Download ppt - Introduction Hitting the Baseball The Flight of the Baseball Pitching the Baseball Summary

GWU Colloquium, October 21, 1999 Page 1

Baseball: It’s Not Nuclear PhysicsBaseball: It’s Not Nuclear Physics(or is it?!)(or is it?!)

Alan M. Nathan Alan M. Nathan

University of IllinoisUniversity of IllinoisGWU Colloquium, October 21, 1999GWU Colloquium, October 21, 1999

IntroductionIntroduction

Hitting the BaseballHitting the Baseball

The Flight of the BaseballThe Flight of the Baseball

Pitching the BaseballPitching the Baseball

Summary


REFERENCESREFERENCES

The Physics of Baseball, Robert K. Adair (Harper Collins,

New York, 1990), ISBN 0-06-096461-8

The Physics of Sports, Angelo Armenti (American Institute of Physics, New York, 1992), ISBN 0-88318-946-1

www.physics.usyd.edu.au/~cross

L. L. Van Zandt, AJP 60, 72 (1991)

www.npl.uiuc.edu/~a-nathan


Hitting the BaseballHitting the Baseball

“...the most difficult thing to do in sports”

--Ted Williams

BA: .344SA: .634OBP: .483HR: 521


Speed of Hit Ball:Speed of Hit Ball:What does it depend on?What does it depend on?

Speed is important: 105 mph gives ~400 ft each mph is worth 5 ft

The basic stuff (“kinematics”)

speed of pitched ball

speed of bat

weight of bat The really interesting stuff (“dynamics”)

“bounciness” of ball and bat

weight distribution of bat

vibrations of bat


What Determines Batted Ball Speed?What Determines Batted Ball Speed?

1. pitched ball speed

2. bat speed

Rigid-Body Kinematics:

Conclusion:Bat Speed Matters More!

V = 0.25 Vball + 1.25 Vbat

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100speed of pitched ball or bat (mph)

vary pitched ball speed

vary bat speed


40

50

60

70

80

90

100

20 30 40 50 60

mass of bat (oz)

constant bat energy

constant bat+batter energy

60

70

80

90

100

110

120

20 30 40 50 60

mass of bat (oz)

constant bat energy

constant bat speed

constant bat+batter energy

What Determines Batted Ball Speed?What Determines Batted Ball Speed?

3. Mass of bat larger mass lower bat speed

Conclusion:mass of bat matters….but not a lot

bat speed vs mass

ball speed vs mass


What Determines Batted Ball Speed? What Determines Batted Ball Speed?

4. Inelasticity Ball compresses

kinetic energy stored in “spring”

Ball expandskinetic energy restored but...

70% of energy is lost!

(heat, deformation,vibrations,...)

Forces are large (>5000 lbs!)

Time is short (<1/1000 sec!)

The hands don’t matter!

0

1000

2000

3000

4000

5000

6000

0 0.2 0.4 0.6 0.8 1

Time in milliseconds

0

2000

4000

6000

8000

1 104

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

force (pounds)

compression (inches)

approx quadratic


Inelasticity: The Inelasticity: The CCoefficient oefficient oof f RRestitutionestitution

COR = Vrel,f/Vrel,I COR2 = KEcm,f /KEcm,i

For baseball, COR=.52-.58 Changing COR by .05 changes V by 7 mph (35 ft!)

How to measure? Bounce ball off hard surface COR2 = hf/hi

0

50

100

150

200

0 0.2 0.4 0.6 0.8 1COR


Energy shared between ball and bat

Ball is inefficient: 25% returned

Wood Bat r~0.02 80% restored

COReff = 0.50-0.51

Aluminum Bat

r~0.10 80% restored COReff = 0.55-0.58

“trampoline effect”

ball flies off the bat!

What About the Bat?What About the Bat?(or, it takes two to tango!)(or, it takes two to tango!)

r Ebat/Eball kball/kbat xbat/ xball

>10% larger!


Properties of BatsProperties of Bats

length, diameter weight position of center of gravity

where does it balance? distribution of weight

moment of inertia center of percussion stiffness and elasticity

vibrational nodes and frequencies


60

70

80

90

100

110

10 15 20 25 30 35

distance from knob (inches)

aluminum

wood

Sweet Spot #1: Sweet Spot #1: MMaximum aximum EEnergy nergy TTransferransfer

Barrel end of bat maximizes bat speed

Center of Mass minimizes angular impulse

MET must be in between

MET COP @ 5” from

knob

Aluminum bat more effective

for inside pitches

CM

Alum Wood

xcm 21.9” 19.6”

kch 9.2” 10.2”

kh 23.8” 22.1”


Sweet Spot #2: Sweet Spot #2: CCenter enter oof f PPercussionercussion

When ball strikes bat... Linear recoil

conservation of momentum Rotation about center of mass

conservation of angular momentum When COP hit

The two motions cancel (at conjugate point) No reaction force felt

x1

x2

x1x2=Icm/M


Sweet Spot #3: “Node” of VibrationSweet Spot #3: “Node” of Vibration

Collision excites bending

vibrations in bat

Ouch!! Energy lost ==>lower COR Sometimes broken bat

Reduced considerably if collision

is a node of fundamental mode

Fundamental node easy to find For an interesting discussion, see

www.physics.usyd.edu.au/~cross


Dynamics of Bat-Ball CollisionDynamics of Bat-Ball Collision

Step 1: Solve eigenvalue problem for free vibrations

Step 2: Model force

Step 3: Expand in normal modes and solve

yA x

yEI

x n

2n2

n2

2

2

x

yEI

x - F

t

yA

2

2

2

2

2

2

0

2000

4000

6000

8000

1 104

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

force (pounds)

compression (inches)

approx quadratic

A

)t(Fq

dt

qd )(y)(q)y( n

n2n2

n2

nn

n xtx,t


-4

-2

0

2

4

0 5 10 15 20 25 30 35

Mode 1Mode 2Mode 3

distance from knob (inches)0

1000

2000

3000

4000

5000

6000

7000

0 2 4 6 8 10

frequency (Hz)

vibrational mode

General ResultsGeneral Results

Excitation of normal mode depends on ... fnT (or T/Tn)

yn at impact point For T 1 ms

only lowest 2 or 3 modes important

(fn=171, 568, 1178, 1851,…)


0

10

20

30

40

50

60

70

rigid recoil

losses in ball

ballvibrations

in bat

30

40

50

60

70

80

90

100

110

18 20 22 24 26 28 30 32

Vf (mph)


rigid bat

realistic bat

RESULTS:

0.15

0.2

0.25

0.3

0.35

0.4

0.45

18 20 22 24 26 28 30 32

|vf/v

i|


rigid bat

realistic bat

vi = 1 m/s

theory vs. experiment (Rod Cross)at low speed

typical speed


Advantages of AluminumAdvantages of Aluminum

Length and weight “decoupled”

Can adjust shell thickness

More compressible => “springier”

Trampoline effect

More of weight closer to hands

Easier to swing

Less rotational energy transferred to bat

More forgiving on inside pitches

Stiffer for bending

Less energy lost due to vibrations


Aerodynamics of a BaseballAerodynamics of a Baseball

Forces on Moving Baseball

No Spin Boundary layer separation DRAG! FD=½CDAv2

With Spin

Ball deflects wake ==>Magnus

force FMRdFD/dv Force in direction front of ball

is turning

Pop

Pbottom


How Large are the Forces?How Large are the Forces?

• Drag is comparable to weight• Magnus force < 1/4 weight)

0

0.5

1

1.5

2

0 25 50 75 100 125 150Dra

g/W

eig

ht

or

Mag

nu

s/W

eig

ht

Speed in mph

Drag/Weight

Magnus/Weight =1800 RPM


The Flight of the Ball:The Flight of the Ball:Real Baseball vs. Physics 101 BaseballReal Baseball vs. Physics 101 Baseball

Role of Drag

Role of Spin

Atmospheric conditions Temperature Humidity Altitude Air pressure Wind

0

50

100

150

200

250

300

350

400

0 100 200 300 400 500 600 700 800

y (ft)

x (ft)

trajectory

vi = 105 mph @350

no drag

-100

0

100

200

300

400

0 20 40 60 80 100

Range (ft)

q (deg)

Range vs. q

100

150

200

250

300

350

400

450

500

40 60 80 100 120 140

Range (ft)

vi (mph)

Range vs. vi

approx linear

Max @ 350


The Role of FrictionThe Role of Friction

Friction induces spin for oblique collisions

Spin Magnus force

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

batball

topspin ==>F down backspin==>F up

sidespin ==> hook

bat

ball


The Home Run SwingThe Home Run Swing

• Ball arrives on 100 downward trajectory

• Big Mac swings up at 250

• Ball takes off at 350

•The optimum home run angle!


Pitching the BaseballPitching the Baseball

“Hitting is timing. Pitching isupsetting timing”

---Warren Spahn

vary speeds manipulate air flow orient stitches


Let’s Get Quantitative!Let’s Get Quantitative!How Much Does the Ball Break?How Much Does the Ball Break?

Kinematics z=vT x=½(F/M)T2

Calibration 90 mph fastball drops 3.5’ due to

gravity alone Ball reaches home plate in ~0.45

seconds Half of deflection occurs in last 15’ Drag: v -8 mph Examples:

“Hop” of 90 mph fastball ~4” Break of 75 mph curveball ~14”

slowermore rpm force larger

3

4

5

6

7

0 10 20 30 40 50 60Ve

rtic

al

Po

sit

ion

of

Ba

ll (

fee

t)

Distance from Pitcher (feet)

90 mph Fastball

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60

Ho

rizo

nta

l Def

lect

ion

of

Bal

l (fe

et)

Distance from Pitcher (feet)

75 mph Curveball


Examples of PitchesExamples of Pitches

Pitch V(MPH) (RPM) T M/W

fastball 85-95 1600 0.46 0.10

slider 75-85 1700 0.51 0.15

curveball 70-80 1900 0.55 0.25

What about split finger fastball?


Effect of the StitchesEffect of the Stitches

Obstructions cause turbulance

Turbulance reduces dragDimples on golf ballStitches on baseball

Asymmetric obstructions

Knuckleball

Two-seam vs. four-seam delivery

Scuffball and “juiced” ball


SummarySummary

Much of baseball can be understood with basic principles of physics

Conservation of momentum, angular momentum, energy

Dynamics of collisions

Excitation of normal modes

Trajectories under influence of forces

gravity, drag, Magnus,….

There is probably much more that we don’t understand

Don’t let either of these interfere with your enjoyment of the game!


What Determines Batted Ball Speed?What Determines Batted Ball Speed?A Simple FormulaA Simple Formula

ibat,iball,fball, vr1

e1 v

r1

r-e v

1.2 1.3 x .15 k

z-z1

m

m r

0.5 nrestitutio oft coefficien e2

CM

bat

ball

Conservation of momentum, energy, and angular momentum:

radius of gyration


How Would a Physicist Design a Bat?How Would a Physicist Design a Bat?

Wood Bat already optimally designed

highly constrained by rules! a marvel of evolution!

Aluminum Bat lots of possibilities exist but not much scientific research a great opportunity for ...

fame fortune


Example 1: FastballExample 1: Fastball

85-95 mph1600 rpm (back)12 revolutions0.46 secM/W~0.1


Example 2: Split-Finger FastballExample 2: Split-Finger Fastball

85-90 mph1300 rpm (top)12 revolutions0.46 secM/W~0.1


Example 3: CurveballExample 3: Curveball

70-80 mph1900 rpm

(top and side)17 revolutions0.55 secM/W~0.25


Example 4: SliderExample 4: Slider

75-85 mph1700 rpm (side)14 revolutions0.51 secM/W~0.15


Note: both ball and racket compress