Babe Ruth, a legend of Major League Baseball
He led the league in home runs during a season 12 times
and made a record of 60 home runs in 1927 season
as well as 714 home runs of his life time record
In the dead-ball era
Ruth's home run and slugging percentage dominated the league
He creates an unchallengeable position in the heart of baseball fans
and was respectfully named "Baseball Demigod"
Roger Maris, an outfielder of New York Yankee
who challenged Ruth's record after 34 years-
- 60 home runs in a season
With his home run number getting close to 60
however, there was no joy for fans
because there can be only one hero in their heart
Roger, up here
Hey it's the voice from above
Hey Maris up here
it's the Babe
Hey you wnat my record? you want my record?
Catch this you ape piece of shit
Stop
In the game 154 in 1961
Maris hit number 59 in the 3rd inning
Two outs, in the top of 9th
and it's the last chance for Maris
Orioles was behind but made an unusual substitution
The closer Hoyt Wilhelm was called on to stop him tying Ruth's record
I hate that knuckleball
In this weather it's gonna be dancin' all over the place
Even if he gets wood on it it's not goin' anywhere
Bushlinger
Wilhelm takes the sign
but we all know what's coming
Does Roger Maris have one last home rum in him
he fouled it back
Roger had a good cut on that ball, but it's dancin' all over the place
Wilhelm's got it
Out
The knuckleball dominated the hitter and Maris failed to tie the record on that day
Knuckleball is a very special pitch in baseball games
It may suddenly change in direction during its flight
and dance all over the place with erratic movement
so that it can fool the batter
In the following we will investigate knuckleball
base on aerodynamics
Furthermore, we will then introduce various pitches of baseball
When it comes to aerodynamics, most people are familiar with Bernoulli's principle
But, to establish enough background knowledge
we need to learn more about the fluid theories
besides the Bernoulli's principle
First let's talk about boundary layer theory
When a fluid flows over a obstacle, they would interact with each other
Let's look into a simplest case
When a fluid passing through a flat plate with velocity V
it causes friction on the surface
In general, fluids are viscous
so the flow speeds decrease gradually when it get close to the surface
and end up at zero
The region where flow speeds gradually decrease
is called the "Boundary layer"
Viscosity dominate physical phenomenon in this thin layer
and the Bernoulli's principle does not apply in this region
However, viscosity has no effect on the region outside the boundary layer
where the fluid behaves like a inviscid ideal fluid
such that the Bernoulli's principle applies
We will back to Bernoulli's principle soon
Now pay attention to the phenomenon inside the boundary layer
If the flow velocity V is not large
The airflow inside the boundary layer is stable
and the streamlines look like a Mille Crepe
This is named a "Laminar boundary layer"
The geometric shape of obstacles have effects on boundary layer
For example, when substitute the plate by a sphere
a new physical phenomenon may occur
Streamlines are drew around the sphere
and the green part represents a boundary layer
which is a important region
Let's study the region outside boundary layer first
where the Bernoulli's principle applies
Observe the motion of a small volume element
Since the speed of a baseball is much lower than sound speed
the compression effect can be ignore
and the volume of this small element keeps unchanged
Imagine it moves in a tube that continuously changes in diameter
When it moves toward A
its speed decreases
A is called the "stagnation point"
around which the fluid separates to both sides
The speed increases from A to B
and reach the highest value at B
After passing through B the speed decreases
and reach the minimum speed when it arrive at C
Bernoulli's principle states that
high speed corresponds to low pressure
and low speed corresponds to high pressure
Therefore, the pressure maximums occurs at A and C
and B has the minimum pressure
Interesting thing occurs in the downstream region
The pressure gradually increases from B to C
Outside the boundary layer, when the external pressure variation is large enough
it can force the downstream boundary layer to move in opposite direction
Once the inverse flow occurs
upstream boundary layer can be pushed away from the surface
This phenomenon is called "Boundary layer separation"
The location of "Separation point" is near B
and the flow is very unstable behind the separation point
Round and round
When various scales of vortices emerge, it is called "Turbulence"
The tail-like turbulent region behind a obstacle is also named a "Wake"
Since the separation points occur close to B and D
the wake width is comparable to the diameter of the sphere
As a whole, wake is a low pressure region
and this is the main reason for air resistance
However, the location of separation points
are sensitive to the geometric shape of obstacles
For a streamlined body
the separation occur near its tail
The smaller the wake width
the smaller the drag
This is the reason why airfoils and solar cars
are designed to be streamlined form
Besides the geometric shape of obstacles
the surface roughness also determines the location of separation points
Take a look at this interesting experiment
The left photo represents a smooth sphere in wind tunnel
and the separation points are close to the largest cross section of the sphere
The right photo shows that with a thin trip wire in the upstream
the separation points move backward and the wake shrinks
What's going on here?
Why a small raised surface leads to this result?
The reason is that the state of boundary layer has been changed
It is originally a laminar flow in the upstream boundary layer
however, since the raised surface creats small scale vortices
the boundary layer becomes turbulent
This is called a "Turbulent boundary layer"
Fluid flows faster outside the boundary layer because no viscous effect there
A turbulent boundary layer is stirred by vortices
which makes it mixed with the outer and faster fluid
Therefore, the average velocities of boundary layer increases
and get more momentum toward downstream
So that the separation points retreat and the wake shrinks
The purpose of dimple design on a golf ball
is to create a turbulent boundary layer and reduce the drag
Baseballs are not smooth spheres, too
The seam lines on the surface of a balseball can disturb boundary layer
In general, baseball seams can cause the retreat of separation points
Baseball seams are not uniformly distributed on the surface
and this could result in the wake deflection
Moreover, the rotations of a baseball are classified into two types
One is four-seamer: There are 4 seams passing by in one full rotation
The other one is two-seamer: There are 2 seams passing by in one full rotation
The changes in attack angle results in non-symmetric force
which can be measured by wind tunnel experiments
This figure shows the force versus attack angle
for a four-seamer at zero rotation speed
As you can see, the force has four periods of variation in one full rotation
The detail explanations for that figure are given in the following computer graphics
The air flows in steadily from the left
In the beginning the attack angle is zero
and the seams are symmetrically distributed
therefore, the wake is right behind the ball without being deflected
But the wake deflects dramatically when the attack angle changes
The largest upward deflection occurs when the attack angle is about 22 degree
What is the reason for wake deflection ?
The air in upstream boundary layer becomes turbulent after passing by the seams
Notice the seam in the lower half part is more close to the stagnation point
and the lower half turbulent boundary leads the upper one
With longer path and more chance for mixing
the boundary layer gets more velocity and momentum moving downward
so the lower half part separation point retreats more
On the contrary
the upper half part separation point is in advance and the average velocity is lower
After mixing of these two sides of boundary layer flow in the wake
the average velocity deviates upward
and this is the explanation for wake deflection
To realize how the force acting on the ball
one can regard the complex interaction processes as a black box
The horizontal air flow deflects upward after passing through the black box
implies that a upward force acting on the air flow
Simultaneously, there must be a reaction force in the black box exert on the ball
which is equal in magnitude and opposite in direction
as specified by Newton's third law
This is the result of non-symmetric seam distribution
Keep on varying the attack angle
the force vanishes at about 45 degree
beyond which the wake deflects to another side
and the force reachs its maximum at about 68 degree
Then the force vanishs again at 90 degree
that it has goes through a period
As a result, there are 4 periods in a full rotation of 360 degree
In the case of two-seam rotation for a knuckleball
there are two main periods of force variation
The maximum force is equivalent to 2/3 of baseball weight
and the period is two times larger than the case of four-seam rotation
Define the coordinate system first before follow-up discussions
Use the rectangular coordinate system
and let X axis pointing toward home plate
Y axis pointing toward first base
Let XY plane be parallel to the ground
so Z axis stands vertically
For this coordinate system
The lift force is in the positive Z direction
the gravity is in the negative Z direction
the lateral force is in the positive/negative Y direction
and the drag force is in the negative X direction
Now consider the rotation axis of a knuckleball is perpendicular to the ground
Observe the trajectory from top view
and assume the ball spins half a rotation during the flight
The wind tunnel experiment implies that
there are two periods of left and right motion for a four-seam knuckleball
In the case of two-seam rotation
the knuckleball trajectory has one period of variation
In practice
a pitcher would try to throw the ball as no spin as possible
The way is to push the ball with fingertip
but it makes the ball hard to be controlled
Since the spin axis may not be fixed in some specific direction
and since the force is sensitive to attack angle
the knuckleball trajectory becomes hard to be predicted
This is a demonstration of knuckleball's movement
by Tim Wakefield in a Japan TV show
From the research of knuckleball we learn that at extremely low spin rate
the seam locations or say attack angle
determines the force on the baseball
In baseball games, however
other kinds of pitches have spin rate over 10 times higher than knuckleball
and a new physical effect of force enters in-
- the "Magnus force"
To get rid of the disturbance by seams
let's begin with a smooth sphere
When it start to spin
The B-side surface and the ambient air move in the same direction
while C-side moves reversely
Since the fluid inside boundary layer is governed by viscosity
the B-side boundary layer flows faster than that of C-side toward downstream
and carries more momentum
Therefore, the B-side separation point becomes relative backward
so the wake deflects downward after mixing of these two sides of air
and the transverse force arises
This is called the "Magnus effect"
that resulted from the rotation of moving object in the fluid
The magnitude of Magnus force is probably proportional to
the air flow speed V and the angular frequency ω of the ball
So the empirical formula for Magnus force is given by
1/2 times "Magnus coefficient CM"
"air density ρ"
"baseball cross-sectional area A"
"radius R"
"angular frequency ω" and the "flow speed V"
The Magnus coefficient is a dimensionless quantity
and is the only one parameter to be determined from experiments
Rotation is the motion characterize with direction
and can be well described by a "Vector"
For example, a spin vector S
Point your right-hand thumb in the direction of the arrow
and the grip of the other four fingers represents gyration of the object
In addition
the length of the arrow is used to represent the spin rate
Therefore, a vector can give a complete description of rotation
Now we can utilize the right-hand rule
to determine the direction of Magnus force
Point your four fingers toward the direction of ball flight
and align the thumb with spin vector S
then you got your palm faces the direction of Magnus force
We've learned the Magnus effect by a spinning smooth ball
What about a spinning ball with seams?
The wake becomes fluttering
The spin of the ball as well as the surface seams result in joint effect
and the Magnus force varies periodically
Although the complex behavior of boundary layer has not been well studied
the average force can still be obtained from experiments
If the spin vector points toward the third base horizontally
the Magnus force will totally contribute to lift
This figure is the measurement results
The horizontal axis represents the spin parameter SP
which is defined as the surface speed Rω of a rotating sphere
divided by the flow speed V
The vertical axis represents the lift coefficient CL
which is defined by Magnus coefficient times SP
The upper curve represents the lift coefficient of 4-seamer
and the lower one represents the lift coefficient of 2-seamer
Taking a 140 km/hr and 20 rev/sec fastball as an example
The lift coefficient for a 4-seamer is about two times larger than that of a 2-seamer
The difference between them decreases
with increasing the spin rate or decreasing the ball speed
In this 140kmh and 20rps case:
The lift for a 4-seamer is about 60% of the weight
while it is about 30% for a 2-seamer
Now the trajectory can be estimated since the lift force is known
Green and red curves denote the trajectories of 4-seamer and 2-seamer, respectively
And the dashed line is a straight line
When the ball arrives at the plate
the 4-seamer drops about 40 cm
while the 2-seamer drops about 70 cm
and the difference is about 30 cm
If there is no lift force
and consider only the gravitational force on the ball
the ball would drop about 1 meter
Another extreme case is a ball with vertical spin axis
such that Magnus force acts totally in the direction of lateral force
Consider the same spin rate and ball speed discussed above
A 2-seamer moves about 30 cm to the left or right
when it arrive at the plate
and the horizontal movement for a 4-seamer is about 60 cm
The ability of horizontal or vertical motion is called "tail-strength"
in Taiwan's baseball terminology
Owing to the surface roughness
the drag coefficient of a baseball is in between that of a smooth ball and a golf ball
General speaking, people think that
2-seamers have larger drag than 4-seamers
But the experiment data show that difference is not big
On the contrary
2-seamers have obviously smaller drag for some special conditions
To pitch a 4-seam fastball
place your index and middle fingertips on the baseball seam
and place your thumb right beneath the ball
At release point, press the fingers downward
and get the ball backspin like this
The spin axis of a 4-seam fast ball is in general oblique
which results in the inside movement for a right-handed batter
4-seam fastball
In the pitcher's view angle
Spin vector S pointing toward lower right
and the right-hand rule tells that
Magnus force M pointing toward the upper right
As for gravity, Fg pointing downward
The resultant force is obtained by making a parallelogram
To pitch a cutter
place the index and middle fingers a little bit outside
and press the fingers downward at release point
and get the ball spin like this
Cutter (Pitcher's view angle)
To pitch a 2-seam fastball
place the index and middle fingertips on the narrow part of the seams
and place the thumb right beneath the ball
then it will rotate as a 2-seamer after delivery
2-seam fastball (Pitcher's view angle)
If push off the index finger at release point and let the ball side spin
the ball will get more lateral movement and sinking
and it becomes a "sinker"
2-seam sinker (Pitcher's view angle)
To pitch a slider
place the index and middle fingers outside of the ball
and rotate your palm a little bit
At release point, press the fingers downward
and let the ball spin in this way
Slider (Pitcher's view angle)
To pitch a curveball
rotate your palm to the left
At release point, rotate the fingers forward
and let the ball spin in this way
Another view angle for curveball
To pitch a forkball
Split your index and middle fingers apart to grip the ball
and place the thumb beneath the ball
It leads to low spin rate and large sink
Forkball (Pitcher's view angle)
To pitch a changeup
make an "OK" gesture
then put the ball in your hand
This results in low spin rate
and is similar to a forkball
Changeup (Pitcher's view angle)
Vertical slider is a special variant of slider
The Magnus force vanishes
since the spin axis aligns with its motion
Gravity and drag are the rest of forces acting on the ball
therefore, it drop fast vertically
The spin axis of a gyroball is in between that of V-slider and cutter
and so does its characteristics
It can move as fast as a cut fastball
and may also sinks like a V-slider
Nevertheless
since every pitch has wide range of physical characteristics
and since there are no standard criteria for classification
some people think that gyroball can just be classified into cutter or slider
To pitch a screwball
Turn your palm inside out to pitch the ball
The spin direction of a screwball for a right-handed pitcher
is similar to that of a left hander's slider or curve
so the pitch moves down and in on a right-handed batter
The screwball pitchers are rare
because it tends to damage pitcher's arms
Conclusions
We have gone into the detail of the physics of boundary layer
and have leaned various baseball pitches
Hope you have fun watching and playing baseball games