1. ODU Colloquium, March 31, 2000 Page 1 The Physics of Baseball Alan M. Nathan University of Illinois ODU Colloquium, March 31, 2000 l Introduction l Hitting the Baseball l The Flight of the Baseball l Pitching the Baseball l Summary

2. ODU Colloquium, March 31, 2000 Page 2 REFERENCES l The Physics of Baseball, Robert K. Adair (Harper Collins, New York, 1990), ISBN 0-06-096461-8 l The Physics of Sports, Angelo Armenti (American Institute of Physics, New York, 1992), ISBN 0-88318-946-1 l www.npl.uiuc.edu/~a-nathan/pob

3. ODU Colloquium, March 31, 2000 Page 3 Hitting the Baseball “...the most difficult thing to do in sports” --Ted Williams BA:.344 SA:.634 OBP:.483 HR: 521 #521, September 28, 1960

4. ODU Colloquium, March 31, 2000 Page 4 Here’s Why….. (Courtesy of Bob Adair)

5. ODU Colloquium, March 31, 2000 Page 5 Description of Ball-Bat Collision l forces large (>8000 lbs!) l time is short (<1/1000 sec!) l ball compresses, stops, expands l kinetic energy  potential energy l bat affects ball….ball affects bat l hands don’t matter! l GOAL: maximize ball exit speed v f v f  105 mph  x  400 ft  x/  v f = 5 ft/mph What aspects of collision lead to large v f ?

6. ODU Colloquium, March 31, 2000 Page 6 l What happens when ball and bat collide? YThe simple stuff c conservation of momentum c conservation of angular momentum c energy dissipation in the ball (compression/expansion) YThe really interesting stuff c vibrations of the bat How to maximize v f ?

7. ODU Colloquium, March 31, 2000 Page 7 The Simple Stuff: Rigid-Body Kinematics V ball,f = 0.25 V ball,i + 1.25 V bat,i Conclusion: v bat much more important than v ball “radius of gyration” e  Coefficient of Restitution  0.5 r  recoil factor =  0.2

8. ODU Colloquium, March 31, 2000 Page 8 Recoil Factor. Translation. Rotation CM. z Important Bat Parameters: m bat, x CM, I CM wood vs. aluminum Conclusion: All things being equal, want m bat, I bat large 0.16 + 0.07 = 0.23

9. ODU Colloquium, March 31, 2000 Page 9 Coefficient of Restitution (e) l “bounciness” of ball Y Bounce ball off massive hard surface Y e 2 = h f /h i l For baseball, e .5 l  3/4 energy lost! l Changing e by.05 changes V by 7 mph (35 ft!) Important Point: the bat matters too!

10. ODU Colloquium, March 31, 2000 Page 10 l Energy shared between ball and bat l Ball is inefficient:  25% returned l Wood Bat Yk ball /k bat ~ 0.02 Y  80% restored Ye eff = 0.50-0.51 l Aluminum Bat Yk ball /k bat ~ 0.10 Y  80% restored Ye eff = 0.55-0.58 c“trampoline effect” c Bat Proficiency Factor  e eff /e c Claims of BPF  1.2 Effect of Bat on COR E bat /E ball  k ball /k bat   x bat /  x ball >10% larger! tennis ball/racket

11. ODU Colloquium, March 31, 2000 Page 11 Rigid-Body Results Aluminum bat more effective for inside pitches CM v ball,I = 90 mph v bat,CM = 54 mph  bat,CM = 51 s -1

12. ODU Colloquium, March 31, 2000 Page 12 l Collision excites bending vibrations in bat YOuch!! Thud!! YSometimes broken bat YEnergy lost  lower v f l Lowest modes easy to find by tapping l Location of nodes important l Modes with f n   1 excited Beyond the Rigid Approximation: A Dynamic Model for the Bat-Ball collision Ref.: AMN, Am. J. Phys, submitted March 2000

13. ODU Colloquium, March 31, 2000 Page 13 20 y z y A Dynamic Model of the Bat-Ball Collision Solve eigenvalue problem for normal modes (y n,  n ) Model ball-bat force F Expand y in normal modes Solve coupled equations of motion for ball, bat

14. ODU Colloquium, March 31, 2000 Page 14 In a bit more detail… impact point ball compression

15. ODU Colloquium, March 31, 2000 Page 15 f 1 = 165 Hz f 2 = 568 Hz f 3 = 1177 Hz f 4 = 1851 Hz Results: 1. Normal Modes Louisville Slugger R161 (34”, 31 oz) Can be measured (modal analysis) nodes

16. ODU Colloquium, March 31, 2000 Page 16 Theory vs. Experiment (Rod Cross) (at 1 m/s) collision time  2.2 ms Results: 2. Low-speed collision

17. ODU Colloquium, March 31, 2000 Page 17 Results: 3. High-speed collision Under realistic conditions… 90 mph, 70 mph at 28” no data (yet)….. CMnodes

18. ODU Colloquium, March 31, 2000 Page 18 Results: 4. The “sweet spot” CMnodes 24” 27” 30” Possible “sweet spots” 1. Maximum of v f (28”) 2. Node of fundamental (27”) 3. Center of Percussion (27”)

19. ODU Colloquium, March 31, 2000 Page 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. ODU Colloquium, March 31, 2000 Page 20 How Would a Physicist Design a Bat? l Wood Bat Yalready optimally designed chighly constrained by rules! Ya marvel of evolution! l Aluminum Bat Ylots of possibilities exist Ybut not much scientific research Ya great opportunity for... cfame cfortune

21. ODU Colloquium, March 31, 2000 Page 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. ODU Colloquium, March 31, 2000 Page 22 Aerodynamics of a Baseball Forces on Moving Baseball No Spin  Boundary layer separation  DRAG!  F D =½C D  Av 2 With Spin  Ball deflects wake ==>Magnus force  F M  RdF D /dv  Force in direction front of ball is turning

23. ODU Colloquium, March 31, 2000 Page 23 How Large are the Forces? Drag is comparable to weight Magnus force < 1/4 weight)  =1800 RPM

24. ODU Colloquium, March 31, 2000 Page 24 The Flight of the Ball: Real Baseball vs. Physics 101 Baseball l Role of Drag l Role of Spin l Atmospheric conditions YTemperature YHumidity YAltitude YAir pressure YWind approx linear Max @ 35 0

25. ODU Colloquium, March 31, 2000 Page 25 The Role of Friction l Friction induces spin for oblique collisions l Spin  Magnus force l Results YBalls hit to left/right break toward foul line YBackspin keeps fly ball in air longer YTopspin gives tricky bounces in infield YPop fouls behind the plate curve back toward field

26. ODU Colloquium, March 31, 2000 Page 26 The Home Run Swing Ball arrives on 10 0 downward trajectory Big Mac swings up at 25 0 Ball takes off at 35 0 The optimum home run angle!

27. ODU Colloquium, March 31, 2000 Page 27 Pitching the Baseball “Hitting is timing. Pitching is upsetting timing” ---Warren Spahn l vary speeds l manipulate air flow l orient stitches

28. ODU Colloquium, March 31, 2000 Page 28 Let’s Get Quantitative! How Much Does the Ball Break? l Kinematics Yz=vT Yx=½(F/M)T 2 l Calibration Y90 mph fastball drops 3.5’ due to gravity alone YBall reaches home plate in ~0.45 seconds l Half of deflection occurs in last 15’ l Drag:  v  -8 mph l Examples: Y“Hop” of 90 mph fastball ~4” YBreak of 75 mph curveball ~14” cslower cmore rpm cforce larger 3 4 5 6 7 0102030405060 Vertical Position of Ball (feet) Distance from Pitcher (feet) 90 mph Fastball 0 0.2 0.4 0.6 0.8 1 1.2 0102030405060 Horizontal Deflection of Ball (feet ) Distance from Pitcher (feet) 75 mph Curveball

29. ODU Colloquium, March 31, 2000 Page 29 Examples of Pitches Pitch V(MPH)  (RPM)TM/W fastball 85-95 16000.460.10 slider 75-85 17000.510.15 curveball 70-80 19000.550.25 What about split finger fastball?

30. ODU Colloquium, March 31, 2000 Page 30 Effect of the Stitches l Obstructions cause turbulance l Turbulance reduces drag YDimples on golf ball YStitches on baseball l Asymmetric obstructions YKnuckleball YTwo-seam vs. four-seam delivery YScuffball and “juiced” ball

31. ODU Colloquium, March 31, 2000 Page 31 Example 1: Fastball 85-95 mph 1600 rpm (back) 12 revolutions 0.46 sec M/W~0.1

32. ODU Colloquium, March 31, 2000 Page 32 Example 2: Split-Finger Fastball 85-90 mph 1300 rpm (top) 12 revolutions 0.46 sec M/W~0.1

33. ODU Colloquium, March 31, 2000 Page 33 Example 3: Curveball 70-80 mph 1900 rpm (top and side) 17 revolutions 0.55 sec M/W~0.25

34. ODU Colloquium, March 31, 2000 Page 34 Example 4: Slider 75-85 mph 1700 rpm (side) 14 revolutions 0.51 sec M/W~0.15

35. ODU Colloquium, March 31, 2000 Page 35 Summary l Much of baseball can be understood with basic principles of physics YConservation of momentum, angular momentum, energy YDynamics of collisions YExcitation of normal modes YTrajectories under influence of forces cgravity, drag, Magnus,…. l There is probably much more that we don’t understand l Don’t let either of these interfere with your enjoyment of the game!

36. ODU Colloquium, March 31, 2000 Page 36 Sweet Spot #2: Center of Percussion l When ball strikes bat... YLinear recoil cconservation of momentum YRotation about center of mass cconservation of angular momentum l When COP hit YThe two motions cancel (at conjugate point) YNo reaction force felt x1x1 x2x2 x 1 x 2 =I cm /M

37. ODU Colloquium, March 31, 2000 Page 37 But… l All things are not equal l Mass & Mass Distribution affect bat speed Conclusion: mass of bat matters….but probably not a lot bat speed vs mass ball speed vs mass