uw colloquium 10/31/05 1 thanks to j. j. crisco & r. m. greenwald medicine & science in...
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UW Colloquium 10/31/05 1
Thanks to J. J. Crisco & R. M. GreenwaldMedicine & Science in Sports & Exercise
34(10): 1675-1684; Oct 2002
Alan M. Nathan,University of Illinoiswww.npl.uiuc.edu/~a-nathan/pob
a-nathan @uiuc.edu
The Physics of Hitting a Home Run
UW Colloquium 10/31/05 2
1927
Solvay Conference:
Greatest physics team
ever assembled
Baseball and Physics
1927 Yankees:
Greatest baseball team
ever assembled
MVP’s
UW Colloquium 10/31/05 4
“Hitting is timing; pitching is
upsetting timing”
Hitting the Baseball:
the most difficult feat in sports
“Hitting is fifty percent above the shoulders”
1955 Topps cards from my personal collection
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Graphic courtesy of Bob Adair and NYT
Hitting and Pitching, Thinking and Guessing
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Example: Tim Wakefield’s Knuckleball
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1. How does a baseball bat work?
2. Why does aluminum outperform wood?
3. How does spin affect flight of baseball?
4. Can a curveball be hit farther than a
fastball?
The Physics of Hitting a Home Run
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Brief Description of Ball-Bat Collision• forces large, time short
– >8000 lbs, <1 ms
• ball compresses, stops, expands– KEPEKE– bat bends & compresses
• lots of energy dissipated (“COR”)– distortion of ball – vibrations in bat
• to hit home run….– large hit ball speed– optimum take-off angle– lots of backspin
Courtesy of CE Composites
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vf = q vball + (1+q) vbat
Conclusion:
vbat matters much more than vball
• q “Collision Efficiency”
• property of ball & bat independent of reference frame ~independent of “end conditions”—more later weakly dependent on vrel
• Superball-wall: q 1• Ball-Bat near “sweet spot”: q 0.2
vf 0.2 vball + 1.2 vbat
vball vbat
vf
Kinematics of Ball-Bat Collision
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Kinematics of Ball-Bat Collision
f ball bat
e-rq =
1+re-r 1+e
v = v v1+r 1+r
r = mball /Mbat,eff : bat recoil factor = 0.25(momentum and angular momentum conservation)
e: “coefficient of restitution” 0.50 (energy dissipation—mainly in ball, some in bat)
vball vbat
vf
q=0.20
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Kinematics of Ball-Bat Collision
f ball bat
e-r 1+ev = v v
1+r 1+r
• r = mball /Mbat,eff: bat recoil factor = 0.25(momentum and angular momentum conservation)
• heavier bat better but…
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The Ideal Bat Weight or Iknob
60
70
80
90
100
110
120
20 30 40 50 60
n=0constant v
bat
n=0.5constant bat KE
vbat
= 65 mph x (32/Mbat
)n
Mbat
(oz)
vf (mph)
n=0.31 (expt)
Observation: Batters prefer lighter bats
Experiments:knob ~ (1/Iknob)0.3
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Accounting for COR:
Dynamic Model for Ball-Bat CollisionAMN, Am. J. Phys, 68, 979 (2000)
• Collision excites bending vibrations in bat
– hurts!
– breaks bats
– dissipates energy • lower COR
• lower vf
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The Details: A Dynamic Model2 2 2
2 2 2
y y A F - EI
t x x:nonuniform beam
-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 ball-bat interaction F(t)
• Step 3: Expand in normal modes and solve
yA x
yEI
x n
2n2
n2
2
2
22n n
n n n n2n
d q F(t) y ( )y( ) q ( )y ( ) q
dt A
zx,t t x
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Modal Analysis of a Baseball Batwww.kettering.edu/~drussell/bats.html
0
0.05
0.1
0.15
0 500 1000 1500 2000 2500
FFT(R)
frequency (Hz)
179
582
1181
1830
2400
frequency
-1.5
-1
-0.5
0
0.5
1
0 5 10 15 20
R
t (ms)
time
0 5 10 15 20 25 30 35
f1 = 179 Hz
f2 = 582 Hz
f3 = 1181 Hz
f4 = 1830 Hz
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Some Interesting Insights:Bat Recoil, Vibrations, COR, and “Sweet Spot”
Evib
vf
e
Node of 1nd mode
+
~ 1 ms only lowest 4 modes excited
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0
20
40
60
80
100
120
0 5 10 15
e
vf (mph)
distance from tip (inches)
nodes4 3 2 1
-30.00
-20.00
-10.00
0.00
10.00
20.00
30.00
0 1 2 3 4 5
v (m/s)
t (ms)
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Experimental Data: Dependence of COR on Impact Location
ball incident on bat at rest
Conclusion: essential physics under control
0.25
0.30
0.35
0.40
0.45
0.50
0.55
23 24 25 26 27 28 29 30 31
e
distance from knob (inches)
flexible bat
rigid bat
Louisville Slugger R161 Wood Batv
i=100 mph
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• handle moves only after ~0.6 ms delay
• collision nearly over by then
• nothing on knob end matters• size, shape• boundary conditions• hands
-30.00
-20.00
-10.00
0.00
10.00
20.00
30.00
0 1 2 3 4 5
v (m/s)
t (ms)
Independence of End Conditions
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0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000
pitcher
catcher
Vibrations and Broken Bats
outside inside
node
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Aluminum has thin shell – Less mass in barrel
–easier to swing and control –but less effective at transferring energy –for many bats cancels
– Hoop modes –trampoline effect –larger COR
Why Does Aluminum Outperform Wood?
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•Two springs mutually compress each other KE PE KE
• PE shared between “ball spring” and “bat spring”
• PE in ball mostly dissipated (~80%!)
• PE in bat mostly restored
• Net effect: less overall energy dissipated...and therefore higher ball-bat COR
…more “bounce”
• Also seen in golf, tennis, …
The “Trampoline” Effect:A Simple Physical Picture
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The Trampoline Effect: A Closer Look
“hoop” modes: cos(2) • k (t/R)3: hoop mode largest in barrel
• f2 (1-3 kHz) < 1/ 1kHz
energy mostly restored (unlike bending modes)
“ping”
Thanks to Dan Russell
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0.40
0.45
0.50
0.55
0.60
0.65
0.70
500 1000 1500 2000
COR-modelCOR-expt
COR
fhoop
(Hz)
Data and Model
to optimize….• kbat small• fhoop > 1
essential physics understood
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Effect of Spin on Baseball Trajectory
Drag: Fd = ½
CDAv2-v direction
“Magnus” or “Lift”: FL = ½ CLAv2
(ω v) direction
v
ω
mg
Fd
FL (Magnus)
CD~ 0.2-0.5CL ~ R/v
(in direction leading edge is turning)
UW Colloquium 10/31/05 25
New Experiment at Illinois
• Fire baseball horizontally from pitching
machine
• Use motion capture to track ball over ~5m
of flight and determine x0,y0,vx,vy,,ay
• Use ay to determine Magnus force as
function of v,
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Motion Capture ExperimentJoe Hopkins, Lance Chong, Hank Kaczmarski, AMN
Two-wheel pitching machine
Baseball with reflecting dot
Motion Capture System
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Experiment: Sample MoCap Datay
z
topspin ay > g
-3000
-2000
-1000
0
1000
2000
-20
0
20
40
60
80
100
120
140
0.00 0.02 0.04 0.06 0.08 0.10 0.12
z (mm)y (mm)
time (sec)
93.6 mph/3040 rpm/1.83g
Z
y
y = ½ ayt2
work in progress
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Some Typical Results
0
0.5
1
1.5
2
0 25 50 75 100 125 150Speed in mph
Drag/Weight
Lift/Weight@1800 rpm
0
20
40
60
80
100
0 50 100 150 200 250 300 350 400
x (ft)
2000 rpm backspin
no spin
200
250
300
350
400
450
10 15 20 25 30 35 40 45 50
2000 rpm backspin
no spin
Lift …--increases range--reduces optimum angle
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Oblique Collisions:Leaving the No-Spin Zone
Friction … • sliding/rolling vs. gripping• transverse velocity reduced, spin increased
vT′ ~ 5/7 vT ~ vT
′/R
Familiar Results• Balls hit to left/right break toward foul line
• Topspin gives tricky bounces in infield
• Pop fouls behind the plate curve back toward field
• Backspin keeps fly ball in air longer
f
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0
50
100
150
200
250
-100 0 100 200 300 400
1.5
0
0.25
0.5 0.75
1.02.0
0.75
Undercutting the ball backspinBall100 downward
Bat 100 upward
D = center-to-center offset
trajectories
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larger for curveball
-1000
0
1000
2000
3000
4000
5000
6000
0 0.2 0.4 0.6 0.8 1A
2000 rpm topspin
2000 rpm backspin
D (in)
(rpm)
Fastball: spin reverses
Curveball: spin doesn’t reverse
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• Bat-Ball Collision Dynamics– A fastball will be hit faster
– A curveball will be hit with more backspin
• Aerodynamics– A ball hit faster will travel farther
– Backspin increases distance
• Which effect wins?
• Curveball, by a hair!
Can Curveball Travel Farther than Fastball?
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Work in Progress
• Collision experiments & calculations to elucidate trampoline effect
• New measurements of lift and drag
• Experiments on oblique collisions– Rod Cross & AMN: rolling almost works at
low speed– AMN: studies in progress at high speed
UW Colloquium 10/31/05 34
Final Summary
• Physics of baseball is a fun application of basic (and not-so-basic) physics
• Check out my web site if you want to know more– www.npl.uiuc.edu/~a-nathan/pob– [email protected]
• Go Red Sox!