the physics of baseball alan m. nathan university of illinois odu colloquium, march 31, 2000

ODU Colloquium, March 31, 2000

The Physics of BaseballThe Physics of Baseball

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

ODU Colloquium, March 31, 2000ODU Colloquium, March 31, 2000

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 Physics of Sports, Angelo Armenti (American Institute of Physics, New York, 1992), ISBN 0-88318-946-1

www.npl.uiuc.edu/~a-nathan/pob


Hitting the BaseballHitting the Baseball

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

--Ted Williams

BA: .344SA: .634OBP: .483HR: 521

#521, September 28, 1960


Here’s Why…..

(Courtesy of Bob Adair)


Description of Ball-Bat CollisionDescription 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 = 5 ft/mph

What aspects of collision lead to large vf?


What happens when ball and bat collide?

The simple stuff conservation of momentum conservation of angular momentum energy dissipation in the ball (compression/expansion)

The really interesting stuff vibrations of the bat

How to maximize vf?


The Simple Stuff: Rigid-Body Kinematics

ibat,iball,fball, vr1e1 v

r1r-e v

Vball,f = 0.25 Vball,i + 1.25 Vbat,i

Conclusion: vbat much more important than vball

“radius of gyration”

e Coefficient of Restitution 0.5

kz-z1

mm 2

CM

bat

ball

r recoil factor

= 0.2


Recoil Factor

.

Translation

.Rotation

CM .

z

• Important Bat Parameters:

mbat, xCM, ICM

• wood vs. aluminum

bat

2ball

bat

ball

Izm

mm r

Conclusion: All things being equal, want mbat, Ibat large0.16 + 0.07 = 0.23


Coefficient of Restitution (e)

“bounciness” of ball Bounce ball off massive hard surface

e2 = hf/hi

For baseball, e .5 3/4 energy lost! Changing e by .05 changes V by 7 mph (35 ft!)

Important Point: the bat matters too!


Energy shared between ball and bat

Ball is inefficient: 25% returned 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

Claims of BPF 1.2

Effect of Bat on COREffect of Bat on COR

Ebat/Eball kball/kbat xbat/ xball

>10% larger!

tennis ball/racket


Rigid-Body ResultsRigid-Body Results

Aluminum bat more effectivefor inside pitches 60

70

80

90

100

110

10 15 20 25 30 35

distance from knob (inches)

aluminum

wood

CM

vball,I= 90 mphvbat,CM = 54 mphbat,CM = 51 s-1

ibat,iball,fball, vr1e1 v

r1r-e v


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

Modes with fn 1 excited

Beyond the Rigid Approximation:

A Dynamic Model for the Bat-Ball collision

Ref.: AMN, Am. J. Phys, submitted March 2000


20

-2 0

-1 5

-1 0

-5

0

5

10

15

20

0 5 10 15 20 25 30 35

y

zy

t)F(z, tyA

zyEI

z 2

2

2

2

2

2

A Dynamic Model of the Bat-Ball Collision

• 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


In a bit more detail…In a bit more detail…

t)(y-t),y(xs t)F(s,- dtydm

At))F(s,(xyq

dtqd

)x(y)t(qt)y(x,

ball02ball

2

ball

02n

n2n2

n2

nn

n

impact pointball compression

0

2000

4000

6000

8000

1 104

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

force (pounds)

compression (inches)

approx quadratic

ODU Colloquium, March 31, 2000 0 5 10 15 20 25 30 35

f1 = 165 Hz

f2 = 568 Hz

f3 = 1177 Hz

f4 = 1851 Hz

Results:1. Normal Modes

Louisville Slugger R161 (34”, 31 oz)

Can be measured (modal analysis)

nodes


0.15

0.2

0.25

0.3

0.35

0.4

0.45

18 20 22 24 26 28 30 32

|vf/v

i|


rigid bat

realistic bat

vi = 1 m/s

Theory vs. Experiment (Rod Cross)(at 1 m/s)

0

10

20

30

40

50

18 20 22 24 26 28 30 32

rigid recoil

losses in ball

ballvibrations

in bat

Vi=1 m/s

COR=0.66

-5

0

5

10

15

20

25

30

35

18 20 22 24 26 28 30 32

Modes >2

Mode 1

Mode 2

Vi=1 m/s

COR=0.66total

collision time 2.2 ms

Results:2. Low-speed collision


Results:3. High-speed collision

• Under realistic conditions…• 90 mph, 70 mph at 28”• no data (yet)…..

20

40

60

80

100

16 20 24 28 32

vf (mph)


flexible bat

rigid bat

Louisville SluggerR161 (33", 31 oz)

CM nodes

0

10

20

30

40

50

60

70

16 20 24 28 32

% Energy

rigid recoil

ball

vibrations

losses inball

(a)

0

5

10

15

20

25

30

16 20 24 28 32distance from knob (cm)

Total

1

3

>3

2

(b)


Results:4. The “sweet spot”

20

40

60

80

100

16 20 24 28 32

vf (mph)


flexible bat

rigid bat

Louisville SluggerR161 (33", 31 oz)

CM nodes

-20

0

20

0 2 4 6 8 10

v (m/s)

t (ms)

Motion of Handle

24”

27”

30”

Possible “sweet spots”

1. Maximum of vf (28”)

2. Node of fundamental (27”)

3. Center of Percussion (27”)

-3

-2

-1

0

1

2

3

0 0.5 1 1.5 2

y (mm)

t (ms)

impact at 27"

13 cm


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

0

20

40

60

80

100

16 20 24 28 32

vf (mph)


wood

aluminum-1

aluminum-2

wood versus aluminum


How Would a Physicist Design a Bat?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


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 BaseballAerodynamics 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 turningPop

Pbottom


How Large are the Forces?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 150Drag

/Wei

ght o

r Mag

nus/

Wei

ght

Speed in mph

Drag/Weight

Magnus/Weight =1800 RPM


The Flight of the Ball:The Flight of the Ball:Real Baseball vs. Physics 101 BaseballReal Baseball vs. Physics 101 Baseball

Role of Drag Role of Spin Atmospheric conditions

Temperature Humidity Altitude Air pressure Wind

approx linear

Max @ 350

-100

0

100

200

300

400

0 20 40 60 80 100

Range (ft)

q (deg)

Range vs. q

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

y (ft)

x (ft)

no drag

with drag

100

200

300

400

500

50 60 70 80 90 100 110 120

Range (ft)

vi (mph)

Range vs. v

0

50

100

150

200

250

-100 0 100 200 300 400

horizontal distance in feet

200

350

500

750900


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


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!


Pitching the BaseballPitching the Baseball

“Hitting is timing. Pitching isupsetting timing”

---Warren Spahn

vary speeds manipulate air flow orient stitches


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

3

4

5

6

7

0 10 20 30 40 50 60Vert

ical

Pos

ition

of B

all (

feet

)

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

Hor

izon

tal D

efle

ctio

n of

Bal

l (fe

et)

Distance from Pitcher (feet)

75 mph Curveball


Examples of PitchesExamples of Pitches

Pitch V(MPH) (RPM) T M/Wfastball 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 StitchesEffect of the Stitches

Obstructions cause turbulance

Turbulance reduces dragDimples on golf ballStitches on baseball

Asymmetric obstructionsKnuckleballTwo-seam vs. four-seam deliveryScuffball and “juiced” ball


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


SummarySummary

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!


Sweet Spot #2: Sweet Spot #2: CCenter enter oof f PPercussionercussion

When ball strikes bat... Linear recoil

conservation of momentum Rotation about center of mass

conservation of angular momentum When COP hit

The two motions cancel (at conjugate point) No reaction force felt

x1

x2

x1x2=Icm/M


But… All things are not equal Mass & Mass Distribution affect bat speed

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

40

50

60

70

80

90

100

20 30 40 50 60

mass of bat (oz)

constant bat energy

constant bat+batter energy

60

70

80

90

100

110

120

20 30 40 50 60

mass of bat (oz)

constant bat energy

constant bat speed

constant bat+batter energy

bat speed vs mass

ball speed vs mass

the physics of baseball alan m. nathan university of illinois odu colloquium, march 31, 2000

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