1. A Physicists Approach to Springboard Diving
4/27/2017
A Physicists Approach to Springboard Diving
Edward N. Roberts
University of the South, Sewanee
March 6th 2002
:Introduction:
-So who’s a better physicist a Cat or a Diver?
I am going to be talking about the actions of both of these, the cat’s comes instinctively and the diver is a trained phenomenon.

2. A Question Posed to Physicists:
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A Question Posed to Physicists:
This is an important question to ask
Is it possible for a somersaulting springboard diver to initiate a twisting motion without any torque being applied to their body? That is, can a diver begin to twist after having left the diving board?
Note: read question…
This is question asked to the graduate, post doctorates, and the faculty in the Physics Department at Cornell University.

3. Answer: Yes Physics Department at Cornell University:
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Answer:
Yes
Physics Department at Cornell University:
Interestingly 56%* of those asked the question answered incorrectly.
-The answer to the question is yes it is possible for a diver to initiate a twisting motion without any torque being applied to their body
-This is what intrigued me…being a diver myself I know that it is possible do the twisting dives without the use of the board…or any torque from the board needed for me to do a twisting dive…Many people are under the false pretense that a diver uses the board in order to twist…however I am going to show that yes it is possible for a diver to use the board to twist, but it is possible to twist without added torque from the board.
*Frohlich, Cliff “Do springboard divers ...”,
Am.J.Phys.47(7), July 1979.

4. Laws of Physics applicable to the sport of Diving
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Laws of Physics applicable to the sport of Diving
Center of Mass
Angular Velocity
Moments of Inertia
Principle of Acceleration
Many more...
-There are many aspects of physics that are involved in diving…However we are going to be examining
-Moments of Inertia
-Angular Velocity
-Angular Momentum
-Center of Mass
-and a few more...

5. Laws of Physics applicable to the sport of Diving
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Laws of Physics applicable to the sport of Diving
Why even talk about the physics of Diving?
We can see conservation laws really at work…
If we can understand the physics of what is going to happen to a diver, we might be able to express to the diver a better approach to executing a dive.

6. Terminology used in Diving:
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Terminology used in Diving:
The Approach
The Hurdle
Categories of dives:
Forward
Back
Reverse
Inward
Twister
Is a diver’s walk down the board where they press the board on the last step.
Hurdle-is the motion of a diver leaping into the air and landing back down on the board. This motion is similar to the way the a hurdler jumps.
Forward - Consists of dives that are preformed with a hurdle and take-off (facing the water) with forward rotation.
Back - Backward take-off (back to the water) with backward rotation.
Reverse - Forward take-off with backward rotation.
Inward - Backward take-off with forward rotation.
Twister - Any of the above groups with 1/2 to 4 twists.
Show video hurdle

7. Terminology used in Diving:
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Terminology used in Diving:
Four positions of dives:
Straight
Pike
Tuck
Free
Straight
Body not bent and as tight as possible.
Pike
The body is only bent at the hips, some divers choose to touch their toes in this position.
Tuck
The body is bent at the hips and the knees.
Free
Combination of the above positions (used for twist dives only).
As we will see in a moment the position that the divers body is in is crucial when we talk about the moments of inertia for the different positions.

8. Flight of a Dive Rotation around Center of Mass
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Flight of a Dive
Rotation around Center of Mass
Parabolic Flight of Dives
What can be determined from this?
Video of Inward 1.5

9. 4/27/2017
Flight of a Dive
Equations of Motion:

10. Difference in the two curves
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This is a plot of the position of a diver’s center of mass through the flight of a dive.
Difference in the two curves
19.26
Voy/vox = 3 degrees
Voy 20 times larger than vox
G = 10.98

11. Parabolic flight of a dive
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Parabolic flight of a dive

12. Parabolic flight of a dive
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Parabolic flight of a dive
G = 9.4

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G = -9.35

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G = 8.6

16. Conservation of Angular Momentum
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Conservation of Angular Momentum
Conservation of Angular Momentum Equation:
Angular Velocity Equation is:
Where Ii is the initial moment of Inertia and, and Ifis the final.
And i is the initial angular velocity and f is the final angular velocity
The delta theta and the delta t are the change in the angle and the change in time.

17. Moments of Inertia: Moments of Inertia must be determined:
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Moments of Inertia:
Moments of Inertia must be determined:
Assumptions:
Rigid Body
Density Distribution equally
14 Separate parts
Represent simple Geometric shapes
In order to talk about the twisting and somersaulting actions of a diver, the moment of Inertia for the diver must be determined for the different positions of the diver’s body.
-Assumptions
-Consider the body as fourteen separate sections.
-Each of the fourteen parts and be represented as geometric shapes.
-The head, feet, and hands of a body can be considered a sphere.
-The arms and legs can be considered thin rod cylinders.
-The trunk of a body can be considered solid cylinders.
-your body is a rigid body- distance between these 14 segments stay fixed
-density is distributed equally through each segment

18. Calculation of the Inertia:
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Calculation of the Inertia:
Thin Rod Cylinder:
Sphere:
Once we have made this assumptions about the shape of the parts of the body, we can use simple moment of inertia about the center of mass equation in order to determine the moment of Inertia for each part of the body…
Where big M is the total mass of the object and big R is the radius of that object
In the solid cylinder the L term is the length of that object.
Solid Cylinder:

19. Calculation of the Mass Chart:
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Calculation of the Mass Chart:
This is a weight percentage table…I took this table from a book about the physics of dance where they were trying to calculate the center of mass for a dancer
Because the moment of inertia is calculated with mass,
It becomes a challenge to weigh these different parts of the human body..
Stanley Plagenhoef, Patterns of Human Motion (Englewood Cliffs, NJ:Prentice-Hall, 1971), chapter 3

20. Calculation of the Mass:
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Calculation of the Mass:
These are the values calculated from my body weight…
I weigh kg
The radii were taken from the circumference of the body part divided by 2 pi.
The lengths were simply measured with a meter stick.

21. Calculation of the Inertia:
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Calculation of the Inertia:
The Parallel-Axis Theorem:
“Relates the moment of inertia about an axis through the center of mass of an object to the moment of inertia about a second parallel axis.”
Because in the human body all segments are not centered at the axis of rotation, the Parallel-Axis Theorem must be used
where big M is again the total mass of the segment, and d is the distance out from the axis of rotation.

22. 14 Separate parts diagram Example Calculation
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14 Separate parts diagram Example Calculation
D = distance from axis of rotation
Calculation for the position with arms straight out.

23. 14 Separate parts diagram Example Calculation
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14 Separate parts diagram Example Calculation

24. Calculation of distance from Axis of Rotation
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Calculation of distance from Axis of Rotation
-In order to calculate the distance each body part is from the axis of rotation, I used a video analysis program called Videopoint…
-This program allows you to mark each part of the body with a different marker.
-Videopoint allows you to Scale
-Moment of Inertia for the straight position.

25. Calculation of distance from Axis of Rotation
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Calculation of distance from Axis of Rotation
-These moments of Inertia for the straight and tuck positions make sense because you would have a greater moment of inertia the further away from the axis of rotation you are…so the straight position would give you a greater moment of inertia in order to conserve angular momentum.
-The moments of inertia were calculated for each of the 14 segments of the body and the all added together to give the total moment of inertia for the body.
-The angular velocity was determined using Videopoint…

26. Videopoint Calculation of 
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Videopoint Calculation of 
Center of Mass used as the origin
Plotted the rotation of the head around the center of mass
Make sure to go to Videopoint to demonstrate how it was used.

27. Example of Tuck  calculation
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Example of Tuck  calculation
Point out the head positions
Rotation around my center of mass
Angular velocity = ∆ø/∆t
Change in angle is 93 degrees
Change in time = .166 seconds

28. Conservation of Angular Momentum
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Conservation of Angular Momentum
Calculated moment of Inertia for the straight position:
I = 15.7 kgm  = 115 °/s = 2.01 rad/s L = 31.5 kgm2/s
Calculated moment of Inertia for the tuck position:
I = 5.30 kgm  = 560 °/s = 9.60 rad/s L = 50.9 kgm2/s
The values for the angular momentum initial and final don’t look like they are matching up…but the amount of error is on the order of about 50%. The reason for this is because there is approximately 20% in the calculation for the Moment of Inertia and then about 40% error in the calculations for the angular velocity.
So when error is added into the equation, the values tend to agree with one another.
When you look at order of magnitudes you can see that the moment of inertia has a multiplier effect of about 3 times, so I am able to decrease my moment of inertia by 3 when I go into a tuck. When looking at the angular momentum, you would expect that you would have the same kind of order of magnitude effect as with the moment of inertia, but I have a multiplier of about 5.
Now that we can see that angular momentum is conserved we can talk about some of the mechanics of the somersault and twist.
31.5 kgm2/s = 50.9 kgm2/s

29. Mechanics of Somersaults
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Mechanics of Somersaults
Angular Velocity:
“Throwing” of arms
“Leaning”
Equal and opposite forces
If we are going to talk about angular velocity we need to understand how a dive establishes this angular velocity.
-Throwing arms
-If a diver were in free space his lower body would rotate in the opposite direction, however the diving board exerts a force that opposes this rotation and provides a torque that makes it possible to perform a somersault.
-2.5 Video
-Leaning - happens even when you throw your arms, but a flip can be performed solely by a diver putting their center of mass far enough out from the board to establish a force, which will cause them to rotate around their center of mass.
The reason that we look at the mechanics of the somersault is in order to perform some twist the angular momentum from flipping is needed

30. “Cat Twists” or Zero Angular Momentum Twist
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Mechanics of a Twist
Three types of Twists:
Torque Twist
“Cat Twists” or Zero Angular Momentum Twist
Torque-free Twist
Here are the three types of twist which I am going to be talking about…
-Torque-free twists are the type of twist that most divers use.

31. The simplest form of a twist Equal and opposite force
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Torque Twist
The simplest form of a twist
Equal and opposite force
Unable to be controlled
The torque twist is the simplest form of twist which is performed while a person is still on the ground
-Do a torque twist
-This occurs because you are pushing off the ground with one leg stronger than the other…the earth gives you a Normal force, equal, but In the opposite direction.
These are not the best type of twist to do for a diver, because there is no graceful way of stopping the twist once you have started it.
Over rotating...

32. “Cat Twists” Why does a cat when dropped land on it’s feet?
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“Cat Twists”
Why does a cat when dropped land on it’s feet?
Conservation of Angular Momentum
How does a cat perform this?
How a diver can do the same twist.
A cat lands on it’s feet because it instinctually performs a series of actions in which the cat conserves an angular momentum of zero.
First the cat pushes it's hind legs out and stretches them as far behind itself as possible. This action causes the rear section of the cat's body to have a larger rotational inertia than the front section of the cat's body. The cat will at the same time bring in its front legs making them have less inertia. This allows the front half of the cat to rotate in one direction, while the rear part rotates very little in the opposite direction. Once the front section of the cat has rotated the 180˚, the cat brings back in the rear legs and stretches out the front legs. This gives the front section a larger rotational inertia in respect to the rear, allowing the cat to rotate the rear half of its body around 180˚. During this the cat keeps a constant angular momentum of zero.
A diver can perform the same kind of a twist by jumping off the board and positioning their arms straight out and rotating them in either the left or right direction…This makes the moment of Inertia of his upper body greater than that of his lower body…The movement of their arms causes the twist to occur.

33. Type of twist which divers perform How a torque-free twist occurs
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Torque-free Twist
Type of twist which divers perform
How a torque-free twist occurs
Possession of Angular Momentum
Not on the board and can twist
Can be controlled
Show a video of 1.5 twister
In order for a torque-free twist to happen a diver must posses angular momentum. This angular momentum most be about the divers somersaulting axis.
When a diver does the twisting action they bring one arm in closer to their center of mass and the other above their head further away from their center of mass. This causes a slight tilt in the direction of the angular momentum. This slight tilt causes the angular momentum to be different than when the arms were symmetric. In order for angular momentum to be conserved an angular velocity component is needed. This comes in the form of twisting the body around the front to back axis.
The same mechanism is used to stop the twist. The diver will change arm positions so the added angular velocity is no longer needed and the twists on the dive stop.
Show video of 1.5 twist.

34. Tilt of a Torque-free Twist
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Tilt of a Torque-free Twist
Here you can see the tilt of the twist in toward the angular momentum.
The reason that a diver would like to use this type of twist is because it can be stopped any where in the dive depending on how many twist you want to do
So can you really leave the board and twist without the added torque of the board?

35. My Experiments Three Camera Angles Timing of each camera Angle.
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My Experiments
Three Camera Angles
Timing of each camera Angle.
In order to see if I really do not use the board to twist I needed to take video tape of the twisting dive.
I needed to get the three angles for the dive in order to look at my hips to see if they were twisting off the board.
The most important of these angles being the angle above the diver.
Hung a Digital video cable over the sprinkle system in fowler.
I had a Front shot, Top shot, and Side shot.
Ball Bounce
What I found

36. Picture of the Overhead Camera
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Picture of the Overhead Camera
In order to see if I really do not use the board to twist I needed to take video tape of the twisting dive.
I needed to get the three angles for the dive in order to look at my hips to see if they were twisting off the board.
The most important of these angles being the angle above the diver.
Hung a Digital video cable over the sprinkle system in fowler.

37. Results of Torque-free Twist
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Results of Torque-free Twist
The next three pictures are of the video I just showed. They three different angles of the same dive at the same time.
In all of the pictures you can see that I have just left the board.
My hips are square coming off the board on all three of them so this shows that I have left the board without twisting.
In the overhead shot you can see that my right arm is starting to drop in order to initiate the twist.

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Torque-free Twist

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Torque-free Twist
So the next thing that intrigued me to look at was the diving board itself.
What is the relationship between the position of the fulcrum of the diving board and how high the diver gets.
I was looking to really see if there is a position that I could put the fulcrum that would cause me to get the most height?
This becomes very important because once you have left the board the only force that is acting on you is gravity.

40. My experiments Diving Board considered a cantilever:
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My experiments
Diving Board considered a cantilever:
-lever arm the distance from the fulcrum to the end of board
Setup for how this was done
Results
Measurement of the length of the board to the fulcrum was measured
Videopoint was used again to see how high above the board my body went.
I just simply did an approach and hurdle and jumped off the board

41. Lever Arm changing Data
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Lever Arm changing Data
As you can see that as the lever arm was increased there is a increasing relationship between he the lever arm and the height that I got.
However the data is not really that good. There are too many dependents on how high you get.
I think that the biggest contributor to this fluctuating data is hurdle and angle that you leave the board.
One of the more interesting things to look at is the flight of the dive. If you are to plot the center of mass as a function of time you get a parabola.

42. Conclusion Divers are able to twist without the diving board
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Conclusion
Divers are able to twist without the diving board
Increasing relationship between the Lever Arm and height received
Questions???
So going back to my original question on who’s a better physicist, a cat or a diver.
Unlike a cat a diver whither they know it or not a diver obeys the laws of physics, conservation of momentum, while a cat instinctively does. Therefore I conclude that a diver would be a better physicist.