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.