A Physicists Approach to A Physicists Approach to Springboard DivingSpringboard Diving

Edward N. Roberts Edward N. Roberts

University of the South, University of the South, SewaneeSewanee

March 6March 6th th 20022002

A Question Posed to Physicists:A Question Posed to Physicists:

Is it possible for a somersaulting springboard Is it possible for a somersaulting springboard diver to initiate a twisting motion without any diver to initiate a twisting motion without any torque being applied to their body? That is, torque being applied to their body? That is, can a diver begin to twist after having left the can a diver begin to twist after having left the diving board?diving board?

Answer:Answer: YesYes

Physics Department at Cornell University:Physics Department at Cornell University:– Interestingly 56%Interestingly 56%** of those asked the question of those asked the question

answered incorrectly.answered incorrectly.

*Frohlich, Cliff “Do springboard divers ...”, Am.J.Phys.47(7), July 1979.

Laws of Physics applicable to the sport of Laws of Physics applicable to the sport of DivingDiving

Center of MassCenter of Mass Angular VelocityAngular Velocity Moments of InertiaMoments of Inertia Principle of AccelerationPrinciple of Acceleration Many more...Many more...

Laws of Physics applicable to the sport of DivingLaws of Physics applicable to the sport of Diving

Why even talk about the Why even talk about the physics of Diving?physics of Diving?

Terminology used in Diving:Terminology used in Diving: The ApproachThe Approach The Hurdle The Hurdle Categories of dives:Categories of dives:

– ForwardForward– BackBack– ReverseReverse– Inward Inward – TwisterTwister

Terminology used in Diving:Terminology used in Diving:

Four positions of dives:Four positions of dives:

– StraightStraight

– PikePike

– TuckTuck

– FreeFree

Flight of a DiveFlight of a Dive

•Rotation around Center of Mass

•Parabolic Flight of Dives

•What can be determined from this?

Flight of a DiveFlight of a Dive

•Equations of Motion:

X =Voxt

Y =Yoy +Yoyt +12

gt2

Parabolic Shape of a Front Dive

y = -2.9327x2 - 19.266x - 28.976

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-5 -4 -3 -2 -1 0

Position (m)

Position (m)

Parabolic flight of a diveParabolic flight of a dive

Parabolic Flight for a Forward Dive x as a function of T

y = -1.3683x + 1.8462

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5

Time (s)

Position (m)

Parabolic flight of a diveParabolic flight of a dive

Parabolic Flight of Front Dive y as a function of T

y = -4.6837x2 + 35.047x - 63.03

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

Time (s)

Position (m)

Parabolic Flight of a Reverse Dive

y = -3.935x2 - 26.32x - 41.673

-1

-0.5

0

0.5

1

1.5

2

2.5

-5 -4 -3 -2 -1 0

Position (m)

Position (m)

Parabolic Flight of a Reverse Dive x as a function of T

y = -1.0906x + 1.1587

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5 6

Time (s)

Position (m)

Parabolic Flight of a Reverse Dive y as a function of T

y = -4.3428x2 + 35.031x - 68.355

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6

Time (s)

Position (m)

Conservation of Angular MomentumConservation of Angular Momentum

Iiωi =I fω f

•Angular Velocity Equation is:

•Conservation of Angular Momentum Equation:

ω =ΔθΔt

Moments of Inertia:Moments of Inertia:

Moments of Inertia must be determined:Moments of Inertia must be determined: Assumptions:Assumptions:

– Rigid BodyRigid Body

– Density Distribution equally Density Distribution equally

– 14 Separate parts14 Separate parts

– Represent simple Geometric shapesRepresent simple Geometric shapes

Calculation of the Inertia:Calculation of the Inertia:

Icm= r2dm∫Thin Rod Cylinder:

Sphere:Icm=

25

MR2

Solid Cylinder:

Icm=12

MR2

Icm=14

MR2 +112

ML2

Calculation of the Mass Chart:Calculation of the Mass Chart:

Stanley Plagenhoef, Patterns of Human Motion (Englewood Cliffs, NJ:Prentice-Hall, 1971), chapter 3

Calculation of the Mass:Calculation of the Mass:

mass kg radius (m) length (m)hands 0.408 0.05feet 1.02 0.065head 4.828 0.0875upper arm 2.244 0.0445 0.3forarm 1.292 0.036 0.26trunk 32.844 0.105 0.56thigh 7.14 0.0811 0.52lower leg 3.06 0.054 0.48

The Parallel-Axis Theorem: The Parallel-Axis Theorem: – ““Relates the moment of inertia about an Relates the moment of inertia about an

axis through the center of mass of an axis through the center of mass of an object to the moment of inertia about a object to the moment of inertia about a second parallel axis.”second parallel axis.”

I =Icm+Md2

Calculation of the Inertia:Calculation of the Inertia:

14 Separate parts diagram14 Separate parts diagramExample CalculationExample Calculation

I Cm (kgm^2) I arms out(kgm^2) d (m)hands 0.0004 0.662 0.90feet 0.0017 2.253 1.05head 0.0148 1.222 0.50upper arm 0.0168 1.156 0.50forarm 0.0073 1.281 0.70trunk 0.9488 4.312 0.32thigh 0.1609 2.172 0.36lower leg 0.0588 3.469 0.74

sum sum1.210 16.526

14 Separate parts diagram14 Separate parts diagramExample CalculationExample Calculation

I Cm (kgm^2) I arm in(kgm^2) d (m)left hand 0.0004 0.3309 0.90feet 0.0017 2.253 1.05head 0.0148 1.222 0.50L upper arm 0.0168 0.5778 0.50L forarm 0.0073 0.6404 0.70trunk 0.9488 4.312 0.32thigh 0.1609 2.172 0.36lower leg 0.0588 3.469 0.74right hand 0.0004 0.0004 0.00R upper arm 0.0168 0.1066 0.20R forarm 0.0073 0.1655 0.35

sum sum1.210 15.2493

Calculation of distance from Axis of RotationCalculation of distance from Axis of Rotation

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Scale1 BScale1 A

Origin 1

head

Foot

lower leg

Thigh

Trunk

upper armforearm

Hand

t = 2.193 [s]

x:-3.57 y:1.89 [m] Point S1

Calculation of distance from Axis of RotationCalculation of distance from Axis of Rotation

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Scale1 B Scale1 A

Origin 1ForearmTrunk

Upper Arm

Head

Feet

ThighHands

CM 2Point S1

t = 3.428 [s]

x:-2.81 y:1.58 [m] Point S1

Videopoint Calculation of Videopoint Calculation of

•Center of Mass used as the origin

•Plotted the rotation of the head around the center of mass

Example of Tuck Example of Tuck calculation calculationOmega chart

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.8 -0.6 -0.4 -0.2 0 0.2

Position (m)

Position (m)

Position of Head

Conservation of Angular MomentumConservation of Angular Momentum

Calculated moment of Inertia for the tuck position:Calculated moment of Inertia for the tuck position:

I = 5.30 kgmI = 5.30 kgm22 = 560 °/s = 9.60 rad/s L = 50.9 kgm = 560 °/s = 9.60 rad/s L = 50.9 kgm22/s/s

Calculated moment of Inertia for the straight position:Calculated moment of Inertia for the straight position:

I = 15.7 kgmI = 15.7 kgm2 2 = 115 °/s = 2.01 rad/s L = 31.5 kgm = 115 °/s = 2.01 rad/s L = 31.5 kgm22/s/s

Iiωi =I fω f•31.5 kgm2/s = 50.9 kgm2/s

Angular Velocity:Angular Velocity:– ““Throwing” of armsThrowing” of arms– ““Leaning”Leaning”

Equal and opposite forcesEqual and opposite forces

Mechanics of SomersaultsMechanics of Somersaults

Three types of Twists:Three types of Twists:

– Torque TwistTorque Twist

– ““Cat Twists” or Zero Angular Momentum Cat Twists” or Zero Angular Momentum TwistTwist

– Torque-free TwistTorque-free Twist

Mechanics of a TwistMechanics of a Twist

The simplest form of a twistThe simplest form of a twist

Equal and opposite forceEqual and opposite force

Unable to be controlled Unable to be controlled

Torque TwistTorque Twist

““Cat Twists”Cat Twists” Why does a cat when dropped land on it’s Why does a cat when dropped land on it’s

feet?feet?–Conservation of Angular MomentumConservation of Angular Momentum

How does a cat perform this?How does a cat perform this?

How a diver can do the same twist.How a diver can do the same twist.

Type of twist which divers performType of twist which divers perform

How a torque-free twist occursHow a torque-free twist occurs– Possession of Angular MomentumPossession of Angular Momentum

Not on the board and can twistNot on the board and can twist

Can be controlledCan be controlled

Torque-free TwistTorque-free Twist

Tilt of a Torque-free TwistTilt of a Torque-free Twist

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Origin 1 [x:498 y:505 (pixels)]

t = 0.5333 [s]

x:4.00 y:1010 (pixels) Origin 1

My ExperimentsMy Experiments

•Three Camera Angles

•Timing of each camera Angle.

Picture of the Overhead Camera Picture of the Overhead Camera

Results of Torque-free TwistResults of Torque-free Twist

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Origin 1

t = 13.76 [s]

x:326 y:982 (pixels) Point S1

Torque-free TwistTorque-free Twist

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Origin 1

t = 10.43 [s]

x:73.0 y:995 (pixels) Point S1

Torque-free TwistTorque-free Twist

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Origin 1

t = 10.85 [s]

x:22.0 y:995 (pixels) Point S1

My experimentsMy 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

Lever arm (m) Height above board (m)

2.71 1.572.73 1.732.75 1.582.77 1.612.79 1.612.82 1.622.84 1.462.87 1.632.90 1.542.93 1.622.95 1.632.97 1.752.99 1.793.05 1.553.08 1.683.10 1.663.12 1.723.15 1.573.18 1.713.20 1.613.25 1.693.28 1.623.31 1.82

Lever Arm changing DataLever Arm changing Data

Hight above diving board as a fuction of lever arm length

1.00

1.20

1.40

1.60

1.80

2.50 3.00 3.50

Lever arm length (m)

ConclusionConclusion Divers are able to twist without the diving Divers are able to twist without the diving

boardboard

Increasing relationship between the Lever Increasing relationship between the Lever Arm and height receivedArm and height received

Questions???Questions???