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The Physics of Baseball Alan M. Nathan University of Illinois ODU Colloquium, March 31, 2000

The Physics of Baseball Alan M. Nathan University of Illinois ODU Colloquium, March 31, 2000. Introduction Hitting the Baseball The Flight of the Baseball Pitching the Baseball Summary. REFERENCES.

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The Physics of Baseball Alan M. Nathan University of Illinois ODU Colloquium, March 31, 2000

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  1. The Physics of BaseballAlan M. Nathan University of IllinoisODU Colloquium, March 31, 2000 • Introduction • Hitting the Baseball • The Flight of the Baseball • Pitching the Baseball • Summary

  2. REFERENCES • The Physics of Baseball, Robert K. Adair (Harper Collins, New York, 1990), ISBN 0-06-096461-8 • The Physics of Sports, Angelo Armenti (American Institute of Physics, New York, 1992), ISBN 0-88318-946-1 • www.npl.uiuc.edu/~a-nathan/pob

  3. #521, September 28, 1960 Hitting the Baseball “...the most difficult thing to do in sports” --Ted Williams BA: .344 SA: .634 OBP: .483 HR: 521

  4. Here’s Why….. (Courtesy of Bob Adair)

  5. Description of Ball-Bat Collision • forces large (>8000 lbs!) • time is short (<1/1000 sec!) • ball compresses, stops, expands • kinetic energy  potential energy • bat affects ball….ball affects bat • hands don’t matter! • GOAL: maximize ball exit speed vf vf 105 mph  x  400 ft x/vf = 5 ft/mph What aspects of collision lead to large vf?

  6. How to maximize vf? • What happens when ball and bat collide? • The simple stuff • conservation of momentum • conservation of angular momentum • energy dissipation in the ball (compression/expansion) • The really interesting stuff • vibrations of the bat

  7.  0.2 r recoil factor = “radius of gyration” The Simple Stuff: Rigid-Body Kinematics e Coefficient of Restitution  0.5 Vball,f = 0.25 Vball,i + 1.25 Vbat,i Conclusion: vbat much more important than vball

  8. . . . CM z Translation Rotation Recoil Factor • Important Bat Parameters: • mbat, xCM, ICM • wood vs. aluminum 0.16 + 0.07 = 0.23 Conclusion: All things being equal, want mbat, Ibat large

  9. Coefficient of Restitution (e) • “bounciness” of ball • Bounce ball off massive hard surface • e2= hf/hi • For baseball, e  .5 •  3/4 energy lost! • Changing e by .05 changes V by 7 mph(35 ft!) Important Point: the bat matters too!

  10. tennis ball/racket Effect of Bat on COR • Energy shared between ball and bat • Ball is inefficient:  25% returned • Wood Bat • kball/kbat ~ 0.02 • 80% restored • eeff = 0.50-0.51 • Aluminum Bat • kball/kbat ~ 0.10 • 80% restored • eeff = 0.55-0.58 • “trampoline effect” • Bat Proficiency Factor eeff/e • Claims of BPF  1.2 Ebat/Eball kball/kbat  xbat/ xball >10% larger!

  11. CM • vball,I= 90 mph • vbat,CM = 54 mph • bat,CM = 51 s-1 Aluminum bat more effective for inside pitches Rigid-Body Results

  12. Beyond the Rigid Approximation: A Dynamic Model for the Bat-Ball collision • Collision excites bending vibrations in bat • Ouch!! Thud!! • Sometimes broken bat • Energy lost  lower vf • Lowest modes easy to find by tapping • Location of nodes important • Modes with fn  1 excited Ref.: AMN, Am. J. Phys, submitted March 2000

  13. y y z A Dynamic Model of the Bat-Ball Collision 20 • Solve eigenvalue problem for normal modes (yn, n) • Model ball-bat force F • Expand y in normal modes • Solve coupled equations of motion for ball, bat

  14. In a bit more detail… impact point ball compression

  15. f1 = 165 Hz f3 = 1177 Hz f2 = 568 Hz f4 = 1851 Hz Results: 1. Normal Modes Louisville Slugger R161 (34”, 31 oz) nodes Can be measured (modal analysis)

  16. Results: 2. Low-speed collision Theory vs. Experiment (Rod Cross) (at 1 m/s) collision time  2.2 ms

  17. Results: 3. High-speed collision CM nodes • Under realistic conditions… • 90 mph, 70 mph at 28” • no data (yet)…..

  18. CM nodes 24” 27” 30” Results: 4. The “sweet spot” Possible “sweet spots” 1. Maximum of vf (28”) 2. Node of fundamental (27”) 3. Center of Percussion (27”)

  19. Wood versus Aluminum • Length and weight “decoupled” • Can adjust shell thickness • Fatter barrel, thinner handle • More compressible • COR larger • Weight distribution more uniform • Easier to swing • Less rotational recoil • More forgiving on inside pitches • Less mass concentrated at impact point • Stiffer for bending • Less energy lost due to vibrations

  20. How Would a Physicist Design a Bat? • Wood Bat • already optimally designed • highly constrained by rules! • a marvel of evolution! • Aluminum Bat • lots of possibilities exist • but not much scientific research • a great opportunity for ... • fame • fortune

  21. Conclusions • The essential physics of ball-bat collision understood • bat can be well characterized • ball is less well understood • the “hands don’t matter” approximation is good • Vibrations play important role • Size, shape of bat far from impact point does not matter • Sweet spot has many definitions

  22. Aerodynamics of a Baseball Forces on Moving Baseball • No Spin • Boundary layer separation • DRAG! • FD=½CDAv2 • With Spin • Ball deflects wake ==>Magnus force • FMRdFD/dv • Force in direction front of ball is turning

  23. How Large are the Forces? =1800 RPM • Drag is comparable to weight • Magnus force < 1/4 weight)

  24. The Flight of the Ball:Real Baseball vs. Physics 101 Baseball • Role of Drag • Role of Spin • Atmospheric conditions • Temperature • Humidity • Altitude • Air pressure • Wind Max @ 350 approxlinear

  25. The Role of Friction • Friction induces spin for oblique collisions • Spin  Magnus force • Results • Balls hit to left/right break toward foul line • Backspin keeps fly ball in air longer • Topspin gives tricky bounces in infield • Pop fouls behind the plate curve back toward field

  26. The Home Run Swing • Ball arrives on 100 downward trajectory • Big Mac swings up at 250 • Ball takes off at 350 • The optimum home run angle!

  27. “Hitting is timing. Pitching is upsetting timing” ---Warren Spahn vary speeds manipulate air flow orient stitches Pitching the Baseball

  28. 7 6 Vertical Position of Ball (feet) 5 90 mph Fastball 4 3 0 10 20 30 40 50 60 Distance from Pitcher (feet) 1.2 1 75 mph Curveball 0.8 0.6 Horizontal Deflection of Ball (feet) 0.4 0.2 0 0 10 20 30 40 50 60 Distance from Pitcher (feet) Let’s Get Quantitative!How Much Does the Ball Break? • Kinematics • z=vT • x=½(F/M)T2 • Calibration • 90 mph fastball drops 3.5’due to gravity alone • Ball reaches home plate in ~0.45 seconds • Half of deflection occurs in last 15’ • Drag: v  -8 mph • Examples: • “Hop” of 90 mph fastball ~4” • Break of 75 mph curveball ~14” • slower • more rpm • force larger

  29. Examples of Pitches Pitch V(MPH)  (RPM) T M/W fastball 85-95 1600 0.46 0.10 slider 75-85 1700 0.51 0.15 curveball 70-80 1900 0.55 0.25 What about split finger fastball?

  30. Effect of the Stitches • Obstructions cause turbulance • Turbulance reduces drag • Dimples on golf ball • Stitches on baseball • Asymmetric obstructions • Knuckleball • Two-seam vs. four-seam delivery • Scuffball and “juiced” ball

  31. Example 1: Fastball 85-95 mph 1600 rpm (back) 12 revolutions 0.46 sec M/W~0.1

  32. Example 2: Split-Finger Fastball 85-90 mph 1300 rpm (top) 12 revolutions 0.46 sec M/W~0.1

  33. Example 3: Curveball 70-80 mph 1900 rpm (top and side) 17 revolutions 0.55 sec M/W~0.25

  34. Example 4: Slider 75-85 mph 1700 rpm (side) 14 revolutions 0.51 sec M/W~0.15

  35. Summary • Much of baseball can be understood with basic principles of physics • Conservation of momentum, angular momentum, energy • Dynamics of collisions • Excitation of normal modes • Trajectories under influence of forces • gravity, drag, Magnus,…. • There is probably much more that we don’t understand • Don’t let either of these interfere with your enjoyment of the game!

  36. x2 x1 Sweet Spot #2: Center of Percussion • When ball strikes bat... • Linear recoil • conservation of momentum • Rotation about center of mass • conservation of angular momentum • When COP hit • The two motions cancel (at conjugate point) • No reaction force felt x1x2=Icm/M

  37. bat speed vs mass ball speed vs mass But… • All things are not equal • Mass & Mass Distribution affect bat speed • Conclusion: • mass of bat matters….but probably not a lot

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