# The Reynolds Number

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```					History
   Newton published an article exploring the
curved flight of tennis balls in 1672.
   New Principles of Gunnery, by B.Robins.
first published in 1742
   About 100 years later G. Magnus gave a
similar explanation.
   The lateral deflecting force of a spinning
sphere ball was named the Magnus Force.
History
 Many people in the last hundred
years have set out to explain
various aspects of baseball.
 In 1987, Robert K Adair was hired
as the Official Physicist of the
National League.
   Published The Physics of Baseball in
1990
The Reynolds Number
   An object’s flow through a fluid (air) is
determined by a Reynolds number –
proportional to the fluid density, the fluid
velocity, and the size of the object, and
inversely proportional to the viscosity of
the fluid.
Vr
Re  

   For baseball Re2200V, with V in mph
Drag
 The retarding force on a baseball
 Drag is proportional to the cross
sectional area of the ball, the square
of the velocity of the ball, the
density of air, and a drag coefficient
(involving the Reynolds number).
2
V
Fd  Cd A
2
Drag
   Turbulence caused
by roughness,
actually lowers the
drag coefficient.
So, the baseball,
both rough and
smooth, is found
in the middle.
Bernoulli’s Principle
   Bernoulli’s principle states that the
sum of the pressure plus the kinetic
energy per unit volume of a flowing
fluid must remain constant:

1 2 TotalEnergy
P  dv 
2     Volume
The Magnus Effect
 When a spinning object moves
through a fluid it experiences a
sideways force.
 The Magnus effect is created by an
imbalance in air pressure.
The Magnus effect on a
spinning baseball
   When an object is
moving through the
air, its surface
interacts with a thin
layer known as the
boundary layer.
   When the boundary
layer peels away from
the surface it creates
a “wake.”
   The amount of Force that a baseball
will curve can be determined by the
equation:

FL  KVCv
   Where FL is the Magnus Force, K is the Magnus Coefficient,
ω is the spin frequency measured in rpm, V is the velocity
of the ball in mph, Cv is the drag coefficient.
The Ole Number One
The Fastball
The Four-seam Fastball
   In the four-seam
fastball or “rising
fastball”, four
seams catch the
air as the ball
rotates, and the
ball tends to float
due to the lift
generated by the
four seams.
The Two-seam grip
   For the two-seam
fastball, there are
only two seams
catching air, thus
the ball tends to
sink.
Four-Seam vs. Two-Seam
   The four-seam grip
is called a “cross-
seams fastball”
   The two-seam grip
is also called “with
the seams fastball”
or “Sinking
Fastball.”
Does a Fastball move like a
Curveball?
   When gripped properly and thrown
with enough backspin a Fastball can
move side to side or even up and
down.
The Sinking Fastball
   The sinking fastball is
gripped on top of the
ball with the narrow
seams exposed.
When releasing this
fastball, you usually
apply pressure
against the seam with
either the index or
middle finger.
The Aerodynamics of a
Curveball

Have you ever wondered
whether a curve ball really
curves, or is it just an optical
illusion?
The Grip
“Choke” the ball
(wedge it down
and forefinger), and
the left; the ball
snaps down and to
the right on release.
The resulting pitch
should drop and
curve to the left.
The flight of a Curveball
The Knuckleball
   There is no
standard for how
to throw a
knuckleball.
   In general the ball
is held as shown,
pushing all of the
fingers evenly out
on the ball results
in very little spin
The Knuckleball
 As the ball is thrown with little or no
spin, the asymmetrical stitches
generate large imbalances of forces
and somewhat unpredictable
trajectory.
 Low resistance turbulent air flow will
be induced by stitches on one side
of ball, while air flows smoothly,
with more resistance on the other.
Knuckleball
Impulse-Momentum
theorem
   Impulse is the total change of
momentum of a body over time.
tf

J   
ti
Fdt
t   
      F ( )dd
1
x(t )  m
0   0
Knuckleball without spin
   If the knuckleball is thrown with no spin it
can only curve laterally in one direction
   However, if ball is thrown in position
where Θ=52° or 310° it will have a
very erratic path, this has been
observed in actual pitches and is a
nightmare for batters and catchers.
Model for no spin knuckle

2
1 F0  d 
x(t )        
2 mv
Knuckleball with spin
   A much more realistic model is
that the ball does spin on the way
home, so the lateral forces it is
exposed to are constantly
changing on the way home.

F0
x(t )        sin(t   )
m  2
Knuckle with spin
 Too much spin will result in a small
deflection, so the spin is critical
 Approximating the curve seen in the
handout, with an Amplitude=.08lbs
we find that a K-ball at 40mph with
2 revolutions on the way home will
only be deflected .048ft or .6 inches
Knuckleball

Ball is pitched at 47mph
Downfalls of the Knuckle
   Catchers must use their glove and their
body to at least block a knuckler, if they
are unsuccessful, the ball will pass them,
resulting in stolen bases or the batters
   If a knuckleball does not break much, due
to too much spin or bad orientation of the
pitch, it will end up in the bleachers
Illegally Modified Baseballs

Possible ways a pitcher can
cheat to gain a bigger
The “Spit-Ball”

When a lubricant is applied to a
baseball it can have the
trajectory of a knuckleball with
the speed of a fastball.
What happens you have a
scuffed baseball
 Scuffing can produce asymmetric
forces on the ball and irregular
trajectories.
 Scuffing the ball can increase the
drag on one side and cause the ball
to move in that direction.
Summary
   People have been studying these
phenomenon since Newton
   The course the baseball takes while it is
in flight is dependant upon the axis of
spin, the amount of spin, and the
orientation of the ball.
   These factors effect trajectories because
of the Magnus Effect and Drag Force

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