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Physics and Baseball:Having Your Cake and Eating it Too

Alan M. Nathana-nathan@uiuc.edu

webusers.npl.uiuc.edu/~a-nathan/pobDepartment of Physics

University of Illinois

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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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A great book to read….

“Our goal is not to reform the game but to understand it.

“The physicist’s model of the game must fit the game.”

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A Physicist’s Approach to Current Issues in Baseball

• Should aluminum bats be banned?

• Corked bats and juiced balls—do they matter?

• What the deal with the gyroball?

• Can steroid use increase home run production?

• Is the game different in Denver?

– and what about that humidor?

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Some Experimental Tools

• Bat testing facility

• High-speed video or motion analysis“You can observe a lot by watching” ---Yogi– swinging the bat – ball-bat collision 1– ball-bat collision 2

• PITCHf/x tracking system

• Trackman radarmore on these later

See http://webusers.npl.uiuc.edu/~a-nathan/pob/video.html for some nifty videos.

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Some Physics Background

• Physics of ball-bat collision

• Aerodynamics of a baseball

• Oblique collisions and spin

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Description of Ball-Bat Collision• forces large, time short

– >8000 lbs, <1 ms

• ball compresses, stops, expands– KEPEKE– bat recoils

• lots of energy dissipated (“COR”)– distortion of ball – vibrations in bat

• to hit home run….– large batted ball speed

• 100 mph~400 ft, each additional mph ~ 5-6’

– optimum take-off angle (300-350)– lots of backspin

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Kinematics of Ball-Bat Collision

ball bat

e-rq =

1+re-r 1+e

BBS = v v1+r 1+r

e: “coefficient of restitution” 0.50 (energy dissipation—mainly in ball, some in bat)

r mballz2/I6 : bat recoil factor = 0.25(momentum and angular momentum conservation)

---heavier is better but…

q=0.20

BBS = q vball + (1+q) vbatvball vbat

BBS

z

9

40

42

44

46

48

50

8.5 9 9.5 10 10.5 11 11.5

I6"

(103 oz-in2)

knob

(rad/s)

y = m1*(9/m0)^m2

ErrorValue

0.392146.218m1

0.0574220.28747m2

NA3.8574Chisq

NA0.93138R

Crisco/Greenwald Batting Cage Study

woodaluminum

Batting cage study show how bat speed depends on I for college baseball players

~ [1/I6]n

0<n<0.5n 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, BBS

• Dynamic model of collision– Treat bat as nonuniform beam– Treat ball as damped spring

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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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Vibrations, COR, and the “Sweet Spot”

Evib

vf

e

+

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

Strike bat here

best performance & feel

@ ~ node 2

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• strike bat in barrel—look at response in handle

• handle moves only after ~0.6 ms delay

• collision nearly over by then

• nothing on knob end matters• size, shape• boundary conditions• hands!

• confirmed experimentally

-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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Aluminum has thin shell – Less mass in barrel

--lower MOI, higher bat speed, easier to control --but lower collision efficiency --partially canceling effects

– “Hoop modes” • trampoline effect • “ping” (~2000 Hz)

Why Is Aluminum Better Than Wood?

demo

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•Two springs mutually compress each other KE PE KE

• PE shared between “ball spring” and “bat spring”

…sharing depends on “kball/kbat”

• 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”—confirmed by experiment

…and higher BBS

• Also seen in golf, tennis, …

The “Trampoline” Effect:A Simple Physical Picture

demo

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Regulating Performance of Non-Wood Bats:A Science-Based Approach Used by NCAA

• Specify maximum q – approx. same as for wood bats of similar wt.– implies bats swung alike will perform alike

• Specify minimum MOI to limit bat speed– smaller than wood

• Together, these determine a maximum BBS– gap between wood and aluminum 5%– does that mean aluminum should be banned?

• an issue many are struggling with

BBS = q vball + (1+q) vbat

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What About Corked Bats?or..What was Sammy thinking?

Conclusion:• No increase in BBS

increase in swing speed decrease in collision efficiency

~ [1/I6]n

0<n<0.5

no trampoline effect!

Wood Bat RW01

unmodified

drilled

corked

0.475

0.480

0.485

0.490

0.495

ball-bat COR

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What About Juiced Baseballs?

Conclusion:No evidence for juiced ball

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Aerodynamics of a Baseball• Gravity• Drag (“air resistance”)• Lift (or “Magnus”)

v

ω

mg

Fd

FM

Courtesy, Popular Mechanics

Fd=½ CDAv2

-v direction

(ω v) direction FM = ½ CMARv

direction leading edge is turning

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Typical values of drag and lift

0

0.5

1

1.5

2

0 25 50 75 100 125 150Speed in mph

Drag/Weight

Lift/Weight@1800 rpm

“Drag crisis?”

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Effect of Drag and Lift on Trajectories

• drag effect is huge

• lift effect is smaller but significant

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

distance (ft)

no drag or lift

drag, no lift drag and lift

v

ω

mg

Fd

FM

220

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

Some Effects of Drag

• Reduced distance on fly ball

• Reduction of pitched ball speed by ~10%

• Asymmetric trajectory:– Total Distance 1.7 x distance

at apex

• Optimum home run angle ~30o-35o

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

distance (ft)

no drag or lift

drag, no lift

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Some Effects of Magnus

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

distance (ft)

no drag or lift

drag, no lift drag and lift

• Backspin makes ball rise– “hop” of fastball

– undercut balls: increased distance, reduced optimum angle of home run

• Topspin makes ball drop– “12-6” curveball

– topped balls nose-dive

• Breaking pitches due to spin– Cutters, sliders, etc.

v

ω

mg

Fd

FM

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The PITCHf/x Tracking SystemA Quantitative Tool to Study Pitched Baseball Trajectories

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How Does PITCHf/x Work?

• Two video cameras track baseball in 1/60-sec intervals – usually “high home” and “high first”– third CF camera used establishes ht. of strike zone

• Pattern-recognition software to identify “blobs”• Camera calibration to convert pixels to (x,y,z)

– 9-parameter fit to trajectory– constant acceleration for x(t),y(t),z(t)

• Use fit to calculate lots of stuff– The full trajectory– The “break”– Drag and Magnus forces

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Example: Drag and Drag Coefficients

20k pitches from Anaheim, 2007

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Using PITCHf/x to Classify PitchesJon Lester, Aug 3, 2007 @ Seattle

I: Nearly overhand fastballII: Slider or cut fastballIII: ¾ FastballIV: Curveball

LHPCatcher’s View

spin axis

break direction = -90o

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What’s the Deal with the Gyroball?

Courtesy, Ryutaro Himeno

Daisuke Matsuzaka:Does he or doesn’t he?

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From PITCHf/x to HITf/xBarry Bond’s 756th Home Run

• PITCHf/x data tracked hit ball over first 20 ft• Precision measurement of endpoint and time-of-flight

• Inferred: v0=112 mph; =27o up; =16o to right of dead center; =1186 rpm (backspin) and 189 rpm (sidespin, breaking to center)

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Baseball Aerodynamics:Things I would like to know better

• Better data on drag– “drag crisis”?– spin-dependent drag?– drag for v>100 mph

• Dependence of drag & Magnus on seam orientation, surface roughness, …

• Is the spin constant?

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Trackman: The Wave of the Futuresee www.trackmangolf.com

• Doppler radar to measure radial velocity• 3-detector array to measure phase

– two angles

• Sidebands gives spin magnitude• Result:

– in principle, full trajectory can be reconstructed, including spin and spin axis

– already in use for golf, currently being adapted for baseball

32thanks to Fredrik Tuxen, CTO of Trackman

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Oblique Collisions:Leaving the No-Spin Zone

Oblique friction spin• still need a good collision model• my model: slide, then roll

Familiar Results:

• Balls hit to left/right break toward foul line

• Topspin gives tricky bounces in infield

• Backspin keeps fly ball in air longer

• Tricky popups to infield

340

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 backspin

Ball100 downward

Bat 100 upward

D = center-to-center offset

trajectories

“vertical sweet spot”

What’s going on here??

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Another familiar result:

Catcher’s View

bat hits under ball:popup to opposite field

bat hits over ball:grounder to pull field

bat tilted downward

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• Steroids increases muscle mass

• Increased muscle mass increases swing speed

• Increased swing speed increase BBS

• Increased BBS means longer fly balls

• Longer fly balls means more home runs

Steroids and Home Run Productonsee Roger Tobin, AJP, Jan. 2008

Home Run Distances, 2007www.hittrackeronline.com

Delta = distance beyond fence (ft)

~4% per foot

Tobin’s Conclusion: increase of BBS by few mph can increase HR rate by 30-50%!

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Steroids and Bat Speed

• Batter supplies energy proportional to M• Energy shared between bat and some fraction

2 of M• Roughly: 2 ~ 0.01 • So roughly, 10% increase in M gives

– ~2.5% increase in vbat

– ~2 mph increase in BBS– ~12 additional ft. on long fly ball– ~20% (Adair) -50% (Tobin) more home runs!

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Work in Progress

• Collision experiments & calculations to elucidate trampoline effect

• New studies of aerodynamics using Trackman and PITCHf/x

• Experiments on high-speed oblique collisions—does slide-then-roll model work?

• A book, with Aussi Rod Cross

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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– webusers.npl.uiuc.edu/~a-nathan/pob– a-nathan@uiuc.edu

• Thanks for your attention and go Red Sox!

2004 2007