by line beginning at the top ; and energy related to rotation is shown by the line beginning at the bottom. When theta equals zero degrees-that is, when the direction of force is through the center of gravity, -all of the energy goes to translation and none goes to rotation. As theta increases, the proportion of energy going to translation decreases as the proportion going to rotation increases. The two are always complementary. As a matter of passing interest, the energy is equally divided beh;een translation and rotation when theta is about 27 degrees. But where good competitive diving performances are concerned, I think thetas of around 15 degrees or less are likely used. This, according to the graph, narrows the range considerably and emphasizes the precision with which a diver must control the direction of his lean and the direction of his force in order to effect consistantly aesthetic performances. I was happy, with the aid of Dr. Bushnell, to arrive at some quantitative description of the partition of energy between rotation and translation. In the language of the Gestalt psychologist, it sort of provided a closure. out solutions are only temporary and seem to have a way of suggesting other problems. And so it is with this description of the partition of energy. If what we have done suggests problems to you I would encourage you to pursue them. But one glaring need impresses me and I sincere1y invite your attention to it. In our computations we have assumed a constant theta for each dive by assuming a constant direction of force and a constant direction of lean. But hard reality tells us that these directions are 1ike1y to not be constant. Force acts through distance in a period of time to suffic1Ent1y accelerate a body to make it a projectile. As the diver rides the board on its upward vibration, either the direction of force, the direction of lean or both are likely changing to some degree and thus causing a change in theta. A quantitative description of this change and its pro234. gressive effect on the partition of energy between translation and rotation would be highly desirable. And if any of you have the interest, the dedication and the technical skill in calculus or computers to pursue that problem, I urgently suggest that you do so. 235.