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The Physics of Hitting a Home Run

Thanks to J. J. Crisco & R. M. Greenwald Medicine & Science in Sports & Exercise 34(10): 1675-1684; Oct 2002. The Physics of Hitting a Home Run. Alan M. Nathan,University of Illinois www.npl.uiuc.edu/~a-nathan/pob a-nathan @uiuc.edu. 1927 Yankees: Greatest baseball team ever assembled.

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The Physics of Hitting a Home Run

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  1. Thanks to J. J. Crisco & R. M. Greenwald Medicine & Science in Sports & Exercise 34(10): 1675-1684; Oct 2002 The Physics of Hitting a Home Run Alan M. Nathan,University of Illinois www.npl.uiuc.edu/~a-nathan/pob a-nathan @uiuc.edu UW Colloquium 10/31/05

  2. 1927 Yankees: Greatest baseball team ever assembled 1927 Solvay Conference: Greatest physics team ever assembled MVP’s Baseball and Physics UW Colloquium 10/31/05

  3. “Hitting is timing; pitching is upsetting timing” “Hitting is fifty percent above the shoulders” Hitting the Baseball: the most difficult feat in sports 1955 Topps cards from my personal collection UW Colloquium 10/31/05

  4. Hitting and Pitching, Thinking and Guessing Graphic courtesy of Bob Adair and NYT UW Colloquium 10/31/05

  5. Example: Tim Wakefield’s Knuckleball UW Colloquium 10/31/05

  6. The Physics of Hitting a Home Run • How does a baseball bat work? • Why does aluminum outperform wood? • How does spin affect flight of baseball? • Can a curveball be hit farther than a fastball? UW Colloquium 10/31/05

  7. Brief Description of Ball-Bat Collision • forces large, time short • >8000 lbs, <1 ms • ball compresses, stops, expands • KEPEKE • bat bends & compresses • lots of energy dissipated (“COR”) • distortion of ball • vibrations in bat • to hit home run…. • large hit ball speed • optimum take-off angle • lots of backspin Courtesy of CEComposites UW Colloquium 10/31/05

  8. vball vbat vf Kinematics of Ball-Bat Collision vf = q vball + (1+q) vbat • q “Collision Efficiency” • property of ball & bat • independent of reference frame • ~independent of “end conditions”—more later • weakly dependent on vrel • Superball-wall: q  1 • Ball-Bat near “sweet spot”: q  0.2 •  vf 0.2 vball + 1.2 vbat Conclusion: vbat matters much more than vball UW Colloquium 10/31/05

  9. vball vbat vf q=0.20 Kinematics of Ball-Bat Collision • r = mball /Mbat,eff :bat recoil factor = 0.25 • (momentum and angular momentum conservation) • e:“coefficient of restitution” 0.50 • (energy dissipation—mainly in ball, some in bat) UW Colloquium 10/31/05

  10. Kinematics of Ball-Bat Collision • r = mball /Mbat,eff:bat recoil factor = 0.25 • (momentum and angular momentum conservation) • heavier bat better but… UW Colloquium 10/31/05

  11. The Ideal Bat Weight or Iknob Experiments: knob ~ (1/Iknob)0.3 Observation: Batters prefer lighter bats UW Colloquium 10/31/05

  12. Accounting for COR: Dynamic Model for Ball-Bat Collision AMN,Am. J. Phys, 68, 979 (2000) • Collision excites bending vibrations in bat • hurts! • breaks bats • dissipates energy • lower COR • lower vf UW Colloquium 10/31/05

  13. y 20 y z The Details: A Dynamic Model • Step 1: Solve eigenvalue problem for free vibrations • Step 2: Nonlinear lossy spring for ball-bat interaction F(t) • Step 3: Expand in normal modes and solve UW Colloquium 10/31/05

  14. f1 = 179 Hz f3 = 1181 Hz f2 = 582 Hz f4 = 1830 Hz frequency time Modal Analysis of a Baseball Bat www.kettering.edu/~drussell/bats.html UW Colloquium 10/31/05

  15. Some Interesting Insights:Bat Recoil, Vibrations, COR, and “Sweet Spot” Node of 1nd mode + e vf Evib ~ 1 ms  only lowest 4 modes excited UW Colloquium 10/31/05

  16. Experimental Data: Dependence of COR on Impact Location ball incident on bat at rest Conclusion:essential physics under control UW Colloquium 10/31/05

  17. Independence of End Conditions • handle moves only after ~0.6 ms delay • collision nearly over by then • nothing on knob end matters • size, shape • boundary conditions • hands UW Colloquium 10/31/05

  18. pitcher catcher Vibrations and Broken Bats inside outside node UW Colloquium 10/31/05

  19. Why Does Aluminum Outperform Wood? Aluminum has thin shell • Less mass in barrel • easier to swing and control  • but less effective at transferring energy  • for many bats  cancels  • Hoop modes • trampoline effect • larger COR  UW Colloquium 10/31/05

  20. The “Trampoline” Effect: A Simple Physical Picture • Two springs mutually compress each other • KE  PE  KE • PE shared between “ball spring” and “bat spring” • PE in ball mostly dissipated(~80%!) • PE in bat mostly restored • Net effect: less overall energy dissipated • ...and therefore higher ball-bat COR • …more “bounce” • Also seen in golf, tennis, … UW Colloquium 10/31/05

  21. The Trampoline Effect: A Closer Look • k  (t/R)3: hoop mode largest • in barrel • f2 (1-3 kHz) < 1/   1kHz •  energy mostly restored • (unlike bending modes) “hoop” modes: cos(2) Thanks to Dan Russell “ping” UW Colloquium 10/31/05

  22. Data and Model • to optimize…. • kbat small • fhoop > 1 essential physics understood UW Colloquium 10/31/05

  23. FL(Magnus)  Drag: Fd = ½ CDAv2 “Magnus” or “Lift”: FL= ½ CLAv2 Fd mg Effect of Spin on Baseball Trajectory (in direction leading edge is turning) CD~ 0.2-0.5 CL ~ R/v UW Colloquium 10/31/05

  24. New Experiment at Illinois • Fire baseball horizontally from pitching machine • Use motion capture to track ball over ~5m of flight and determine x0,y0,vx,vy,,ay • Use ay to determine Magnus force as function ofv,  UW Colloquium 10/31/05

  25. Motion Capture System Two-wheel pitching machine Baseball with reflecting dot Motion Capture ExperimentJoe Hopkins, Lance Chong, Hank Kaczmarski, AMN UW Colloquium 10/31/05

  26. y z Experiment: Sample MoCap Data topspin  ay > g y = ½ ayt2 work in progress UW Colloquium 10/31/05

  27. Some Typical Results • Lift … • --increases range • --reduces optimum angle UW Colloquium 10/31/05

  28. Oblique Collisions:Leaving the No-Spin Zone Friction … • sliding/rolling vs. gripping • transverse velocity reduced, spin increased vT′ ~ 5/7 vT  ~ vT′/R Familiar Results • Balls hit to left/right break toward foul line • Topspin gives tricky bounces in infield • Pop fouls behind the plate curve back toward field • Backspin keeps fly ball in air longer f UW Colloquium 10/31/05

  29. Ball100 downward D = center-to-center offset Bat 100 upward Undercutting the ball  backspin trajectories UW Colloquium 10/31/05

  30. Fastball: spin reverses Curveball: spin doesn’t reverse   larger for curveball UW Colloquium 10/31/05

  31. Can Curveball Travel Farther than Fastball? • Bat-Ball Collision Dynamics • A fastball will be hit faster • A curveball will be hit with more backspin • Aerodynamics • A ball hit faster will travel farther • Backspin increases distance • Which effect wins? • Curveball, by a hair! UW Colloquium 10/31/05

  32. Work in Progress • Collision experiments & calculations to elucidate trampoline effect • New measurements of lift and drag • Experiments on oblique collisions • Rod Cross & AMN: rolling almost works at low speed • AMN: studies in progress at high speed UW Colloquium 10/31/05

  33. Final Summary • Physics of baseball is a fun application of basic (and not-so-basic) physics • Check out my web site if you want to know more • www.npl.uiuc.edu/~a-nathan/pob • a-nathan@uiuc.edu • Go Red Sox! UW Colloquium 10/31/05

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