Transcript Baseball: It's Not Nuclear Physics (or is it?!)

The Physics of Hitting a Home Run
Alan M. Nathan,University of Illinois
www.npl.uiuc.edu/~a-nathan/pob
a-nathan @uiuc.edu
Thanks to J. J. Crisco & R. M. Greenwald
Medicine & Science in Sports & Exercise 34(10): 1675-1684;
Oct 2002
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Baseball and Physics
1927 Yankees:
Greatest baseball team
ever assembled
1927
Solvay Conference:
Greatest physics team
ever assembled
MVP’s
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Hitting the Baseball:
the most difficult feat in sports
“Hitting is fifty percent
above the shoulders”
“Hitting is timing; pitching is
upsetting timing”
1955 Topps cards from my personal collection
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Hitting and Pitching, Thinking and Guessing
Graphic courtesy of Bob Adair and NYT
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Example: Tim Wakefield’s Knuckleball
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The Physics of Hitting a Home Run
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?
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Brief Description of Ball-Bat Collision
• forces large, time short
– >8000 lbs, <1 ms
• ball compresses, stops, expands
– KEPEKE
– 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
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Courtesy of CE Composites
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Kinematics of Ball-Bat Collision
vball vbat
vf = q vball + (1+q) vbat
vf
• 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
Conclusion:
vbat matters much more than vball
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Kinematics of Ball-Bat Collision
e-r
q=
1+r
 e-r 
vf = 
v ball 

1+r 
vball vbat
1+e 
 1+r  v bat
vf
r = mball /Mbat,eff : bat recoil factor =  0.25
(momentum and angular momentum conservation)
e: “coefficient of restitution”  0.50
q=0.20
(energy dissipation—mainly in ball, some in bat)
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Kinematics of Ball-Bat Collision
 e-r 
1+e 
vf = 
vball  
vbat


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
v (mph)
f
120
110
n=0.5
constant bat KE
n=0.31 (expt)
100
90
n=0
constant v
80
70
bat
v
bat
60
Experiments:
knob ~ (1/Iknob)0.3
20
= 65 mph x (32/M )
n
bat
30
M
bat
40
(oz)
50
60
Observation: Batters prefer lighter bats
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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 Model
nonuniform beam:
2 y
2  2 y 
 A 2  F - 2  EI 2 
t
x  x 
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• Step 1: Solve eigenvalue
problem for free vibrations
   yn 
2


EI


A

n yn
2 
2 
x  x 
2
2
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y
20
01
5
0
5-
y
• Step 2: Nonlinear lossy spring
z
for ball-bat interaction F(t)
• Step 3: Expand in normal
modes and solve
d 2q n
F(t) yn ( z )
2
y(x,t )   q n (t )y n ( x)
 n q n 
2
dt
A
n
-10
-15
-20
0
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01
51
02
52
03
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Modal Analysis of a Baseball Bat
www.kettering.edu/~drussell/bats.html
f1 = 179 Hz
f3 = 1181 Hz
f2 = 582 Hz
f4 = 1830 Hz
FFT(R)
1
0.15
582
0.5
0
R
5
10
15
20
25
30
1181
-0.5
-1
1830
179
0.05
-1.5
35
0.1
0
frequency
time
0
5
10
t (ms)
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2400
20
0
0
500
1000
1500
frequency (Hz)
2000
2500
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Some Interesting Insights:
Bat Recoil, Vibrations, COR, and “Sweet Spot”
Node of 1nd mode
v (mph) v (m/s)
nodes
0.5
0.4
e
1
e
0.4
+
f
4 3 2
vf
0.3
120
30.00
100
20.00
80
10.00
60
0.00
40
-10.00
20
-20.00
0.3
0.2
Evib
0.2
0.1
0
5
10
distance from tip (inches)
0
15
-30.00
0
1
2
3
4
5
t (ms)
~ 1 ms  only lowest 4 modes excited
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Experimental Data: Dependence of COR
on Impact Location
ball incident on bat at rest
e
0.55
0.50
rigid bat
0.45
flexible bat
0.40
0.35
0.30
0.25
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Louisville Slugger R161 Wood Bat
v =100 mph
i
24
25 26 27 28 29 30
distance from knob (inches)
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Conclusion: essential physics under control
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Independence of End Conditions
• handle moves only after ~0.6 ms delay
• collision nearly over by then
• nothing on knob end matters
• size, shape
• boundary conditions
• hands
v (m/s)
30.00
20.00
10.00
0.00
-10.00
-20.00
-30.00
0
1
2
3
4
5
t (ms)
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Vibrations and Broken Bats
inside
outside
pitcher
0.000
5.000
10.000
15.000
20.000
node
25.000
30.000
35.000
catcher
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Why Does Aluminum Outperform Wood?
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 
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The “Trampoline” Effect:
A Simple Physical Picture
•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, …
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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)
Thanks to Dan Russell
“ping”
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Data and Model
COR
0.70
0.65
COR-model
COR-expt
0.60
0.55
to optimize….
• kbat small
• fhoop > 1
0.50
0.45
0.40
500
1000
1500
f
hoop
2000
(Hz)
essential physics understood
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Effect of Spin on Baseball Trajectory
FL (Magnus)
v

Fd
mg
ω
Drag: Fd = ½ CDAv2
-v direction
“Magnus” or “Lift”: FL = ½ CLAv2
(ω  v) direction
(in direction leading edge is turning)
CD~ 0.2-0.5
CL ~ R/v
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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 Experiment
Joe Hopkins, Lance Chong, Hank Kaczmarski, AMN
Motion Capture System
Two-wheel pitching machine
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Baseball with reflecting dot
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Experiment: Sample MoCap Data
y
93.6 mph/3040 rpm/1.83g
2000
z
140
120
100
1000
80
z (mm)
y
0
topspin  ay > g
60
y (mm)
40
-1000
-2000
-3000
0.00
y=½
0.02
0.04
0.06
20
Z
ayt2
0
0.08
0.10
-20
work in progress
0.12
time (sec)
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Some Typical Results
2
1.5
Drag/Weight
2000 rpm backspin
100
1
80
0.5
Lift/Weight
@1800 rpm
60
0
0
25
50
75
100
40
Speed in mph
no spin
150
125
20
0
450
0
50
2000 rpm backspin
100 150 200 250 300 350 400
x (ft)
400
Lift …
--increases range
--reduces optimum angle
350
no spin
300
250
200
10
15
20
25
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
35
40
45
50
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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
f
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
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Undercutting the ball  backspin
Ball100 downward
D = center-to-center offset
Bat 100 upward
250
200
150
1.5
1.0
2.0
100
0.5
50
0
-100
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trajectories
0.75
0.75
0.25
0
100
0
200
300
400
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Fastball: spin reverses
Curveball: spin doesn’t reverse
 (rpm)
6000
5000
  larger for curveball
4000
2000 rpm topspin
3000
2000
1000
2000 rpm backspin
0
-1000
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0
0.2
0.4
0.6
DA (in)
0.8
1
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Can Curveball Travel Farther than
Fastball?
• 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!
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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!
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