The Physics of Baseball (or…Just How Did McGwire Hit 70. ) Alan M 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 Personal note: physics all my professional life; baseball much longer; connection only recently. But you also know that you really don’t have to be Einstein to understand or play the game. After all, it isn’t rocket science, or brain surgery, or even nuclear physics. But there is more to the game than meets the untrained eye. In my day job as an nuclear physicist, I do experiments involving the collision of subatomic particles. It really is no exaggeration that some of the same physics principles that apply in those collisions also applies in a very different type of collision, that between the baseball and bat. So, in this talk, I hope to give you an appreciation of some of the ways that physics can help describe the game of baseball and in the process give you a whole different way of looking at the game. It won’t make you another Mark McGwire but it should take you up a notch in your level of appreciation. So, with that in mind, here is an outline of my talk. Now, let’s play hardball! 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 Introduction: Adair book very nice. Lots of interesting facts about the game. Good insight as to how a physicist thinks about the world around him. Intellectually honest: tells you what we know and understand very well, what we can make an educated guess about, and what we don’t even have a clue about. Goal is not to use physics to improve the game. Through over 100 years of evolution, ball players know how to play the game, even though they may not really know why they do what they do. They know what works. Our goal is to understand the game. So…let’s begin

“...the most difficult thing to do in sports” Hitting the Baseball “...the most difficult thing to do in sports” --Ted Williams, Professor Emeritus of Hitting Round ball, round bat, squarely <0.5 s; half of break after decision reached; timing (.01s -->1’) Physiological problem…energy from muscles to bat Lots of ways to fail (early, late, up, down, along the bat…) Compare with golf, where ball is not even moving! Hard! Compare with subatomic collisions--fascinating

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 HOLD BAT & BALL Broad subject Focus on 1 ms during actual collision What physics can we bring to bear …or, what aspects of collision lead to well-hit ball If you want to optimize your chances of getting a hit, you want to maximize the speed of the hit ball. whether swing for fences or line drive through infield neglect place hitting, which is another art by itself What features of collision maximize speed of hit ball as it leaves the bat? NOTE: for home runs….105 mph/400’, 1mph/5’ GO TO JHS DEMO 80,60-->102 90,60-->105 90,70-->118

What Determines Batted Ball Speed? How does batted ball speed depend on ... pitched ball speed? bat speed? V = 0.25 Vball + 1.25 Vbat 1. SHOW GRAPHS: Bat speed more important than ball speed, although both are relevant. You can hit home from lobbed pitch (or fungo), perhaps with some difficulty, but you surely cannot hit homer by bunting. This agrees with your intuition. How can we understand the physics of this? Formula: 10 mph of ball speed is worth 2.5 mph 10 mph of bat speed is worth 12.5 mph 2. DISCUSS: Principle is conservation of momentum. Momentum is mass times velocity. Bat has much more of it than ball because it is so much heavier. During the collision, some of the bats momentum is transferred to the ball, turning it around and sending it out with a larger speed than it had coming in. The bat loses an equal amount of momentum, but the total momentum of bat plus ball is the same before as after the collision. It is because the bat is so much heavier than the ball that it’s speed is much more important than the pitched ball speed. Our Physics 111 students actually do this situation as a homework problem, so the physics is fairly basic. Conclusion: Bat Speed Matters More!

What Determines Batted Ball Speed? Mass of bat Conclusion: mass of bat matters ...but not a lot Insect sitting on bat (90+70=160 mph). Violent collision. Ball reverses direction, goes out at 100 mph. Bat exerts large force on ball. Ball exerts equal and opposite force on bat. Bat recoil backwards. Energy of bat is energy robbed from ball. Heavier the bat, less the recoil. Billiard balls… Billiard ball on bowling ball… Billiard ball on brick wall…. Diminishing returns Moreover…speed of bat depends on weight of bat. Tradeoff. Tendency to use lighter bats (30-35 oz): more control; can wait longer; ... NCAA (L-5; L-3)

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! SO FAR: mass,speed of ball,bat; conservation of momentum; before, after collision NOT ENOUGH: superball NEED TO LOOK MICROSCOPICALLY (DURING) The interesting stuff: where does energy go? How does this affect what happens? What are implications for bat/ball design? Nuclear physics analogy! Important point: time is very short. Ball doesn’t know the hands are there. You could let go of the bat 1 msec before collision and it would make no difference! (a controversial statement)

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! SO FAR: mass,speed of ball,bat; conservation of momentum; before, after collision NOT ENOUGH: superball,nurf ball (SHOW THESE) NEED TO LOOK MICROSCOPICALLY (DURING) The interesting stuff: where does energy go? How does this affect what happens? What are implications for bat/ball design? Nuclear physics analogy! Important point: time is very short. Ball doesn’t know the hands are there. You could let go of the bat 1 msec before collision and it would make no difference! (a controversial statement)

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-------> Conservation of momentum not full story. Try hitting superball! All other things being equal, superball will go much further! Bounciness, or elasticity, of ball. Has to do with amount of kinetic energy--energy of motion--lost in the collision. DEMO: DROP BASEBALL FROM FLOOR; DROP SUPERBALL FROM FLOOR. Whatever this bounciness is, superball has more of it. MODEL (the way physicists work): springs, friction, lost energy. COR. Same model tells us size of forces and time over which they act (has to do with vibrational motion of springs). Test procedures: 85 mph on ash/concrete. Not sensitive to “inside”. General principle…deep interior-->higher speed More typical: 160 mph!

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 Result: “trampoline effect” BPF ~ 1.2 Ball flies off the bat! A more efficient bat and/or ball DEMO: BASEBALL ON FLOOR, TENNIS RACKET DEAD BALL ON FLOOR, TENNIS RACKET SUPERBALL ON FLOOR, SPONGE For well-hit ball (near sweet spot)... The issue: springiness of surface. Bat compresses ball, ball compresses bat--same size force--which one gives depends on how easily compressed. Wood bat, for given force, compresses 1/50 of ball. Or…only 1/50 of energy in collision is stored in bat, rest is in ball. Tennis racket--effective BPF>1, just like Al bat. Al bat: 10% in COR==>7 mph==>35’ or more COST! new rule changes in NCAA nails! Better bat: a tennis racket Better ball: spherical metallic shell

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 DEMO with real bat. Distribution of weight: 1. How far from handle-->affects swing (NCAA: specifies weight but ignores I) 2. How weight is distributed about CM--->affects energy lost to rotation Stiffness affects energy lost to vibrations in bat

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 CM: define Hit at CM: recoils (momentum conservation) Hit anywhere else: recoils and rotates (two separate motions) Angular Momentum conversation: Amount of recoil independent of where you hit it. Amount of rotation get larger the further from CM hi whack it DEMO with swinging bat. REMARKS: tennis racket golf putter

Sweet Spot #2: Maximum Energy Transfer Barrel end of bat maximizes bat speed Center of Mass minimizes angular impulse MET must be in between Not on COP! Aluminum bat more effective for inside pitches CM COP DISCUSS: Wood vs. Al For Al, more uniform wt. Distribution-->CM closer to handle For same reason, higher I about CM (sort of same reason wider head tennis racket is more effective over a larger region) Both conspire to keep curve higher for inside, lower for outside

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 DEMO: wiggle nurf bat strike bat (hear and feel): 160 Hz, 560 Hz, … NOTE: nuclear physics analogy Principle: time/frequency Al bat: stiff to bending, esp in handle region--> higher frequencies (200 Hz vs 165 Hz)-->less effectively excited in collision. More forgiving for inside pitch! NOTE: no hands approx probably breaks down for balls hit far from node. I.e., firm grip helps for mishit balls.

So you think bats cannot bend…..

So you think bats cannot bend…..

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 Wood: Not much one can do, given constraints imposed by rules. Aluminum: NCAA!

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

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 Force in direction front of ball is turning DRAG: Ball has to push air out of the way Air follows contours, then breaks off in turbulant wake Result…high pressure in front, low pressure in back-->drag Separation further to front as v increases--> Magnus Ball pulls air on top further around than air on bottom.

The Flight of the Balll Role of Drag Role of Spin Atmospheric conditions Temperature Humidity Altitude Air pressure Wind ROLE OF DRAG: factor of 2 in distance optimum angle from 45 deg to 35 deg not parabolic orbit (outfielders know this! ball goes up, then “dies” and just sort of falls MAGNUS: keeps ball in air longer. Esp impt for golf. 100’ altitude +7’ 10 deg air temp +4’ 10 deg ball temp +4’ 1” drop in barometer +6’ 1 mph following wind +3’ ball at 100% humidity -30’ hit along foul line +11’

The Home Run Swing The optimum home run angle! Ball arrives on 100 downward trajectory Big Mac swings up at 250 Ball takes off at 350 The optimum home run angle! So how does he do it? Strong==>ability to get high bat speed quickly; good technique Show movie of #70 NOTE: 10o is ideal “contact” angle

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

Pitching the Baseball Don Larsen, 1956 World Series Last pitch of perfect game “Hitting is timing. Pitching is upsetting timing” ---Warren Spahn vary speeds manipulate air flow orient stitches

Let’s Get Quantitative! I. How Large are the Forces? Terminal velocity Drag is comparable to weight Magnus force < 1/4 weight)

Let’s Get Quantitative! II. How Much Does the Ball Break? 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 Skip; show JHS trajectories instead.

Example 1: Fastball 85-95 mph 1600 rpm (back) 12 revolutions 0.46 sec M/W~0.1 Gravity: falls ~3.5 ft. Magnus is 0.1 W and up Hence…falls 4” less…enough to cause a problem in a game of inches NOTE: no rising fastball! TOSS STYROFOAM BALL

Example 2: Split-Finger Fastball 85-90 mph 1300 rpm (top) 12 revolutions 0.46 sec M/W~0.1 Falls ~4” more than gravity, 8” more than normal fastball! TOSS STYROFOAM BALL

Example 3: Curveball 70-80 mph 1900 rpm (top and side) 17 revolutions 0.55 sec M/W~0.25 Magnus is larger (spin,vel) Velocity is smaller Total break can be upwards of 18” Usually drops/breaks TOSS STYROFOAM BALL

Example 4: Slider 75-85 mph 1700 rpm (side) 14 revolutions 0.51 sec M/W~0.15 Faster than curveball, less break, side break

Examples of Trajectories 3 4 5 6 7 10 20 30 40 50 60 Vertical Position of Ball (feet) Distance from Pitcher (feet) 90 mph Fastball 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 Horizontal Deflection of Ball (feet) Distance from Pitcher (feet) 75 mph Curveball skip

Obstructions cause turbulance Turbulance reduces drag 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 skip

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!