Physics of Baseball: Page 1 The Physics of Baseball (or…Just How Did McGwire Hit 70?) Alan M....

Physics of Baseball:

The Physics of BaseballThe Physics of Baseball

(or…Just How Did McGwire Hit 70?)(or…Just How Did McGwire Hit 70?)

Alan M. Nathan Alan M. Nathan University of IllinoisUniversity of Illinois

February 5, 1999February 5, 1999

IntroductionIntroduction

Hitting the BaseballHitting the Baseball

The Flight of the BaseballThe Flight of the Baseball

Pitching the BaseballPitching the Baseball

Summary


REFERENCESREFERENCES

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! » [email protected]» www.npl.uiuc.edu/~nathan


Hitting the BaseballHitting the Baseball

“...the most difficult thing to do in sports”

--Ted Williams,

Professor Emeritus of

Hitting


Speed of Hit Ball:Speed of Hit Ball:What does it depend on?What does it depend on?

Speed is important: 105 mph gives 400 fteach 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


What Determines Batted Ball Speed?What Determines Batted Ball Speed?

How does batted ball speed depend on ...pitched ball speed?bat speed?

V = 0.25 Vball + 1.25 Vbat

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100speed of pitched ball or bat (mph)

vary pitched ball speed

vary bat speed

Conclusion:Bat Speed Matters More!


What Determines Batted Ball Speed?What Determines Batted Ball Speed?

Mass of bat

Conclusion:mass of bat matters ...but not a lot

90

95

100

105

110

115

120

125

20 25 30 35 40 45 50Mass of Bat (oz)

Vary Bat Mass and Bat Speed


Dynamics of Ball-Bat CollisionDynamics of Ball-Bat Collision

Ball compresseskinetic energy stored in “spring”

Ball expandskinetic 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!

0

1000

2000

3000

4000

5000

6000

0 0.2 0.4 0.6 0.8 1

Time in milliseconds


Dynamics of Ball-Bat CollisionDynamics of Ball-Bat Collision

Ball compresseskinetic energy stored in “spring”

Ball expandskinetic 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!

0

1000

2000

3000

4000

5000

6000

0 0.2 0.4 0.6 0.8 1

Time in milliseconds

after during before


The The CCoefficient oefficient oof f RRestitutionestitution

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------->


What About the Bat?What About the Bat?(or, it takes two to tango!)(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


Properties of BatsProperties 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


Sweet Spot #1: Center of PercussionSweet Spot #1: Center of Percussion

When ball strikes bat...Linear recoil

» conservation of momentumRotation about center of mass

» conservation of angular momentum When CP hit

The two motions cancel at handleNo reaction force felt at handle


Sweet Spot #2: Sweet Spot #2: MMaximum aximum EEnergy nergy TTransferransfer

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

20

40

60

80

100

120

10 15 20 25 30 35

distance from handle in inches

wood bat

aluminum bat

CM COP


Sweet Spot #3: “Node” of VibrationSweet Spot #3: “Node” of Vibration

Collision excites bending

vibrations in bat

Ouch!!Energy lost ==>lower CORSometimes 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


So you think bats cannot bend…..So you think bats cannot bend…..

0

20

40

60

80

100

120

10 15 20 25 30 35distance from knob (inches)

rigid bat

real bat


So you think bats cannot bend…..So you think bats cannot bend…..


How Would a Physicist Design a Bat?How Would a Physicist Design a Bat?

Wood Batalready optimally designed

» highly constrained by rules!a marvel of evolution!

Aluminum Bat lots of possibilities existbut not much scientific researcha great opportunity for ...

» fame

» fortune


Advantages of AluminumAdvantages 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 BaseballAerodynamics of a Baseball

Forces on Moving Baseball

No SpinBoundary layer separationDRAG!Grows with v2

With Spin

Ball deflects wake

action/reaction==>Magnus

force

» Force grows with rpm

» Force in direction front of ball is turning

Pop

Pbottom


The Flight of the BalllThe Flight of the Balll

Role of Drag

Role of Spin

Atmospheric conditionsTemperatureHumidityAltitudeAir pressureWind

0

50

100

150

200

250

-100 0 100 200 300 400

horizontal distance in feet

200

350

500

750900


The Home Run SwingThe 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!


The Role of FrictionThe 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

batball

topspin ==>F down backspin==>F up

sidespin ==> hook

bat

ball


Pitching the BaseballPitching the Baseball

“Hitting is timing. Pitching isupsetting timing”

---Warren Spahn

vary speeds manipulate air flow orient stitches

Don Larsen, 1956 World Series Last pitch of perfect game


Let’s Get Quantitative!Let’s Get Quantitative!I. How Large are the Forces?I. How Large are the Forces?

• Drag is comparable to weight• Magnus force < 1/4 weight)

0

0.5

1

1.5

2

0 25 50 75 100 125 150Dra

g/W

eig

ht

or

Mag

nu

s/W

eig

ht

Speed in mph

Drag/Weight

Magnus/Weight


Let’s Get Quantitative!Let’s Get Quantitative!II. How Much Does the Ball Break?II. How Much Does the Ball Break?

Depends on…Magnitude and direction of forceTime over which force acts

Calibration90 mph fastball drops 3.5’ due to

gravity aloneBall 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

0

0.5

1

1.5

2

0 25 50 75 100 125 150Dra

g/W

eig

ht

or

Mag

nu

s/W

eig

ht

Speed in mph

Drag/Weight

Magnus/Weight


Example 1: FastballExample 1: Fastball

85-95 mph1600 rpm (back)12 revolutions0.46 secM/W~0.1


Example 2: Split-Finger FastballExample 2: Split-Finger Fastball

85-90 mph1300 rpm (top)12 revolutions0.46 secM/W~0.1


Example 3: CurveballExample 3: Curveball

70-80 mph1900 rpm

(top and side)17 revolutions0.55 secM/W~0.25


Example 4: SliderExample 4: Slider

75-85 mph1700 rpm (side)14 revolutions0.51 secM/W~0.15


Examples of TrajectoriesExamples of Trajectories

3

4

5

6

7

0 10 20 30 40 50 60

Ve

rtic

al

Po

sit

ion

of

Ba

ll (

fee

t)

Distance from Pitcher (feet)

90 mph Fastball

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

Ho

rizo

nta

l Def

lect

ion

of

Bal

l (fe

et)

Distance from Pitcher (feet)

75 mph Curveball


Effect of the StitchesEffect of the Stitches

Obstructions cause turbulance

Turbulance reduces dragDimples on golf ballStitches on baseball

Asymmetric obstructions

Knuckleball

Two-seam vs. four-seam delivery

Scuffball and “juiced” ball


SummarySummary

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!

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