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Physics of the Curveball
Jason Pelc
October 31, 2007
(Submitted as coursework for Physics 210, Stanford University, Autumn
2007)
Introduction
The curveball is now a central component in the game of baseball. Thrown by imparting
spin onto the baseball during release, the force arising from the interaction of the
baseball's spin with the surrounding air often causes deflections of more than a foot from a
free-fall trajectory. This paper reviews the experimental and analytical work done on
understanding the physics behind what makes the curveball curve.
History and Basics of Curveball Science
Invention of the curveball is credited to William "Candy" Cummings [1]. In 1863,
Cummings discovered that he could cause a baseball to curve downward by rolling the
ball off his second fingertip and accompanying his usual throwing motion with a hard
torque of the wrist to impart topspin on the baseball. Cummings spent the next several
years improving his control over the new pitch, and, according to baseball lore, used it for
the first time in a game in which his club, the Brooklyn Excelsiors, defeated Harvard
College in 1867. Cummings went on to have a successful baseball playing career,
including playing during the inaugural season of the National League in 1876, and is
enshrined in Baseball's Hall of Fame.
However, long before people knew how to throw curveballs, it was recognized that the
spin of a spherical projectile can substantially alter its trajectory. In fact, Newton
recognized that the fact that tennis balls curve is due to spin imparted upon them in 1671
[2]. In 1877, Lord Rayleigh, also discussing curving tennis balls, credited the German
engineer G. Magnus with the first explanation of the lateral deflection of a spinning ball,
and as a result the effect is generally called the Magnus effect [3]. Historians of science
now understand that Magnus' explanation had its origins in 1742 in a book by a British
scientist B. Robins in his book New Principles of Gunnery in which he analyzed the
effects of spin on flying cannonballs and musket balls.
The elements of the Magnus' and Robins' explanation are quite simple. A spinning ball
causes a whirlpool of air around it. Air flying past the ball will flow more quickly on the
side of the ball in which the flow velocity is parallel to the spin velocity of the ball, and
will flow more slowly on the side in which it opposes the spin. Bernoulli's principle states
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that the sum of the kinetic energy and the pressure of a fluid is constant, so that there will
be an increase (decrease) in pressure on the side of the ball in which the spin opposes
(reinforces) the airflow. This pressure imbalance causes a net overall force perpendicular
to the ball's velocity. In the case of a spinning baseball with topspin, the pressure is higher
above the ball than below it, causing a net downward force on the ball.
The essential physics involved with this
explanation can be visualized in Figure 1,
which consists of a photograph of a spinning
baseball in the presence of smoke. The smoke
streams are released with equal spacing at their
source to the left of the ball, and the
counterclockwise spin of the ball causes the
smoke streamlines to bunch at the bottom edge
of the ball. This closer spacing is indicative of
increased air velocity, which as discussed
above, indicates a pressure deficit and a
Fig. 1: Airflow near a spinning
downward force on the ball. A downward
baseball. Ball is spinning
force can also be explained by noting that the
counterclockwise at 30 rev/s and the
smoke wake is deflected upwards, which by
wind speed is 47 mph to the right [4].
Newton's Third Law must correspond to an
equal-but-opposite downward force on the ball.
Experiments with Baseballs
Formalism of ball dynamics
Before discussing the experimental work on baseball trajectories, it is useful to consider
the general properties of the motion of an object through a fluid. The fluid can impart two
general types of forces on a moving body, which are defined relative to the translational
velocity u of center of mass of the baseball. Firstly, the drag force D opposes u and is a
frictional force proportional to u. The force we shall generally be concerned with is the lift
force FL, which acts perpendicularly to both u and the ball's spin vector &Omega. The
general theory of the aerodynamics of a moving body dictates that the lift force is given by
where &rho is the mass density of the air, A is the cross-sectional area of the ball, and the
lift coefficient CL, a dimensionless constant, will be a function of several parameters:
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Professor Robert B. Laughlin, Department of Physics, Stanford University
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where S, Re, and &kappa are dimensionless variables.
S is the spin parameter of the ball, and relates the
speed of a point on the ball's equator to the
translational speed and is given by &Omegar/u. Re is
the Reynolds number, a parameter which relates the
inertial forces to the viscous forces acting on the
baseball, and is given by 2ur/&nu, where &nu is the
kinematic viscosity of air. Because &nu can change
depending on environmental conditions, weather and
altitude can affect the ball dynamics through a
dependence of CL on Re. The final factor &kappa
describes the ball's roughness, and is given by
&Deltar/r, where &Deltar is a measure of the
maximum excursion of the surface from a true sphere.
&kappa conceivably could depend on the orientation
of the seams of the baseball relative to the spin axis,
and also can be used to compare sports balls of various
types (golf balls, soccer balls, etc). Several
experiments on the aerodynamics of spinning balls Fig. 2: Diagram of Briggs'
report a near-linear dependence of CL on S, only a experimental setup for tests of
very weak dependence on Re, and critical behavior in lateral deflection of baseballs in
which C switches sign as &kappa approaches 0. a wind tunnel.
L
These experiments are discussed in the following
sections.
Preliminary Experiments
L. Briggs, in 1959, did the first systematic study of baseball deflection as a function of u
and &Omega published in mainstream scientific literature [5]. His experiments, a
schematic of which are shown in Figure 2, were done by dropping a spinning baseball into
a wind tunnel. Briggs spun the baseball using an electric motor in which a rotating shaft
was attached vertically to the ball by a valve-controlled suction cup such that the spin axis
was parallel to gravity. The ball, coated in a colored lubricant, was dropped into the wind
stream and the deflection from vertical was recorded twice, once with the ball spinning
each direction relative to the wind. The average deflection D was measured for several u
and &Omega.
Briggs' data has been shown to be
consistent with later experiments, but he
came to some erroneous conclusions.
Firstly, Briggs' extrapolation that a
nonspinning
ball (&Omega
= 0)
experiences no deflection has been shown
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Professor Robert B. Laughlin, Department of Physics, Stanford University
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to be inconsistent with an experiment by
Watts and Sawyer in which strong (FL/mg ~
0.3) orientation-dependent lift forces were
observed [6]. In addition, Briggs observed
that D varied with the square of u, a result
that agreed with C. Drury's reports of
experiments done by Sikorsky [7]. This
result, however, does not agree with the
Kutta-Zhukovsky
theorem
from
aerodynamics, which states that a net
circulation of an inviscid fluid about a twodimensional object results in a force
proportional to the product of the velocity
and the circulation [8]. Therefore, it is
reasonable to expect that the lift force
would be proportional to &Omegau, as had
Fig. 3: Conglomeration of experimental
results on curveballs [11]. (Reproduced by been seen in a study on golf balls [9], rather
permission of LeRoy Alaways.)
than &Omegau2, as Briggs suggests.
Further Work
In 1986, Watts and Ferrer extended Briggs' methods to explore this controversy, and
presented the first discussion of baseball dynamics phrased in terms of the three
dimensionless parameters above [watts]. Watts also used Briggs' wind tunnel technique,
and performed experiments for the largest S values in the scientific literature, out to S
&asymp 1.0. Another very systematic study was completed by Alaways in 1998, using a
more modern technique of high speed video capture of pitches thrown by both humans an
pitching machines. He used a parameter estimation by fitting to an analytical model of
baseball trajectories including lift and drag forces, and plots these trajectories along with
those where the pitches in comparison to those without aerodynamic forces (free-fall)
[11].
Figure 3 shows a conglomeration
of experimental results including
the work discussed by Briggs,
Sikorsky, Watts and Ferrer, and
Alaways. Note that the 2-seam or
4-seam orientations correspond
to the number of seams
intersecting the equatorial plane
of the spinning baseball. Figure 4
shows an example trajectory of a
pitched curveball and compares it
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Professor Robert B. Laughlin, Department of Physics, Stanford University
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to the arc taken by a ball with the
same initial velocity but without
spin. It is worth noting that
CL is independent of S for
values of S above 0.25.
For lower values of S, the 4seam orientation has higher
lift than the 2-seam
orientation.
CL &asymp S for S > 0.25.
When the spin axis is
misaligned from the
horizontal as in Fig. 4, the
ball undergoes a sideways
force as well.
Other data of Alaways shows
that CL is approximately
Fig. 4:Trajectory of a machine-thrown curveball
with initial velocity of 77 mph with a spin of 24 rev/s
[11]. (Reproduced by permission of LeRoy
Alaways.)
independent of Re for all ball velocities studied. Briggs' data and the Watts/Ferrer data
also confirm this.
Smooth Spheres
Work on the aerodynamics of smooth spheres shows an anomalous Magnus effect of
opposite sign. This counterintuitive effect was also investigated experimentally by Briggs,
but was not studied analytically until 2003 [12]. Borg et al. studied the forces on a
spinning sphere in the limit of rarefied gases. In this limit of low density, and
correspondingly low viscous forces and large Re, the gas is treated as a set of ballistic
particles, and the theory is only valid in the regime in which the mean free path of gas
particles is larger than the size of the sphere. They find a negative lift coefficient
proportional only to the product &Omegar. In order for CL to be independent of u, it must
be linearly dependent on both Re and S, which is exactly what Briggs' data showed.
Just as there is an intuitive explanation of the Magnus effect in terms of Bernoulli's
principle, Borg gives an explanation of this inverse Magnus effect in terms of momentum
conservation. Gas particles preferentially hit the windward face of the sphere, and are
primarily deflected along the motion of the spinning surface. In the case of a ball traveling
forward with topspin, this indicates a deflection of gas particles downwards, which must
correspond to a net upward force on the sphere, opposite to the Magnus force which tends
to push a curveball downward.
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Professor Robert B. Laughlin, Department of Physics, Stanford University
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Discussion
Because of the complexity of the surface of the baseball and the mechanism of boundary
layer separation for turbulent flow, it is impossible to analytically solve the functional
form of the dependence of CL on S, Re, and κ. Experimental work completed since the
1950s shows that
1. For values of S larger than 0.25, CL rose linearly with S, and was independent of the
orientation of the ball. This implies that for large S, the ball acts as a uniformly rough
sphere independent of seam position.
2. For values of S smaller than 0.25, the influence of the orientation of the ball is
stronger. Four-seam pitches experience a larger deflecting force than two-seam
pitches. This is most likely due to the fact that the seams are the cause of the surface
roughness, and when more seams tumble across the face of the ball, the ball has a
larger effective roughness. In fact, fairly large, orientation-dependent forces are
observed for nonspinning baseballs, accounting for the unpredictable behavior of the
knuckleball, a pitch deliberately thrown without spin. Interestingly, pitchers are
generally taught to throw a 2-seam curveball under the assumption that it will curve
more. This is apparently never true if the spin rate is held constant, but it is possible
that aligning the index finger with the seams as occurs in the 2-seam orientation
allows the application of a larger torque on the ball during release, and therefore a
larger S value.
3. CL is nearly independent of Re for usual baseball speeds between 70 and 100 mph,
for rough balls. This means that contrary to popular belief among baseball players,
humidity and altitude do not have a significant effect on the deflection of curveballs.
4. As κ is decreased, CL changes sign, and develops a linear variation with Re for low
values of S. There is an analogy between this critical roughness and critical Reynolds
number in which the Magnus force changes sign. This phase transition is still poorly
understood, but is of little interest to the field of sports ball aerodynamics, in which
one generally seeks to maximize the lift force [12].
Most baseball players report that a curveball is often seen to "break," or suddenly alter its
trajectory. This effect can only be an optical illusion, as studies of baseball trajectories
indicate that the Magnus force acts downward during the entire flight of the ball, giving it
a parabolic trajectory [11]. From a batter's perspective, however, the majority of the
deviation from a straight line occurs in the last moments as the ball approaches home
plate.
There have been extensive studies of other sports balls in which the effects of spin on
lateral deflection are studied. The same effects that force a ball with topspin downwards
also act to keep a golf ball with backspin in the air for longer than a non-spinning ball. An
excellent review of sports ball aerodynamics has been given by Mehta [4].
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© 2007 Jason Pelc. The author grants permission to copy, distribute and display this work
in unaltered form, with attribution to the author, for noncommercial purposes only. All
other rights, including commercial rights, are reserved to the author.
References
[1] D. Fleitz, "Candy Cummings", in The Baseball Biography Project. Accessed 22
October 2007.
[2] I. Newton, Phil. Trans. R. Soc. 7, 3078 (1771-2).
[3] L. Rayleigh, Messenger of Mathematics 7, 14 (1877).
[4] R. Mehta, Ann. Rev. Fluid Mech. 17, 151 (1989).
[5] L. Briggs, Am. J. Phys. 27, 589 (1959).
[6] R. Watts and E. Sawyer, Am. J. Phys. 43, 960 (1975).
[7] J. Drury, "The Hell It Don't Curve," from The Fireside Book of Baseball, ed. by
Charles E. Einstein (Simon and Schuster, 1956).
[8] R. Granger, Fluid Mechanics (Holt, Rinehart, and Winston, New York, 1985).
[9] P. Bearman and J. Harvey, Aeronaut. Q. 27, 112 (1976).
[10] R. Watts and R. Ferrer, Am. J. Phys. 55, 40 (1987).
[11] L. Alaways, Aerodynamics of the Curve-Ball: Investigation of the Effects of Angular
Velocity on Baseball Trajectories, Ph.D. Thesis, University of California, Davis (1998).
[12] K. Borg, L. Soderholm, and H. Essen, Phys. Fluids 15, 736 (2003).
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