the physics of baseball alan m. nathan university of illinois odu colloquium, march 31, 2000
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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. - PowerPoint PPT PresentationTRANSCRIPT
ODU Colloquium, March 31, 2000 Page 1
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
ODU Colloquium, March 31, 2000 Page 2
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
ODU Colloquium, March 31, 2000 Page 3
Hitting the BaseballHitting the Baseball
“...the most difficult thing to do in sports”
--Ted Williams
BA: .344SA: .634OBP: .483HR: 521
#521, September 28, 1960
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Here’s Why…..
(Courtesy of Bob Adair)
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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?
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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?
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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
ODU Colloquium, March 31, 2000 Page 8
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
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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!
ODU Colloquium, March 31, 2000 Page 10
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
ODU Colloquium, March 31, 2000 Page 11
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
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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
ODU Colloquium, March 31, 2000 Page 13
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
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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
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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
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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|
distance from knob (inches)
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
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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)
distance from knob (inches)
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)
ODU Colloquium, March 31, 2000 Page 18
Results:4. The “sweet spot”
20
40
60
80
100
16 20 24 28 32
vf (mph)
distance from knob (inches)
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
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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)
distance from knob (inches)
wood
aluminum-1
aluminum-2
wood versus aluminum
ODU Colloquium, March 31, 2000 Page 20
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
ODU Colloquium, March 31, 2000 Page 21
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
ODU Colloquium, March 31, 2000 Page 22
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
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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
ODU Colloquium, March 31, 2000 Page 24
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
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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
ODU Colloquium, March 31, 2000 Page 26
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!
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Pitching the BaseballPitching the Baseball
“Hitting is timing. Pitching isupsetting timing”
---Warren Spahn
vary speeds manipulate air flow orient stitches
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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
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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?
ODU Colloquium, March 31, 2000 Page 30
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
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Example 1: FastballExample 1: Fastball
85-95 mph1600 rpm (back)12 revolutions0.46 secM/W~0.1
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Example 2: Split-Finger FastballExample 2: Split-Finger Fastball
85-90 mph1300 rpm (top)12 revolutions0.46 secM/W~0.1
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Example 3: CurveballExample 3: Curveball
70-80 mph1900 rpm
(top and side)17 revolutions0.55 secM/W~0.25
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Example 4: SliderExample 4: Slider
75-85 mph1700 rpm (side)14 revolutions0.51 secM/W~0.15
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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!
ODU Colloquium, March 31, 2000 Page 36
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
ODU Colloquium, March 31, 2000 Page 37
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