the physics of hitting a home run
DESCRIPTION
Thanks to J. J. Crisco & R. M. Greenwald Medicine & Science in Sports & Exercise 34(10): 1675-1684; Oct 2002. The Physics of Hitting a Home Run. Alan M. Nathan,University of Illinois www.npl.uiuc.edu/~a-nathan/pob a-nathan @uiuc.edu. 1927 Yankees: Greatest baseball team ever assembled. - PowerPoint PPT PresentationTRANSCRIPT
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
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1927 Solvay Conference:
Greatest physics team ever assembled
Baseball and Physics
1927 Yankees:Greatest baseball team
ever assembled
MVP’s
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“Hitting is timing; pitching isupsetting 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+r
e-r 1+ev = 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 v1+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 Collision
AMN, 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 xyEI
x n
2n2
n2
2
2
22n n
n n n n2n
d q F(t) y ( )y( ) q ( )y ( ) qdt 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)
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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 400x (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
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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!