1. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 1 The Physics of Hitting a Home Run Colloquium, St. Mary’s University Halifax, October 4, 2002 Alan M. Nathan University of Illinois at Urbana-Champaign a-nathan@uiuc.edu http://www.npl.uiuc.edu/~a-nathan/pob l Introduction l Model for colinear ball-bat collisions  some applicatons l Beginners guide to aerodynamics l Model for oblique ball-bat collisions  some applications l Summary & Conclusions

2. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 2 1927 Solvay Conference: Greatest physics team ever assembled Baseball and Physics 1927 Yankees: Greatest baseball team ever assembled MVP’s

3. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 3 Hitting the Baseball “... the most difficult thing to do in sports” -- Ted Williams: 1918-2002 BA:.344 SA:.634 OBP:.483 HR: 521 #521, September 28, 1960

4. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 4 Introduction: Description of Ball-Bat Collision l forces large (>8000 lbs!) l time short (<1/1000 sec!) l ball compresses, stops, expands  kinetic energy  potential energy  lots of energy lost l bat is flexible  hands don’t matter! l to hit a home run...  large hit ball speed  optimum take-off angle  backspin

5. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 5 The Ball-Bat Collision: Kinematics v f = e A v ball + (1+e A ) v bat Conclusion: v bat matters much more than v ball v ball v bat vfvf “Lab” Frame v rel e A v rel Bat Rest Frame e A  “Collision Efficiency” property of ball & bat weakly dependent on v rel Superball-wall: e A  1 Ball-Bat near “sweet spot”: e A  0.2  v f  0.2 v ball + 1.2 v bat

6. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 6 What Does e A Depend On? Kinematics: recoil of bat (r) Dynamics: energy dissipation (COR) Small r is best r  0.25 typical…depends on …. mass of bat mass distribution of bat impact location.. CM. b = + Heavier bat is better but….

7. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 7 What is Ideal Bat Weight? Note: Batters seem to prefer lighter bats! Actually, Scaling with I knob better

8. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 8  v BAT (6”) = 1.2 mph/(1000 oz-in 2) (  v f =1.5  0.3 mph) v bat  I -0.3 v bat  I -0.5

9. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 9 Energy Dissipation: the ball-bat COR Coefficient Of Restitution in CM frame: E f /E i = COR 2 ball on hard floor: COR 2 = h f /h i  0.25  COR  0.5  ~3/4 CM energy dissipated! depends (weakly) on impact speed the bat matters too!  vibrations  “trampoline” effect

10. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 10 Aside: Effect of “Juiced” Ball MLB: COR = 0.546  0.032 @ 58 mph on massive rigid surface 10% increase in COR  ~30-35 ft increase in distance

11. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 11 l Collision excites bending vibrations in bat  Ouch!! Thud!! Sometimes broken bat  Energy lost  lower e, v f l Find lowest mode by tapping l Reduced considerably if  Impact is at a node  Collision time (~0.6 ms) > T N see AMN, Am. J. Phys, 68, 979 (2000) Accounting for Energy Dissipation: Dynamic Model for Ball-Bat Colllision

12. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 12 ball bat Mass= 1 2 4  >> 1 m on M a +M b (1 on 6)  << 1 m on M a (1 on 2) rigid flexible The Essential Physics: A Toy Model

13. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 13 The Details: A Dynamic Model 20 y z y l Step 1: Solve eigenvalue problem for free vibrations l Step 2: Nonlinear lossy spring for F l Step 3: Expand in normal modes and solve

14. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 14 Normal Modes of the Bat Louisville Slugger R161 (33”, 31 oz) f 1 = 177 Hz f 2 = 583 Hz f 3 = 1179 Hz f 4 = 1821 Hz Can easily be measured: Modal Analysis

15. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 15 frequency barrel node Expt Calc 179 17726.526.6 582 58327.828.2 1181117929.029.2 1830182130.029.9 Measurements via Modal Analysis Louisville Slugger R161 (33”, 31 oz) Conclusion: free vibrations of bat can be well characterized FFT

16. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 16 Theory vs. Experiment: Louisville Slugger R161 33-inch/31-oz. wood bat Conclusion: essential physics understood only lowest mode excited lowest 4 modes excited

17. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 17 Time evolution of the bat T= 0-1 ms T= 1-10 ms Ball leaves bat Conclusion: Knob end doesn’t matter

18. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 18 Effect of Bat on COR: Vibrations COR maximum near 2 nd node

19. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 19 Putting Everything Together... “sweet spot” depends on collision efficiency *recoil factor *COR how bat is swung CM v f = e A v ball + (1+e A ) v bat

20. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 20 Conclusion: ideal ball-bat collision can be simulated

21. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 21 Wood versus Aluminum Kinematics Length, weight, MOI “decoupled” shell thickness, added weight fatter barrel, thinner handle weight distribution more uniform CM closer to handle less mass at contact point  easier to swing Dynamics Stiffer for bending Less vibrational energy More compressible COR larger Question: Is the “sweet spot” larger for Al bat?

22. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 22 The “Trampoline” Effect l Compressional energy shared between ball and bat  PE bat /PE ball = k ball /k bat  PE ball mostly dissipated (75%) l Wood Bat  k ball /k bat <<1  little effect on COR: “BPF”  1 l Aluminum Bat  compressible through “shell” modes  k ball /k bat ~ 0.10 (more or less)  PE bat mostly restored (more on this later)  COR larger: “ BPF”  1.1-1.2

23. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 23 The Trampoline Effect: A Closer Look Bending Modes vs. Shell Modes k  R 4 : large in barrel  little energy stored f (170 Hz, etc) > 1/   energy goes into vibrations k  (t/R) 3 : small in barrel  more energy stored f (2-3 kHz) < 1/   energy mostly restored

24. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 24 Wood versus Aluminum: Dynamics of “Trampoline” Effect “bell” modes: “ping” of bat Want k small to maximize stored energy Want  >>1 to minimize retained energy Conclusion: there is an optimum 

25. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 25 Where Does the Energy Go?

26. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 26 “Corking” a Wood Bat (illegal!) Drill ~1” diameter hole along axis to depth of ~10” Smaller mass larger recoil factor (bad) higher bat speed (good) Is there a trampoline effect?

27. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 27 Not Corked DATA Corked COR: 0.445  0.005 0.444  0.005 Conclusions: no tramopline effect! corked bat is WORSE even with higher v bat Baseball Research Center, UML, Sherwood & amn, Aug. 2001 calculation

28. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 28 Aerodynamics of a Baseball Forces on Moving Baseball No Spin  Boundary layer separation  DRAG!  F D =½ C D  Av 2 With Spin  Ball deflects wake ==>”lift”  F M ~  RdF D /dv  Force in direction front of ball is turning Drawing courtesty of Peter Brancazio

29. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 29

30. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 30 approx linear: The Flight of the Ball: Real Baseball vs. Physics 101 Baseball

31. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 31 Summary of Aerodynamics l 108 mph  ~400 ft l each mph  ~5 ft l optimum angle ~35 0 l 2000 rpm backspin  Increases range ~27 ft  Decreases optimum angle ~3 0 l these number are only estimates!

32. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 32 Oblique Collisions: The Role of Friction l Friction halts v T  spin, “lift” l 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

33. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 33 Model for Oblique Collisions:  v N treated as before v Nf = e A (v ball +v bat ) N + v bat,N  Angular momentum conserved about contact point (!)  Friction reduces v T, increases   Rolls when v T = R   Horizontal: v Tf  (5/7)v T  Vertical: a bit more complicated  Not the way a superball works! vNvN vTvT vN|vN| vT|vT|

34. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 34 Oblique Collisions: Horizontal Plane Initial takeoff angle down the line power alley

35. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 35 Oblique Collisions: Vertical Plane optimum: D  0.75”  3000 rpm   33 0 Ball10 0 downward Bat 10 0 upward D = center-to-center offset

36. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 36 Typical Trajectories Ball10 0 downward Bat 10 0 upward D = center-to-center offset

37. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 37 Some Practical/Interesting Questions l Does more friction help? l Can a curveball be hit further than a fastball?

38. The Physics of Hitting a Home Run St. Mary’s University Colloquium October 4, 2002 Page 38 Summary and Conclusions l Some aspects of baseball are amenable to physics analysis  Kinematic and dynamics of ball-bat collision  Trajectory of a ball with drag and lift l Can understanding these things improve our ability to play the game?  Almost surely NOT l Can understanding these things enhance our own enjoyment of the game  For me, a resounding YES  I hope for you also