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Fermilab Colloquium, July 12, 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
H. Brody, AJP 54, 640 (1986); AJP 58, 756 (1990) P. Kirkpatrick, AJP 31, 606 (1963) L. Van Zandt, AJP 60, 172 (1992) R. Cross, AJP 66, 772 (1998); AJP 67, 692 (1999) AMN, AJP 68, to appear in Sept. 2000 L. Briggs, AJP 27, 589 (1959) R. Mehta, Ann. Ref. Fluid Mech. 17, 151 (1985) www.npl.uiuc.edu/~a-nathan/pob
Fermilab Colloquium, July 12, 2000 Page 3
A Philosophical Note:“…the physics of baseball is not the clean, well-defined
physics of fundamental matters but the ill-defined physics of the complex world in which we live, where elements are not ideally simple and the physicist must make best judgments on matters that are not simply calculable…Hence conclusions about the physics of baseball must depend on approximations and estimates….But estimates are part of the physicist’s repertoire…a competent physicist should be able to estimate anything ...”
“The physicist’s model of the game must fit the game.”
“Our aim is not to reform baseball but to understand it.”
---Bob Adair in “The Physics of Baseball”, May, 1995 issue of Physics Today
Fermilab Colloquium, July 12, 2000 Page 4
Hitting the BaseballHitting the Baseball
“...the most difficult thing to do in sports”
--Ted Williams
BA: .344SA: .634OBP: .483HR: 521
#521, September 28, 1960
Fermilab Colloquium, July 12, 2000 Page 6
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 = 4-5 ft/mph more later
What aspects of collision lead to large vf?
Fermilab Colloquium, July 12, 2000 Page 7
What happens when ball and bat collide?
The simple stuff: kinematics conservation of momentum conservation of angular momentum
The really interesting stuff: energy dissipation compression/expansion of ball vibrations of the bat
How to maximize vf?
Fermilab Colloquium, July 12, 2000 Page 8
The Simple Stuff: Kinematics
ibat,iball,fball, vr1
e1 v
r1
r-e v
vball,f = 0.2 vball,i + 1.2 vbat,i
Conclusion: vbat much more important than vball
Question: what bat/ball properties make vball,f large?
e Coefficient of Restitution 0.5
r recoil factor 0.25
Fermilab Colloquium, July 12, 2000 Page 9
Sosa’s 500’ Blast(s) in Home Run Derby:
A Numerical Analysis
• D = 500
* vf 127 mph
• vball,i 60 mph
* vbat,i 96 mph!
• if vball,i were 90 mph
* D = 530
Fermilab Colloquium, July 12, 2000 Page 10
Energy in Bat Recoil
.
Translation
.Rotation
CM .
z
• Important Bat Parameters:
mbat, xCM, ICM
• wood vs. aluminum
bat
2ball
bat
ball
I
zm
m
m r
Conclusion: All things being equal, want mbat, Ibat large
0.17 + 0.07 = 0.24
Want r small to mimimizerecoil energy
Fermilab Colloquium, July 12, 2000 Page 11
But… All things are not equal Mass & Mass Distribution affect bat speed
Conclusion:mass of bat matters….but probably not a lot
see Watts & Bahill, Keep Your Eye on the Ball, 2nd edition, ISBN 0-7167-3717-5
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
Fermilab Colloquium, July 12, 2000 Page 12
• in CM frame: Ef/Ei = e2
• massive rigid surface: e2 = hf/hi
• typically e 0.5~3/4 CM energy dissipated!
• depends on ball, surface, speed,...
• is the ball “juiced”?
Energy Dissipated:
Coefficient of Restitution (e):“bounciness” of ball
i rel,
f rel,
v
v e
Fermilab Colloquium, July 12, 2000 Page 13
COR and the “Juiced Ball” Issue
MLB: e = 0.546 0.032 @ 58 mph on massive rigid surface
Conclusion: more systematic studies needed
0.40
0.45
0.50
0.55
0.60
40 60 80 100 120 140equivalent impact speed (mph)
COR
Briggs, 1945
UML/BHM
Lansmont BBVC
MLB specs
MLB/UML
COR Measurements
320
360
400
440
0.4 0.45 0.5 0.55 0.6
R (ft)
cor
*
*~ 35 '
Distance vs. COR "90+70" collision
Fermilab Colloquium, July 12, 2000 Page 14
CM energy shared between ball and bat
Ball is inefficient: 75% dissipated
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
Effect of Bat on COR: Local CompressionEffect of Bat on COR: Local Compression
Ebat/Eball kball/kbat xbat/ xball
>10% larger!
tennis ball/racket
Recent BPF data:(Lansmont BBVC/Trey Crisco)
0.99 wood 1.12 aluminum
More later on wood vs. aluminum
Fermilab Colloquium, July 12, 2000 Page 15
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
Beyond the Rigid Approximation:
A Dynamic Model for the Bat-Ball collision
see AMN, Am. J. Phys, 68, in press (2000)
Fermilab Colloquium, July 12, 2000 Page 16
20
-2 0
-1 5
-1 0
-5
0
5
10
15
20
0 5 10 15 20 25 30 35
y
z
y
t)F(z, t
yA
z
yEI
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
‡ Note for experts: full Timoshenko (nonuniform) beam theory used
Euler-Bernoulli Beam Theory‡
Fermilab Colloquium, July 12, 2000 Page 17
Normal Modes of the Bat
Louisville Slugger R161 (33”, 31 oz)
Can easily be measured (modal analysis)0 5 10 15 20 25 30 35
f1 = 177 Hz
f2 = 583 Hz
f3 = 1179 Hz
f4 = 1821 Hznodes
Fermilab Colloquium, July 12, 2000 Page 18
-1.5
-1
-0.5
0
0.5
1
0 5 10 15 20
R
t (ms)
0
0.05
0.1
0.15
0 500 1000 1500 2000 2500
FFT(R)
frequency (Hz)
179
582
1181
1830
2400
frequency barrel nodeExpt Calc Expt Calc 179 177 26.5 26.6 582 583 27.8 28.21181 1179 29.0 29.21830 1821 30.0 29.9
Measurements via Modal Analysis
Louisville Slugger R161 (33”, 31 oz)
Conclusion: free vibrationsof bat can be well characterized
Fermilab Colloquium, July 12, 2000 Page 19
0.4
0.8
1.2
1.6
2
0 20 40 60 80 100 120 140
t (ms)
impact speed (mph)
collision time versus impact speed
Model for the Ball
3-parameter problem:
k t
n v-dependence of t
m COR
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
F=kxn
F=kxm
0
2000
4000
6000
8000
10000
0 0.2 0.4 0.6 0.8
Force (lb)
time (ms)
160 mph
80 mph
Fermilab Colloquium, July 12, 2000 Page 20
t)(y-t),y(xs t)F(s,- dt
ydm
A
t))F(s,(xyq
dt
qd
)x(y)t(qt)y(x,
ball02ball
2
ball
02n
n2n2
n2
nn
n
impact pointball compression
Putting it all together….
Expectation: only modes with fn t < 1 strongly excited
Fermilab Colloquium, July 12, 2000 Page 21
Results: Ball Exit SpeedLouisville Slugger R16133-inch/31-oz. wood bat
Conclusion: essential physics under control
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
23 24 25 26 27 28 29 30 31
vfinal
/vinitial
distance from knob (inches)
data from Lansmont BBVCbat pivoted about 5-3/4"
vinitial
=100 mph
rigid bat
flexible bat
nodes
only lowest mode excited lowest 4 modes excited
0
0.1
0.2
0.3
0.4
16 20 24 28 32
vfinal
/vinitial
distance from knob (inches)
rigid bat
flexible bat
CM node
data from Rod Crossfreely suspended bat
vi = 2.2 mph
Fermilab Colloquium, July 12, 2000 Page 22
• Under realistic conditions…
• 90 mph, 70 mph at 28”
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 32
distance from knob (cm)
Total
1
3
>3
2
(b)
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
Fermilab Colloquium, July 12, 2000 Page 23
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.05 0.1 0.15 0.2 0.25 0.3
eeff
/e
distance from barrel (m)
Trey Crisco's Batting Cage Data(wood)
calculation
Fermilab Colloquium, July 12, 2000 Page 24
Results:The “sweet spot”
30
40
50
60
70
80
90
100
110
20 22 24 26 28 30 32
vf (mph)
x (inches)
flexible (free or pivoted)
rigid pivoted
rigid free
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
Hands don’t matter!
Fermilab Colloquium, July 12, 2000 Page 25
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
Fermilab Colloquium, July 12, 2000 Page 26
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
Fermilab Colloquium, July 12, 2000 Page 27
Things I would like to understand betterThings I would like to understand better
Relationship between bat speed and bat weight and weight distribution
Effect of “corking” the bat Location of “physiological” sweet spot Better model for the ball FEA analysis of aluminum bat Why is softball bat different from baseball bat?
Fermilab Colloquium, July 12, 2000 Page 28
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
Fermilab Colloquium, July 12, 2000 Page 29
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 turning
Pop
Pbottom
Drawing courtesty of Peter Brancazio
Fermilab Colloquium, July 12, 2000 Page 30
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 150Dra
g/W
eig
ht
or
Mag
nu
s/W
eig
ht
Speed in mph
Drag/Weight
Magnus/Weight =1800 RPM
Fermilab Colloquium, July 12, 2000 Page 31
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
Fermilab Colloquium, July 12, 2000 Page 32
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
Fermilab Colloquium, July 12, 2000 Page 33
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!
Fermilab Colloquium, July 12, 2000 Page 34
Pitching the BaseballPitching the Baseball
“Hitting is timing. Pitching isupsetting timing”
---Warren Spahn
vary speeds manipulate air flow orient stitches
Fermilab Colloquium, July 12, 2000 Page 35
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 60Ve
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
Fermilab Colloquium, July 12, 2000 Page 36
Examples of PitchesExamples 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?
Fermilab Colloquium, July 12, 2000 Page 37
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
Fermilab Colloquium, July 12, 2000 Page 38
Example 1: FastballExample 1: Fastball
85-95 mph1600 rpm (back)12 revolutions0.46 secM/W~0.1
Fermilab Colloquium, July 12, 2000 Page 39
Example 2: Split-Finger FastballExample 2: Split-Finger Fastball
85-90 mph1300 rpm (top)12 revolutions0.46 secM/W~0.1
Fermilab Colloquium, July 12, 2000 Page 40
Example 3: CurveballExample 3: Curveball
70-80 mph1900 rpm
(top and side)17 revolutions0.55 secM/W~0.25
Fermilab Colloquium, July 12, 2000 Page 41
Example 4: SliderExample 4: Slider
75-85 mph1700 rpm (side)14 revolutions0.51 secM/W~0.15
Fermilab Colloquium, July 12, 2000 Page 42
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!