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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 H. Brody, AJP 54 , 640 (1986); AJP 58 , 756 (1990)

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**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 • H. Brody, AJP 54, 640 (1986); AJP 58, 756 (1990) • P. Kirkpatrick, AJP 31, 606 (1963) • L. Van Zandt, AJP 60, 172 (1992) • R. Cross, AJP 66, 772 (1998); AJP 67, 692 (1999) • AMN, AJP 68, to appear in Sept. 2000 • L. Briggs, AJP 27, 589 (1959) • R. Mehta, Ann. Ref. Fluid Mech. 17, 151 (1985) • www.npl.uiuc.edu/~a-nathan/pob**A Philosophical Note:**“…the physics of baseball is not the clean, well-defined physics of fundamental matters but the ill-defined physics of the complex world in which we live, where elements are not ideally simple and the physicist must make best judgments on matters that are not simply calculable…Hence conclusions about the physics of baseball must depend on approximations and estimates….But estimates are part of the physicist’s repertoire…a competent physicist should be able to estimate anything ...” “The physicist’s model of the game must fit the game.” “Our aim is not to reform baseball but to understand it.” ---Bob Adair in “The Physics of Baseball”, May, 1995 issue of Physics Today**#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**Here’s Why…..**(Courtesy of Robert K. Adair)**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 = 4-5 ft/mph more later What aspects of collision lead to large vf?**How to maximize vf?**• What happens when ball and bat collide? • The simple stuff: kinematics • conservation of momentum • conservation of angular momentum • The really interesting stuff: energy dissipation • compression/expansion of ball • vibrations of the bat**The Simple Stuff: Kinematics**r recoil factor 0.25 e Coefficient of Restitution 0.5 vball,f = 0.2 vball,i + 1.2 vbat,i Conclusion: vbatmuch more important than vball Question: what bat/ball properties make vball,flarge?**Sosa’s 500’ Blast(s) in Home Run Derby:**A Numerical Analysis • D = 500 • vf 127 mph • vball,i 60 mph • vbat,i 96 mph! • if vball,i were 90 mph • D = 530**.**. . CM z Translation Rotation Energy in Bat Recoil • Important Bat Parameters: • mbat, xCM, ICM • wood vs. aluminum Want rsmall to mimimize recoil energy 0.17 + 0.07 = 0.24 Conclusion: All things being equal, want mbat, Ibat large**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 • see Watts & Bahill, Keep Your Eye on the Ball, 2nd edition, ISBN 0-7167-3717-5**“bounciness” of ball**Energy Dissipated: Coefficient of Restitution (e): • in CM frame: Ef/Ei = e2 • massive rigid surface: e2 = hf/hi • typically e 0.5 • ~3/4 CM energy dissipated! • depends on ball, surface, speed,... • is the ball “juiced”?**COR and the “Juiced Ball” Issue**• MLB:e = 0.546 0.032 @ 58 mph on massive rigid surface Conclusion: more systematic studies needed**tennis ball/racket**Ebat/Eball kball/kbat xbat/ xball >10% larger! Effect of Bat on COR: Local Compression • CM energy shared between ball and bat • Ball is inefficient: 75% dissipated • 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 Recent BPF data: (Lansmont BBVC/Trey Crisco) 0.99wood 1.12 aluminum More later on wood vs. aluminum**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 see AMN, Am. J. Phys, 68, in press (2000)**y**y z A Dynamic Model of the Bat-Ball Collision 20 Euler-Bernoulli Beam Theory‡ • 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 ‡Note for experts: full Timoshenko (nonuniform) beam theory used**f1 = 177 Hz**f3 = 1179 Hz f2 = 583 Hz f4 = 1821 Hz nodes Normal Modesof the Bat Louisville Slugger R161 (33”, 31 oz) Can easily be measured (modal analysis)**Measurements via Modal Analysis**Louisville Slugger R161 (33”, 31 oz) frequencybarrel node ExptCalcExptCalc 17917726.526.6 58258327.828.2 1181117929.029.2 1830182130.029.9 Conclusion: free vibrations of bat can be well characterized**F=kxn**F=kxm Model for the Ball 3-parameter problem: k nv-dependence of m COR**ball compression**impact point Putting it all together…. Expectation: only modes with fn < 1 strongly excited**Results: Ball Exit Speed**Louisville Slugger R161 33-inch/31-oz. wood bat only lowest mode excited lowest 4 modes excited Conclusion:essential physics under control**CM**nodes • Under realistic conditions… • 90 mph, 70 mph at 28”**nodes**24” 27” 30” Results: The “sweet spot” Possible “sweet spots” 1. Maximum of vf (~28”) 2. Node of fundamental (~27”) 3. Center of Percussion (~27”) Hands don’t matter!**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**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**Things I would like to understand better**• Relationship between bat speed and bat weight and weight distribution • Effect of “corking” the bat • Location of “physiological” sweet spot • Better model for the ball • FEA analysis of aluminum bat • Why is softball bat different from baseball bat?**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**Aerodynamics of a Baseball**Forces on Moving Baseball • No Spin • Boundary layer separation • DRAG! • FD=½CDAv2 • With Spin • Ball deflects wake ==>Magnus force • FMRdFD/dv • Force in direction front of ball is turning Drawing courtesty of Peter Brancazio**How Large are the Forces?**=1800 RPM • Drag is comparable to weight • Magnus force < 1/4 weight)**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**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**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!**“Hitting is timing. Pitching is**upsetting timing” ---Warren Spahn vary speeds manipulate air flow orient stitches Pitching the Baseball**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**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?**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**Example 1: Fastball**85-95 mph 1600 rpm (back) 12 revolutions 0.46 sec M/W~0.1**Example 2: Split-Finger Fastball**85-90 mph 1300 rpm (top) 12 revolutions 0.46 sec M/W~0.1**Example 3: Curveball**70-80 mph 1900 rpm (top and side) 17 revolutions 0.55 sec M/W~0.25**Example 4: Slider**75-85 mph 1700 rpm (side) 14 revolutions 0.51 sec M/W~0.15**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!

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