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1927 Yankees: Greatest baseball team ever assembled. 1927 Solvay Conference: Greatest physics team ever assembled. MVP’s. Baseball and Physics. As smart as he was, Albert Einstein could not figure out how to handle those tricky bounces at third base. Philosophical Notes:. - PowerPoint PPT Presentation
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The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 2
1927
Solvay Conference:
Greatest physics team
ever assembled
Baseball and Physics
1927 Yankees:
Greatest baseball team
ever assembled
MVP’s
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 3
As smart as he was, Albert Einstein could not figure out how to handle those tricky bounces at third base.
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 4
Philosophical Notes:“…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 ...”
---Bob Adair in “The Physics of Baseball”, May, 1995 issue of Physics Today
“The physicist’s model of the game must fit the game.”
“Our aim is not to reform baseball but to understand it.”
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 5
Hitting the Baseball
“...the most difficult thing to do in sports”
--Ted Williams: 1918-2002
BA: .344SA: .634OBP: .483HR: 521 #521, September 28, 1960
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 6
Introduction: Description of Ball-Bat Collision
forces large (>8000 lbs!) time short (<1/1000 sec!) ball compresses, stops, expands
kinetic energy potential energy
lots of energy lost
bat is flexible hands don’t matter
to hit a home run... large hit ball speed
optimum take-off angle
backspin
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 7
The Ball-Bat Collision: Kinematics
vf = eA vball + (1+eA) vbat
Conclusion:
vbat matters much more than vball
vball vbat
vf
“Lab” Framevrel
eAvrel
Bat Rest Frame
eA “Collision Efficiency”
• property of ball & bat• weakly dependent on vrel
• Superball-wall: eA 1• Ball-Bat near “sweet spot”: eA 0.2
vf 0.2 vball + 1.2 vbat
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 8
What Does eA Depend On?
r1
r-e eA • Kinematics: recoil of bat (r)
• Dynamics: energy dissipation (e)
I
bm
m
m
m
m r
CMbat,
2ball
bat
ball
effbat,
ball
Small r is best
r 0.25 typical…depends on….
• mass of bat
• mass distribution of bat
• impact location
. .CM .
b
= +
Heavier bat is better but….
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 9
What is Ideal Bat Weight?
60
70
80
90
100
110
120
20 30 40 50 60
vf (mph)
Mbat
(oz)
vbat
= 65 mph x (32/Mbat
)n
n=0.5constant KE
n=0.31experiment
n=0constant v
bat
Note: Batters seem to prefer lighter bats!
Actually,
Scaling with Iknob better
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 10
vBAT(6”) = 1.2 mph/(1000 oz-in2) (vf=1.5 0.3 mph)
40
42
44
46
48
50
1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9
Iknob
(104 oz-in2)
knob
(rad/s)
Crisco/Greenwald Batting Cage Study
vbat I-0.3
vbat I-0.5
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 11
Energy Dissipation: the ball-bat COR (e)
Coefficient Of Restitution
• in CM frame: Ef/Ei = e2
• ball on hard floor: e2 = hf/hi 0.25
e 0.5 (note: r=0.25, e=0.5 eA =0.2)
~3/4 CM energy dissipated!
• depends (weakly) on impact speed
• the bat matters too! vibrations “trampoline” effect
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 12
Aside: Effect of “Juiced” Ball
MLB: e = 0.546 0.032 @ 58 mph on massive rigid surface
320
360
400
440
0.4 0.45 0.5 0.55 0.6
R (ft)
cor
*
*~ 35 '
Distance vs. COR "90+70" collision
0.40
0.45
0.50
0.55
0.60
60 80 100 120 140equivalent impact speed (mph)
COR
Briggs, 1945
UML/BHM
Lansmont
MLB specs
MLB/UML
COR Measurements
Lansmont/CPD
10% increase in COR ~30-35 ft increase in distance
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 13
Collision excites bending vibrations in
bat
Ouch!! Thud!! Sometimes broken bat
Energy lost lower e, vf
Find lowest mode by tapping
Reduced considerably if
Impact is at a node
Collision time (~0.6 ms) > TN
see AMN, Am. J. Phys, 68, 979 (2000)
Accounting for Energy Dissipation:
Dynamic Model for Ball-Bat Colllision
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 14
Dynamic Model
x
yEI
x - F
t
yA
2
2
2
2
2
2
-2 0
-1 5
-1 0
-5
0
5
10
15
20
0 5 10 15 20 25 30 35
20
y
z
y
Step 1: Solve eigenvalue problem for free vibrations
Step 2: Nonlinear lossy spring for F
Step 3: Expand in normal modes and solve
yA x
yEI
x n
2n2
n2
2
2
A
)t(Fq
dt
qd )(y)(q)y( n
n2n2
n2
nn
n xtx,t
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 15
Normal Modes of the Bat
Louisville Slugger R161 (33”, 31 oz)
0 5 10 15 20 25 30 35
f1 = 177 Hz
f2 = 583 Hz
f3 = 1179 Hz
f4 = 1821 Hz
Can easily be measured: Modal Analysis
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 16
-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
FFT
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 17
Theory vs. Experiment:Louisville Slugger R16133-inch/31-oz. wood bat
Conclusion: essential physics understood
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
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 18
Time evolution
of the bat
0.6 ms
hands don’t matter
-4
-2
0
2
4
6
8
10
displacement (mm)
0.1 ms intervals
impact point
pivot point
-50
0
50
100
150
200
0 5 10 15 20 25 30distance from knob (inches)
1 ms intervals
impact point
pivot point
T= 0-1 ms
T= 1-10 ms
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 19
Effect of Bat on COR: Vibrations
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0
10
20
30
40
50
60
70
80
0 2 4 6 8 10 12 14
COR % Energy Dissipated
inches from barrel
Ball
Vibrations
Nodes
COR
COR maximum near 2nd node
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 20
Putting Everything Together...
“sweet spot” depends on•collision efficiency
*recoil factor*COR
•how bat is swung
20
40
60
80
100
0.00
0.05
0.10
0.15
0.20
0.25
0 2 4 6 8 10 12 14
eAv (mph)
distance from tip (inches)
vf
eA
vbat
CM
vf = eA vball + (1+eA) vbat
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 21
60
70
80
90
100
110
0 2 4 6 8 10
Vf (mph)
distance from barrel (inches)
Crisco/Greenwald Data vs. Calculation
Conclusion: ideal ball-bat collision can be simulated
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 22
Wood versus Aluminum
Kinematics
Length, weight, MOI “decoupled”
shell thickness, added weight
fatter barrel, thinner handle
weight distribution more uniform
CM closer to handle
less mass at contact point easier to swing
Dynamics
Stiffer for bending
Less vibrational energy
More compressible
COR larger
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 23
The “Trampoline” Effect
Compressional energy shared between ball and bat PEbat/PEball = kball/kbat << 1 PEball mostly dissipated (75%)
Wood Bat hard to compress little effect on COR: “BPF” 1
Aluminum Bat
compressible through “shell” modes
kball/kbat ~ 0.10 (more or less)
PEbat mostly restored (more on this later)
COR larger: “BPF” 1.1 (more or less)
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 24
The Trampoline Effect: A Closer Look
Bending Modes vs. Shell Modes
k R4: large in barrel little energy stored
f (170 Hz, etc) > 1/ energy goes into
vibrations
k (t/R)3: small in barrel
more energy stored
f (2-3 kHz) < 1/ energy mostly restored
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 25
Wood versus Aluminum:
Dynamics of “Trampoline” Effect
“bell” modes: 3
2 R
t k
R
t ω
“ping” of bat
• Want k small to maximize stored energy
• Want >>1 to minimize retained energy
• Conclusion: there is an optimum
0 1000 2000 3000 4000frequency (Hz)
bending modes
bell modes
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 26
Where Does the Energy Go?
0
50
100
150
200
250
300
350
400
0 0.2 0.4 0.6 0.8 1
Wood Bat
Ball KE
Ball PE
Bat Recoil KE
Bat Vibrational E
Energy (J)
t (ms)
0
50
100
150
200
250
300
350
400
0 0.2 0.4 0.6 0.8 1
Aluminum Bat
Ball KE
Ball PE
Bat Recoil KE
Bat Vibrational E
Energy (J)
t (ms)
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 27
“Corking” a Wood Bat (illegal!)
Drill ~1” diameter hole along axis to depth of ~10”
• Smaller mass
• larger recoil factor (bad)
• higher bat speed (good)
• Is there a trampoline effect?
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 28
Not Corked DATA Corked COR: 0.445 0.005 0.444 0.005
Conclusions:
• no tramopline effect!
• corked bat is WORSE even with higher vbat
Baseball Research Center, UML, Sherwood & amn, Aug. 2001
70
80
90
2 3 4 5 6 7 8 9
vf (mph)
distance from knob (inches)
uncorked
corked
calculation
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 29
Aerodynamics of a Baseball
Forces on Moving Baseball
No SpinBoundary layer separationDRAG!FD=½ CDAv2
With Spin
Ball deflects wake ==>”lift”FM ~ RdFD/dvForce in direction front of ball
is turning
Pop
Pbottom
Drawing courtesty of Peter Brancazio
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 30
0
0.5
1
1.5
2
0 25 50 75 100 125 150Speed in mph
Drag/Weight
Magnus/Weight@1800 rpm
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 31
0
50
100
150
200
250
300
350
400
10 20 30 40 50 60 70 80 90
Range (ft)
(deg)
Range vs.
2000 rpm
0 rpm
0
20
40
60
80
100
120
0 100 200 300 400 500 600 700
y (ft)
x (ft)
no drag
with drag
approx linear: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 400horizontal distance in feet
200
350
500
750900
The Flight of the Ball:Real Baseball vs. Physics 101 Baseball
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 32
Summary of Aerodynamics
108 mph ~400 ft each mph ~5 ft optimum angle ~350
2000 rpm backspinIncreases range ~27 ftDecreases optimum angle ~30
these number are only estimates!
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 33
Oblique Collisions:The Role of Friction
Friction halts vT
spin, “lift”
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 Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 34
Model for Oblique Collisions:
vN treated as before
vNf = eA(vball+vbat)N + vbat,N
Angular momentum conserved about contact point (!)
Friction reduces vT, increases
Rolls when vT = R
Horizontal: vTf (5/7)vT
Vertical: a bit more complicated Not the way a superball works!
vN
vT
vN|
vT|
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 35
Oblique Collisions: Horizontal Plane
0
20
40
60
80
0 2 4 6 8 10 12 14
f (deg)
t (ms)
foul
fair
~11 inches
Initial takeoff angle
down the line
0
20
40
60
80
100
120
0 100 200 300 400
height vs. distance
angle vs. distanze
0
20
40
60
80
100
120
0 100 200 300 400
height vs. distance
angle vs. distanze
power alley
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 36
Oblique Collisions: Vertical Plane
-2000
0
2000
4000
6000
8000
10000
12000
-50
0
50
100
150
-0.5 0 0.5 1 1.5 2 2.5 3
(rpm) (deg)
D (inches)
optimum:
D 0.75”
3000 rpm
330
Ball100 downward
Bat 100 upward
D = center-to-center offset
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 37
Typical Trajectories
Ball100 downward
Bat 100 upward
D = center-to-center offset
0
50
100
150
200
250
0 50 100 150 200 250 300 350 400
1.5
0
0.25
0.50.75
1.0
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 38
Some Practical/Interesting Questions
Does more friction help? Can a curveball be hit further than a
fastball?
The Physics of Hitting a Home Run ANL Colloquium September 20, 2002 Page 39
Summary and Conclusions Some aspects of baseball are amenable to
physics analysis Kinematic and dynamics of ball-bat collision Trajectory of a ball with drag and lift
Can understanding these things improve our ability to play the game? Almost surely NOT
Can understanding these things enhance our own enjoyment of the game For me, a resounding YES I hope for you also