The Physics of Baseball

The Physics of Baseball (or…Just How Did McGwire Hit 70?) Alan M. Nathan University of Illinois February 5, 1999   Introduction Hitting the Baseball The Flight of the Baseball Pitching the Baseball    Summary Physics of Baseball: Page 1 REFERENCES  The Physics of Baseball, Robert K. Adair (Harper Collins, New York, 1990), ISBN 0-06-096461-8  The Sporting Life, Davis and Stephens (Henry Holt and Company, New York, 1997), ISBN 0-8050-4540-6 http://www.exploratorium.edu/sports ME!   » a-nathan@uiuc.edu » www.npl.uiuc.edu/~nathan Physics of Baseball: Page 2 Hitting the Baseball “...the most difficult thing to do in sports” --Ted Williams, Professor Emeritus of Hitting Physics of Baseball: Page 3 Speed of Hit Ball: What does it depend on?  Speed is important: 105 mph gives 400 ft each mph is worth 5 ft The basic stuff (“kinematics”) speed of pitched ball speed of bat  weight of bat  The really interesting stuff (“dynamics”) “bounciness” of ball and bat weight distribution of bat vibrations of bat Physics of Baseball: Page 4 What Determines Batted Ball Speed?  How does batted ball speed depend on ... pitched ball speed? bat speed? V = 0.25 Vball + 1.25 Vbat Conclusion: Bat Speed Matters More! Physics of Baseball: Page 5 What Determines Batted Ball Speed?  Mass of bat Conclusion: mass of bat matters ...but not a lot  Physics of Baseball: Page 6 Dynamics of Ball-Bat Collision  Ball compresses kinetic energy stored in “spring” Ball expands kinetic energy restored but...  70% of energy is lost! (heat, deformation,vibrations,...)    Forces are large (>5000 lbs!) Time is short (<1/1000 sec!) The hands don’t matter! Physics of Baseball: Page 7 Dynamics of Ball-Bat Collision after  during before Ball compresses kinetic energy stored in “spring” Ball expands kinetic energy restored but...  70% of energy is lost! (heat, deformation,vibrations,...)    Forces are large (>5000 lbs!) Time is short (<1/1000 sec!) The hands don’t matter! Physics of Baseball: Page 8 The Coefficient of Restitution     COR measures “bounciness” of ball Final speed/Initial speed For baseball, COR=.52-.58 Changing COR by .05 changes V by 7 mph (35 ft!)  How to measure? This is square of COR-------> Physics of Baseball: Page 9 What About the Bat? (or, it takes two to tango!)  Wood Bat Efficiently restores energy But only 2% energy stored Bat Performance Factor (BPF) ~1 .02  Aluminum Bat Stores ~ 20% energy Efficiently restores energy Result: “trampoline effect” » BPF ~ 1.2 » Ball flies off the bat!  A more efficient bat and/or ball Physics of Baseball: Page 10 Properties of Bats  length, diameter   weight position of center of gravity where does it balance? distribution of weight “moment of inertia”    center of percussion stiffness and elasticity vibrational nodes and frequencies Physics of Baseball: Page 11 Sweet Spot #1: Center of Percussion   When ball strikes bat... Linear recoil » conservation of momentum Rotation about center of mass » conservation of angular momentum When CP hit The two motions cancel at handle No reaction force felt at handle Physics of Baseball: Page 12 Sweet Spot #2: Maximum Energy Transfer  Barrel end of bat maximizes bat speed Center of Mass minimizes angular impulse CM COP    MET must be in between Not on COP! Aluminum bat more effective for inside pitches Physics of Baseball: Page 13 Sweet Spot #3: “Node” of Vibration  Collision excites bending vibrations in bat Ouch!! Energy lost ==>lower COR Sometimes broken bat  Reduced considerably if collision is a node of fundamental mode Fundamental node easy to find For an interesting discussion, see www.physics.usyd.edu.au/~cross   Physics of Baseball: Page 14 So you think bats cannot bend….. Physics of Baseball: Page 15 So you think bats cannot bend….. Physics of Baseball: Page 16 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 Physics of Baseball: Page 17 Advantages of Aluminum  Length and weight “decoupled” Can adjust shell thickness  More compressible => “springier” Trampoline effect  More of weight closer to hands Easier to swing Less rotational energy transferred to bat More forgiving on inside pitches  Stiffer for bending Less energy lost due to vibrations Physics of Baseball: Page 18 Aerodynamics of a Baseball Forces on Moving Baseball  No Spin Boundary layer separation DRAG! Grows with v2  With Spin Ball deflects wake action/reaction==>Magnus force » Force grows with rpm Pop Pbot tom » Force in direction front of ball is turning Physics of Baseball: Page 19 The Flight of the Balll    Role of Drag Role of Spin Atmospheric conditions Temperature Humidity Altitude Air pressure Wind Physics of Baseball: Page 20 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! Physics of Baseball: Page 21 Physics of Baseball: Page 22 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 Physics of Baseball: Page 23 Pitching the Baseball   “Hitting is timing. Pitching is upsetting timing” ---Warren Spahn   Don Larsen, 1956 World Series Last pitch of perfect game  vary speeds manipulate air flow orient stitches Physics of Baseball: Page 24 Let’s Get Quantitative! I. How Large are the Forces? Drag/Weight or Magnus/Weight 2 1.5 1 0.5 0 Magnus/Weight Drag/Weight 0 25 50 75 100 Speed in mph 125 150 • Drag is comparable to weight • Magnus force < 1/4 weight) Physics of Baseball: Page 25 Let’s Get Quantitative! II. How Much Does the Ball Break?  Drag/Weight or Magnus/Weight     Depends on… Magnitude and direction of force Time over which force acts 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 reduces fastball by about 8 mph Examples: Hop of 90 mph fastball: ~4” Break of 70 mph curveball ~16” » slower » force larger 2 1.5 1 0.5 0 Magnus/Weight Drag/Weight 0 25 50 75 100 Speed in mph 125 150 Physics of Baseball: Page 26 Example 1: Fastball 85-95 mph 1600 rpm (back) 12 revolutions 0.46 sec M/W~0.1 Physics of Baseball: Page 27 Example 2: Split-Finger Fastball 85-90 mph 1300 rpm (top) 12 revolutions 0.46 sec M/W~0.1 Physics of Baseball: Page 28 Example 3: Curveball 70-80 mph 1900 rpm (top and side) 17 revolutions 0.55 sec M/W~0.25 Physics of Baseball: Page 29 Example 4: Slider 75-85 mph 1700 rpm (side) 14 revolutions 0.51 sec M/W~0.15 Physics of Baseball: Page 30 Examples of Trajectories Vertical Position of Ball (feet) 7 6 5 90 mph Fastball 4 3 0 10 20 30 40 50 60 Distance from Pitcher (feet) Horizontal Deflection of Ball (feet) 1.2 1 75 mph Curveball 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 Distance from Pitcher (feet) Physics of Baseball: Page 31 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 Physics of Baseball: Page 32  Summary  Much of baseball can be understood with basic principles of physics Conservation of momentum, angular momentum, energy Dynamics of collisions 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! Physics of Baseball: Page 33