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The Physics of Baseball Alan M. Nathan University of Illinois ODU Colloquium, March 31, 2000. Introduction Hitting the Baseball The Flight of the Baseball Pitching the Baseball Summary. REFERENCES.

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The physics of baseball alan m nathan university of illinois odu colloquium march 31 2000
The Physics of BaseballAlan M. Nathan University of IllinoisODU Colloquium, March 31, 2000

  • Introduction

  • Hitting the Baseball

  • The Flight of the Baseball

  • Pitching the Baseball

  • Summary


References
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

  • www.npl.uiuc.edu/~a-nathan/pob


Hitting the baseball

#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 Bob 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 = 5 ft/mph

What aspects of collision lead to large vf?


How to maximize 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


0.2

r recoil factor =

“radius of gyration”

The Simple Stuff: Rigid-Body Kinematics

e Coefficient of Restitution  0.5

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

Conclusion: vbat much more important than vball


.

.

.

CM

z

Translation

Rotation

Recoil Factor

  • Important Bat Parameters:

  • mbat, xCM, ICM

  • wood vs. aluminum

0.16 + 0.07 = 0.23

Conclusion: All things being equal, want mbat, Ibat large


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!


Effect of bat on cor

tennis ball/racket

Effect of Bat on COR

  • 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

Ebat/Eball kball/kbat  xbat/ xball

>10%

larger!


CM

  • vball,I= 90 mph

  • vbat,CM = 54 mph

  • bat,CM = 51 s-1

Aluminum bat more effective

for inside pitches

Rigid-Body Results


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

  • Modes with fn  1 excited

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


y

y

z

A Dynamic Model of the Bat-Ball Collision

20

  • 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…

impact point

ball compression


f1 = 165 Hz

f3 = 1177 Hz

f2 = 568 Hz

f4 = 1851 Hz

Results:

1. Normal Modes

Louisville Slugger R161 (34”, 31 oz)

nodes

Can be measured (modal analysis)


Results:

2. Low-speed collision

Theory vs. Experiment (Rod Cross)

(at 1 m/s)

collision time  2.2 ms


Results:

3. High-speed collision

CM

nodes

  • Under realistic conditions…

  • 90 mph, 70 mph at 28”

  • no data (yet)…..


CM

nodes

24”

27”

30”

Results:

4. The “sweet spot”

Possible “sweet spots”

1. Maximum of vf (28”)

2. Node of fundamental (27”)

3. Center of Percussion (27”)


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
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 baseball
Aerodynamics of a Baseball

Forces on Moving Baseball

  • No Spin

    • Boundary layer separation

    • DRAG!

    • FD=½CDAv2

  • With Spin

    • Ball deflects wake ==>Magnus force

    • FMRdFD/dv

    • Force in direction front of ball is turning


How large are the forces
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
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
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
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!


Pitching the baseball

“Hitting is timing. Pitching is

upsetting timing”

---Warren Spahn

vary speeds

manipulate air flow

orient stitches

Pitching the Baseball


Let s get quantitative how much does the ball break

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
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
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
Example 1: Fastball

85-95 mph

1600 rpm (back)

12 revolutions

0.46 sec

M/W~0.1


Example 2 split finger fastball
Example 2: Split-Finger Fastball

85-90 mph

1300 rpm (top)

12 revolutions

0.46 sec

M/W~0.1


Example 3 curveball
Example 3: Curveball

70-80 mph

1900 rpm

(top and side)

17 revolutions

0.55 sec

M/W~0.25


Example 4 slider
Example 4: Slider

75-85 mph

1700 rpm (side)

14 revolutions

0.51 sec

M/W~0.15


Summary
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!


Sweet spot 2 c enter o f p ercussion

x2

x1

Sweet Spot #2: Center of Percussion

  • 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

x1x2=Icm/M


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


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