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University Physics I

Honors Project Nathan Hemby

Instructor: Dr. Benjamin Davis

Teaching Assistants: Mr. David French and Mr. Jeb Stacy

Lab Time: MW 1:30p-3:20p

Due Date: November 24, 2015

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The Underlying Physics of Pitching a Baseball

Since the first professional games were played in the early 1850’s, baseball has been one

of America’s favorite pastimes. The game of baseball captures the imagination of those who

follow it because of the unique balance of athleticism and finesse that players must exhibit while

playing in order to be the best in the sport. One of the best examples of this distinct blend of skill

is on display at the beginning of every play. The pitcher and batter matchup that begins every

play exemplifies the blend of these distinct skill sets. Locked in a duel every pitch, the pitcher

throws a wide variety of pitches to the batter, attempting to throw off the batter’s timing and

produce a favorable result for the defense. The batter, in defense, attempts to put the ball in play

by making contact with the barrel of his bat. While both sides of this matchup provide intriguing

examples of applied physics in the real world, the pitching side of the matchup will be of

particular interest for this paper. In this paper we will examine the underlying physics of pitching

a baseball. We will examine the different aspects of the physics behind a pitch from the pitcher’s

first motion to when the ball crosses home plate, including the mechanics of a pitcher preparing

to pitch the ball, the delivery of the ball, and the interaction between the ball and the air

surrounding it as the ball travels to home plate.

A Brief History of the Baseball

Before beginning an analysis of the pitcher and the interactions

between the ball and the air, it is necessary to understand the

ball with which baseball is played. An example of today’s

baseball can be found in Figure 1.1, which shows the balls

used from 1998 to present. Today’s baseballs, according to

Figure 1.1- Today’s regulation

MLB baseball, in use since 1998

(Mitchell 2013).

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the Major League Baseball 2015 Official Rules, must be, “a sphere formed by yarn wound

around a small core of cork, rubber or similar material, covered with two strips of white

horsehide or cowhide, tightly stitched together. It shall weigh not less than five nor more than 5¼

ounces avoirdupois and measure not less than nine nor more than 9¼ inches in circumference”

(Official Baseball Rules 2015 Edition 2015). It is also important to note the “tightly stitched”

cover. This cover is two pieces of figure-eight shaped leather stitched together so that the stitches

form a raised seam on the cover of the ball. This seam will become important later in the paper

when discussing how the ball interacts with the air.

Pitcher Mechanics When Preparing to Deliver the Baseball

When a pitcher begins to deliver the baseball, his main objective is to impart as much

momentum to the ball as possible. The momentum that the pitcher generates and imparts to the

ball is converted to kinetic energy upon release. The total kinetic energy of the baseball will

contain a combination of rotational kinetic energy (spin) and translational kinetic energy

(velocity). In order to maximize the amount of momentum the pitcher can impart on the ball, he

goes through a specific routine before each pitch. This routine can be broken into six parts: wind-

up, stride, arm cocking, arm acceleration, arm deceleration, and follow-through (Werner et al

1993). During wind-up and stride, the pitcher turns his body so that his torso is in line with home

plate and raises his leg facing the batter to about waist height. He then strides out toward home

plate, preparing to deliver the ball. This motion is illustrated in Figure 1.2 (A) through (E).

During arm cocking, the pitcher rotates his torso to face home plate while also cocking his arm to

a nearly perpendicular angle between his forearm and upper arm and rotating his shoulder so that

the wrist lags behind the elbow. This motion is illustrated in Figure 1.2 (F) through (H). During

arm acceleration, the pitcher rotates his torso further and at a downward angle, pulling with it his

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shoulder, elbow, and wrist. This motion pulls the

entire arm forward, causing the wrist to be rotated

over the top of the elbow and past the elbow until

the elbow reaches full extension, as illustrated in

Figure 1.2 (G) through (I). Finally, the ball is

delivered to the plate and as the pitcher decelerates

his arm and follows through into a fielder’s position

(Werner et al 1993).

According to Putnam (1993), these phases

are achieved due to the initial forward drive of the

proximal segments of the pitcher’s body (i.e. from

the torso down) causing the distal segments of the

pitcher’s body (i.e. the shoulder and arm) to lag behind in the pitching motion. This lag is the

cause of the arm cocking phase in the pitching motion. The lag sets up a whip-like acceleration

as the higher mass (and therefore higher momentum) proximal segment decelerates and transfers

momentum up the kinetic chain to the lower mass distal segments, leading to a high generation

of angular momentum into the end of the distal segment during the arm acceleration phase

(Putnam 1993). The generated momentum is then transferred to the ball as the pitcher’s arm

reaches maximum extension and releases the ball.

Figure 1.2- The throwing motion of a

pitcher. (Werner et al 1993)

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Interaction Between the Baseball and the Air When Pitched

Once a baseball leaves the pitcher’s hand it is immediately subjected to several forces

that continue to act on the ball until it is hit by the batter or lands in the catcher’s mitt. These

forces determine the path that the ball travels on its way to the batter. The most identifiable force

is the force of gravity. However, there are two other forces that act upon the ball due to its

interaction with the earth’s atmosphere. The drag force and the Magnus force play a key role in

helping the pitcher throw a wide array of pitches that deviate from their expected path.

Drag Force

The drag force is the force that is exerted by the air on a baseball as it moves through the air and

is exerted in the direction directly opposite the path of the baseball. The drag force provides a

negative acceleration on the baseball due to the air opposing the ball’s motion. Beginning with

Newton’s second law of motion:

𝐹𝐷 = 𝑚𝑎𝑦 (1)

with ay equal to acceleration opposite the ball’s velocity, m is equal to the ball’s mass. The given

equation for drag force is modeled by the following:

𝐹𝐷 =1

2𝐶𝐷𝐴𝜌𝑣

2 (2)

where CD is the drag coefficient, A is the cross-sectional area of the ball, ρ is the density of the

air, and v is the velocity of the ball. Setting both equations equal to each other makes it possible

to solve for the drag coefficient, which yields the following equation:

𝐶𝐷 = 𝛿𝑎𝑦

𝑣𝑦2 (3)

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where δ is called the drag length. The drag length is the distance that the object would need to

travel before its velocity slows down by a factor of e. The drag length is given by the equation:

𝛿 =2𝑚

𝜌𝐴 (4)

where m is the mass of the baseball. When examining the drag length, it must be observed that

the density of the air can have a widely ranging value that can vary by as much as 30%,

depending on the atmospheric conditions at the time the ball is thrown. The variables that

contribute to this variation include air pressure, temperature, humidity, and altitude (Kagan

2014). After solving for the drag coefficient, the range of values yielded is around 0.5 for a ball

travelling at 60 miles per hour and 0.3 for a ball travelling at 120 miles per hour (Adair 1995).

Another important component of the drag on a baseball is the Reynold’s number. The

Reynold’s number is representative of how a fluid will flow around an object as the object passes

through it. It is defined as the ratio of two times the density of the air times the velocity of the

ball times the radius of the ball to the viscosity of the air, and is modeled by the equation:

𝑅𝐸 =2𝜌𝑉𝑏

𝜇 (5)

where 𝜌 is the density of the air, V is the velocity of the ball, b is the radius of the ball, and 𝜇 is

the viscosity of the air. The Reynold’s number is a dimensionless unit (“Drag on a Baseball"

2015). When related to a baseball, it describes the thickness of the layer of air that is displaced

by the surface of the ball. This is determined by the interactions between the surface of the ball

and the molecules of air directly in contact with the ball, called the boundary layer. In the

boundary layer, the air molecules closest to the surface of the ball stick to the ball, then collide

with the molecules in close proximity to them. This reaction occurs out to a certain distance

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away from the surface of the ball, determined by the Reynold’s Number, until the air molecules

do not have any energy to interact with surrounding air particles. This interaction within the

boundary layer can be described as being either laminar or turbulent, depending on how the air

behaves after colliding with the ball. The air

near the surface of the ball flows smoothly over

the surface of the ball when the boundary layer

is laminar. On the contrary, a turbulent

boundary layer is characterized by the random

displacement of the surrounding air, resulting

in the boundary layer yielding a swirling

motion near the ball, which creates a thicker

boundary layer near the ball. A ball will

maintain a laminar boundary layer

until it reaches the critical Reynold’s

number (“Boundary Layer” 2015).

For a baseball, the critical Reynold’s

number lies at a lower threshold than

for that of a smooth sphere, as

illustrated in Figure 2.1. This split

between a baseball and a smooth

sphere is caused by the seams creating

a rough surface, which disturbs the boundary layer, sending it into a turbulent state sooner than a

smooth sphere (“Drag on a Baseball” 2015). Once the ball reaches the critical Reynold’s number,

Figure 2.1- Illustration of the graph of the

Reynold’s number for a baseball versus a

smooth sphere. (“Drag on a Baseball”

2015)

Figure 2.2- Reaction of air flow in a laminar

boundary layer versus a turbulent boundary layer (

"Boundary Layer” 2015)

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the boundary layer begins transitioning from laminar to turbulent. By transitioning to a turbulent

boundary layer, the drag on the ball is decreased, which allows the ball to travel with less

resistance through the air, maintaining its velocity for a greater distance to the plate. The

difference between laminar and boundary layers is illustrated in Figure 2.2.

Magnus Force

When a ball is pitched, the natural tendency of the pitcher is to impart spin to the ball.

This spin causes a change in the interactions between the surface of the ball and the boundary

layer. By altering this interaction, another force acts upon the ball during its trip to home plate,

called the Magnus force. The Magnus force is caused by the change in air pressure in the

boundary layer on the top and half and bottom half of the ball and is modelled by the equation:

𝐹𝑀 =1

2𝐶𝐿𝜌𝐴𝑣

2 (6)

where CL is the coefficient of lift, ρ is the density of the air, A is the cross-sectional area of the

ball, and v is the velocity of the ball (Nathan 2008). The Magnus force is caused by the pitcher

giving the ball a high amount of spin, which alters the ball’s interaction with the boundary layer,

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which causes a change in the pressure surrounding

the ball. This spin rotates the ball in the same

direction as the airflow over the top half of the

sphere, which increases the velocity of the air over

that hemisphere of the ball. The opposite happens

on the bottom half of the ball, where the rotation

of the ball opposes the airflow, which decreases

the velocity of the air over the bottom hemisphere

of the ball. An illustration of the air flow around

the ball is shown in Figure 2.3. This change in

velocity causes a change in pressure on each hemisphere, according to Bernoulli’s principle.

Bernoulli’s principle is modelled by the equation:

𝑝 +1

2𝜌𝑉2 + 𝜌𝑔ℎ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (7)

where p is pressure, ρ is density of the air, V is velocity, g is acceleration due to gravity, and h is

elevation. However, it can be assumed that the change in height is negligible for a pitched

baseball, so Bernoulli’s equation for a pitched baseball can be rewritten as:

𝑝 +1

2𝜌𝑉2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (8)

because the term for the relatively small change in height causes the 𝜌𝑔ℎ term of the equation to

become negligibly small during calculations. Because Bernoulli’s equation is constant for a

given situation, the variables in the equation must increase when others decrease. Therefore, the

pressure from the surrounding air on the top hemisphere of the ball decreases with the increase in

flow velocity of the air, while the pressure from the surrounding air increases on the bottom

Figure 2.3- Visualization of the air flow

around a baseball that contributes to the

Magnus force (“Science of Cricket –

Drift and Dip” 2009).

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hemisphere of the ball due to the decrease in the flow velocity of the air. The difference in

pressures creates an uneven force on the ball, which makes the ball deviate from its assumed

path (“Bernoulli’s Equation” 2015).

Also a factor of the Magnus force is the lift coefficient, CL. The lift coefficient is a value

that is shown to be mainly dependent upon the spin factor of the ball, which is the radius times

the angular velocity divided by the velocity of the ball, S=Rω/v. This implies that the Magnus

force on a baseball will be greater for larger rotational velocities ω (Nathan 2008).

The Magnus force is the main reason that pitchers are able to command a wide arsenal of

pitches. Pitchers are able to use the Magnus force in order to cause the ball to deviate from the

path it would be expected to take if it traveled its initial path. Below are some discussions on

how the Magnus force affects several common pitches as well as a discussion of the knuckleball.

Fastball

The main objective of the pitcher when throwing a fastball is to get the ball to the plate

with as much velocity as possible. To achieve this, he grips the ball with his fingertips in order to

transfer as much momentum as possible to the ball. During his release, the pitcher imparts a

linear velocity as well as a backward rotation to the

ball. The ball rotates clockwise when viewed from

the first base side of the field. The backspin of the

ball causes the ball to “rise” upwards when

travelling to the plate. This “rise” is actually an

illusion caused by the ball falling a shorter vertical

distance than the batter expects due to the large

Figure 2.4- Shows the spin of a

fastball as viewed from the first base

side of the field (Petchesky 2013).

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upward Magnus force counteracting some of the force due to gravity. The Magnus force acts on

a fastball as shown in figure 2.3. Ideally, the pitcher will throw a fastball with a rotational axis

perpendicular to the direction of travel as shown in Figure 2.4, which would maximize the

upward Magnus force on the ball. However, most pitchers throw the ball with an axis of rotation

that is slightly tilted off of the perpendicular, which causes a slight deviation from the expected

linear path of the ball due to the small horizontal component of the Magnus force that would then

be present.

Curveball

A curveball is meant to fool the batter by making the ball curve downwards on its path to

the plate. In order to achieve this, the pitcher gives a rotation to the ball in the opposite direction

as that of a fastball, so that the ball has topspin as it travels to the plate. The ball rotates in a

counterclockwise direction about the rotational axis when viewed from the first base side of the

field, as shown in Figure 2.5 (Petchesky 2013). This causes the effects of the Magnus force to be

reversed compared to a fastball (shown in Figure 2.6), causing the curveball to fall a greater

vertical distance than a fastball (Kennel 2012). The Magnus force acts on a curveball as shown in

Figure 2.6, where the pressure regions are reversed from those of a fastball. An area of high

pressure forms above the ball and an area of low pressure

forms below the ball, which causes a Magnus force that

aids the force of gravity. Ideally, the pitcher will throw a

curveball with a rotational axis perpendicular to the

direction of travel as shown in Figure 2.5, which would

maximize the downward Magnus force on the ball.

However, most pitchers throw a curveball that has an axis

Figure 2.5- Shows the spin of a

curveball as viewed from the first

base side of the field (Petchesky

2013).

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of rotation that is slightly off of the perpendicular,

which causes a horizontal deviation in addition to the

vertical drop due to the horizontal component of the

Magnus force that would then be present (Petchesky

2013).

Other Breaking Pitches

Along with the fastball and curveball, certain pitchers also throw other types of breaking

pitches, such as the slider and the screwball. These pitches are thrown by varying the ball grip

and release motion when throwing, but the physics behind them are similar to the curveball. The

only difference is in the variance of the rotational axis of the ball. The slider is thrown so that the

axis of rotation is perpendicular to the axis of rotation of a curveball and perpendicular to the

initial velocity vector. This orientation causes the Magnus force to act on either side of the ball,

causing a horizontal displacement as well as a small vertical drop due to gravity. The screwball

has an axis of rotation that is similar to the axis of rotation of the slider, but a the axis of a

screwball is tilted so that it has a horizontal displacement in the opposite direction of a slider.

Figure 2.6- The Magnus force on a

curveball acts opposite of the Magnus

force on a fastball due to the change in

pressure regions (Kennel 2012).

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Special Case- The Knuckleball

The knuckleball is somewhat of an anomaly in the realm of baseball pitches. While other

pitches rely heavily on the use of the Magnus force in order to achieve their variance in motion,

the knuckleball does not. The knuckleball utilizes the asymmetry of the seams on the ball in

order to produce a pitch that follows an unpredictable flight path that can confuse the batter.

When thrown correctly, a knuckleball is released relatively slowly (between 50mph and 85 mph)

and with a slow forward rotation near one rotation per second (much slower than a typical pitch).

The ball rotates slowly enough that the seams interact with the boundary layer as they cross it,

influencing the separation of the boundary layer from the ball. The separation of the boundary

layer is altered by the seams in two ways: by the seams tripping the boundary layer, inducing a

turbulent boundary and delaying separation, or by directly forcing separation to occur earlier

than it would if a smooth portion of the ball was involved in the separation, as shown in Figure

2.7. This interaction between the seams and the air results in a net force acting on the ball,

generating a lift that causes the ball to deviate from its projected path (Borg and Morrissey

2014). Borg and Morrissey tested this phenomenon by exposing different orientations of the ball

to a wind stream and measured the lift

coefficient. They took measurements at

stationary orientations as well as taking

measurements while the ball was slowly

rotating. The lift coefficient is seen roughly as a

periodic function when plotted against the angle

of displacement from seam orientation that

produces no drag. Their results showed that the

Figure 2.7- Shows boundary layer

interaction and types of separation that

occur when pitching a knuckleball (Borg

and Morrissey 2014).

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greatest positive lift coefficients were achieved

when the ball exposed a seam to the separation

point of the boundary layer. When the ball was

in this orientation, results showed a lift

coefficient between 0.1 and 0.3. The greatest

negative lift coefficients were achieved when

the ball exposed a seam to the boundary layer

at an angle of 30° below the neutral position. At

this orientation, the results showed a lift

coefficient between -0.15 and -0.3. The full

graph of their results is shown in Figure

2.8 (Borg and Morrissey 2014). The data

suggests that if the ball were thrown

with no rotation, the lift force would

cause the ball to have a constant lateral

acceleration in one direction, while a

slowly spinning ball has a constantly

changing force acting upon it. Given the

inconsistent nature of pitching that

makes it nearly impossible to reproduce

identical initial conditions, the knuckleball

travels a different path to home plate each

pitch as even minute changes to any of the

Figure 2.8- Graph of the lift coefficient as

a function of ball orientation for spinning

and non-spinning balls (Borg and

Morrissey 2014).

Figure 2.9- Demonstration of chaotic dynamics,

showing how a change of 0.5 revolutions leads to

an 8-inch difference in the final horizontal

position of the ball. The black patch is the width

of home plate (Nathan 2011).

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initial variables can yield widely varying and constantly changing forces acting within the

boundary layer (Watts and Sawyer 1975, Borg and Morrissey 2014). This is a phenomenon is

known as chaotic dynamics. According to Alan Nathan, the principles of chaotic dynamics can

cause changes in the end location of a knuckleball by as much as one foot. He analyzed one

professional knuckleball and found that changing the total revolution of the ball from 1.5

revolutions to 1.0 revolution led to an eight-inch difference in the final horizontal position of the

ball as it crossed home plate, as shown in Figure 2.9 (Nathan 2011). The improbability of

identical initial variables when throwing a knuckleball leads to a remarkably unpredictable pitch

that confuses batters and entertains spectators.

Conclusion

The nuances of a baseball pitch are a wealth of applied physics just waiting to be

examined. From the wind-up to the flight of the ball, it is possible to quantify and calculate the

interactions of a baseball and its surroundings within the scope of University Physics I

knowledge. Just a small fraction of the current research into the biomechanics and aerodynamics

of pitching a baseball are presented in this paper, but this paper aims to provide a brief overview

of the applied physics behind pitching a baseball.

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Works Cited

“3.01 The Ball.” 2015. In Official Baseball Rules 2015 Edition, ed. Tom Lepperd, Accessed

November 9, 2015. http://mlb.mlb.com/mlb/downloads/y2015/official_baseball_rules.pdf

Adair, Robert. 1995. “The Physics of Baseball.” Physics Today, May 1995. Accessed November

23, 2015. http://baseball.physics.illinois.edu/Adair_PhysicsToday_May95.pdf.

“Bernoulli’s Equation.” 2015. NASA Glenn Research Center. Ed. Nancy Hall. Accessed

November 20, 2015. https://www.grc.nasa.gov/WWW/K-12/airplane/bern.html.

Borg, John P., and Michael P. Morrissey. 2014. “Aerodynamics of the Knuckleball Pitch:

Experimental Measurements on Slowly Rotating Baseballs.” American Journal of

Physics 82 (10): 921-927. Accessed November 23, 2015. doi: 10.1119/1.4885341.

“Boundary Layer.” 2015. NASA Glenn Research Center. Ed. Nancy Hall. Accessed November

15, 2015. http://www.grc.nasa.gov/WWW/k-12/airplane/boundlay.html.

“Drag on a Baseball." 2015. NASA Glenn Research Center. Ed. Nancy Hall. Accessed

November 15, 2015. http://www.grc.nasa.gov/WWW/K-12/airplane/balldrag.html.

Kagan, David, and Alan M. Nathan. 2014. “Simplified Models for the Drag Coefficient of a

Pitched Baseball.” The Physics Teacher 159 (5): 278-280. Accessed November 14, 2015.

doi: 10.1119/1.4872406.

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Kennel, Elliot. 2012. “The Rising Fastball-Myth or Fact?.” The Village Elliot. Accessed

November 21, 2015. http://elliotkennel.blogspot.com/2012/05/rising-fastball-myth-or-

fact.html.

Mitchell, Jeffery. 2013. "Five Obscure Facts About Baseballs." Gunaxin.com. Accessed

November 10, 2015. http://sports.gunaxin.com/facts-about-baseballs/164352.

Nathan, Alan. 2011. “Anatomy of a Really Nasty Pitch Update: Why is Knuckleball Movement

so Erratic?.” Physics of Baseball. Accessed November 23, 2015.

http://baseball.physics.illinois.edu/DickeyPitch103a.html.

Nathan, Alan M. 2008. “The Effect of Spin on the Flight of a Baseball.” American Journal of

Physics 76 (2): 119-124. Accessed November 21, 2015. doi: 10.1119/1.2805242.

Petchesky, Barry. 2013. “Unspinning The Mythical Gyroball, The Demon Miracle Pitch That

Wasn't.” Deadspin.com. Accessed November 21, 2015. http://deadspin.com/unspinning-

the-mythical-gyroball-the-demon-miracle-pit-1451016294.

Putnam, Carol A. 1993. “Sequential Motions of Body Segments in Striking and Throwing Skills:

Descriptions and Explanations.” In Proceedings of the XIIIth Congress of the

International Society of Biomechanics. Abstract. Halifax: Elsevier Ltd. Accessed

November 10, 2015. doi:10.1016/0021-9290(93)90084-R.

“Science of Cricket – Drift and Dip.” 2009. Rushed Behind. Accessed November 21, 2015.

http://rushedbehind.blogspot.com/2009/09/science-of-cricket-drift-and-dip.html.

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Watts, Robert G., and Eric Sawyer. 1975. “Aerodynamics of a Knuckleball.” American Journal

of Physics 43 (11): 960-963. Accessed November 23, 2015. doi: 10.1119/1.10020.

Werner, Sherry L. et al. 1993. “Biomechanics of the Elbow During Baseball Pitching” Journal of

Orthopaedic & Sports Physical Therapy 17 (6): 274-278. Accessed November 10, 2015.

doi: 10.2519/jospt.1993.17.6.274.

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Figure 2.1- Illustration of the graph of the

Reynold’s number for a baseball versus a

smooth sphere (Drag on a Baseball 2015).

Figure 2.2- Reaction of air flow in a laminar boundary

layer versus a turbulent boundary layer (Drag on a

Baseball 2015).

Figure 1.2- The throwing

motion of a pitcher (Werner

et al 1993).

Figure 2.3- Visualization of the air flow

around a baseball that contributes to the

Magnus force (“Science of Cricket –

Drift and Dip” 2009.)

Figure 2.4- Shows the spin of a

fastball as viewed from the first base

side of the field (Petchesky 2013).

Figure 2.5- Shows the spin of a

curveball as viewed from the first

base side of the field (Petchesky

2013).

Figure 1.1- Today’s regulation

MLB baseball, in use since 1998

(Mitchell 2013).

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Figure 2.6- The Magnus force on a

curveball acts opposite of the Magnus

force on a fastball due to the change in

pressure regions (Kennel 2012).

Figure 2.7- Shows boundary layer

interaction and types of separation that

occur when pitching a knuckleball (Borg

and Morrissey 2014).

Figure 2.8- Graph of the lift coefficient as

a function of ball orientation for spinning

and non-spinning balls (Borg and

Morrissey 2014). Figure 2.9- Demonstration of chaotic dynamics,

showing how a change of 0.5 revolutions leads to

an 8-inch difference in the final horizontal

position of the ball. The black patch is the width

of home plate (Nathan 2011).

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Omitted Pieces

The most notable subject of the delivery is the immense amount of rotation that takes

place. The reason that rotation is so heavily utilized by the pitcher is because the rotation creates

a large amount of torque by transferring energy up the kinetic chain. The pitcher’s kinetic chain

is the name for the series of movements that the pitcher makes in order for the pitcher to throw

the ball. This chain stretches from the pitcher’s foot that is driving off of the mound to the

pitcher’s fingertips when he throws the ball (Blake 2015). This chain allows the pitcher to

generate a force by pushing off the earth and transferring that force to the ball, while adding

force generated by musculoskeletal activity through the kinetic chain. The result is a searing

fastball that can reach speeds up to 100mph.

Blake, Chris. 2015. “The Kinetic Chain: Strength and Conditioning of the Baseball Athlete.”

University of Connecticut Orthopaedic Surgery Sports Medicine. Accessed November

10, 2015. http://uconnsportsmed.uchc.edu/injury/prevention/kinetic_chain.html.

This motion was modeled by Feltner and Dapena using 3D analysis of several pitches.

Using their analysis, they were able to derive equations for the forces exerted on the upper arm

and lower arm, and using those forces, derive equations for the torques acting on each segment

of the arm (Feltner and Dapena 1989). They used these derivations

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In order to calculate the forces, they assumed that the distal segment was under the force

of its own weight (WD), a proximal force (FD), and a torque from the upper arm at the elbow

(TD). They also assumed that the upper arm was under the force of its own weight (WU), a

negative distal force (-FD) and the negative torque of the distal segment (-TD) exerted by the

distal segment, as well as the force (FU) and torque (TU) both exerted at the shoulder. They

begin by calculating the acceleration of the center of mass of the upper arm, shown below. Note

the table of nomenclature or Figure 1.3 for explanation of symbols.

𝑎𝐺𝑢 = 𝑎𝑠 + (𝛼𝑈 × 𝑟𝑈) + (𝜔𝑈 × (𝜔𝑈 × 𝑟𝑈)) (1)

After calculating aGu, the acceleration of the center of mass of the distal segment is calculated:

𝑎𝐺𝑑 = 𝑎𝑆 + (𝑎𝑈 × 𝑙𝑈) + (𝜔𝑈 × (𝜔𝑈 × 𝑙𝑈)) + (𝛼𝐷 × 𝑟𝐷) + (𝜔𝐷 × (𝜔𝐷 × 𝑟𝐷)) (2)