Biomechanics of Throwing a Frisbee Jesse Rappaport BIOL 438 4/12/12

Types of Frisbee Throws Backhand Forehand Overhead All Include flick of the wrist/arm to rotate the frisbee Focus on rotation Research Focus What is the rotational kinetic energy in a frisbee throw? What is the angular velocity needed to produce a steady frisbee toss? What are the major differences in slow versus fast throws?

Frisbee Flight Forces in flight include: Gyroscopic rotation Drag Lift Rotation is needed to keep frisbee in flight Difference between Center of Mass and Center of Pressure creates wobble Wobble is diminished by increased angular velocity Hammond, pg. 6 Figure 1: COM: Center of Mass; COP: Center of Pressure; D (Drag); L (Lift); v (velocity); mg (gravity)

How? Solution: Must be able to see the rotation of the arm and rotation of the frisbee Must be able to see overall velocities of arm movement and frisbee flight Take view from above!

Technique overview Two throws First: attempt a quick throw with adequate spin Second: attempt a slower throw with more wobble Analyze direction of arm and wobble of frisbee qualitatively Analyze rotation of the arm and frisbee quantitatively Measure/calculate the following: Mass Moment of Inertia Angular Momentum Angular Acceleration Torque Kinetic Energy

Anatomy Involved Normal throw Focused throw Use of whole body Step with legs Turn with torso Turn with arm Flick with wrist Focused throw Use of arm Extend arm Rotate forearm Flick wrist

Overall linear velocity is shown to increase with distance from the shoulder. This produces the initial velocity of the frisbee.

Biggest Observational Difference: Slower throw uses more of an arc in the arm and less of a rotational change by the end. Why is this important? - Quick rotation allows for increased velocity over short amount of time, allowing for a faster resulting angular KE during flight

Equations/Calculations Mass of Frisbee: 0.175 grams Radius of Frisbee: 0.27305 meters Mass of Arm: 2.52% of body mass = 1.886 grams Radius of Arm: 0.4402 meters Mass of Hand: 0.65% of body mass = 0.486 grams Radius of Hand: 0.137 meters Moment of Inertia for throwing system (frisbee and arm) Treat frisbee as thin disk and arm as rod with rotation about the elbow Use parallel axis theorem (Iz= Icm + m(rarmlength)2) for frisbee where Icm= ½m(rfrisbee)2 Add Iarm= ⅓m(Larm)2 Moment of Inertia using arm: 0.162 kg•m2

Equations/Calculations Using digital protractor, angular displacement, θ, was measured with the initial point where angular velocity, ω0, was zero. The following equations were used to calculate angular acceleration (α), angular velocity (ω), torque (τ), and Kinetic Energy (KE). θ=ωot+½αt2 ω=ωo+αt τ=Iα KE = ½Iω2 or ½mv2

Throw by the numbers Fast throw Slow throw Angular displacement θ (radians) 1.571 Time t (seconds) 0.12 Angular acceleration α (radians/s2) 109.083 Torque t (newtons) 15.269 Angular velocity w (radians/s) 13.09 KE (rotational) J (Joules) 47.97 Angular displacement θ (radians) 1.134 Time t (seconds) 0.2 Angular acceleration α (radians/s2) 56.723 Torque t (newtons) 9.204 Angular velocity w (radians/s) 11.344 KE (rotational) J (Joules) 10.441

Post-release spin Slow Throw Fast Throw Angular Velocity: 15.3 rad/sec Angular KE: 0.764 J Linear Velocity: 6.996 m/s Linear KE: 4.282 J Angular Velocity: 34.9 rad/sec Angular KE: 3.973 J Linear Velocity: 11.258 m/s Linear KE: 11.09 J

Results The increased angular acceleration lead to a larger final angular velocity The fast throw added significantly more angular velocity and therefore more angular kinetic energy than the slow throw or the overall movement of the faster throw. The lack of angular velocity lead to wobble in the slow throw, whereas the fast throw had initially no wobble. Not all KE was converted to frisbee- arm still continues forward and along same path

Conclusions In order to have a more effective frisbee throw, be sure to generate a high angular acceleration to create a strong angular velocity in the frisbee during flight Creating a more drastic rotation in less time towards the end of the throw will help to generate the KE needed Full body usage will aid to the amount of force generated in both the x-direction and rotation

Future Considerations Camera angles Having multiple cameras taking images of the same shot (or one camera with mirrors rigged up) would allow for further analysis of relationship between the degree of wobbling and the angular velocity generated Compute many more trials to gauge the direct relationship between angular velocity and amount of wobble Calculate elasticity in arm as it whips forward Calculate rotational and linear KE using entire body to throw

References Plagenhoef, S., Evans, F.G. and Abdelnour, T. (1983) Anatomical data for analyzing human motion. Research Quarterly for Exercise and Sport 54, 169-178. http://biosport.ucdavis.edu/research-projects/frisbee-flight- simulation-and-throw-biomechanics/8thISCSB_Frisbee_throws.pdf http://biosport.ucdavis.edu/research-projects/frisbee-flight- simulation-and-throw-biomechanics/HummelThesis.pdf