October 31, 2007

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.

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 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.

Fig. 1: Airflow near a spinning baseball. Ball is
spinning counterclockwise at 30 rev/s and the wind speed is
47 mph to the right [4]. |

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 downward force on the ball. A downward force can also be explained by noting that the smoke wake is deflected upwards, which by Newton's Third Law must correspond to an equal-but-opposite downward force on the ball.

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 **F**_{L}, 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
*C _{L}*, a dimensionless constant, will be a function of
several parameters:

Fig. 2: Diagram of Briggs' experimental setup for
tests of lateral deflection of baseballs in a wind
tunnel. |

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 &Omega*r*/*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 2*ur*/&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 *C _{L}* on Re. The final
factor &kappa describes the ball's roughness, and is given by
&Delta

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.

Fig. 3: Conglomeration of experimental results on
curveballs [11]. (Reproduced by permission of
LeRoy Alaways.) |

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 to be inconsistent with an experiment by
Watts and Sawyer in which strong (*F _{L}*/

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].

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.) |

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 to the arc taken by a ball with the same initial velocity but without spin. It is worth noting that

*C*is independent of_{L}*S*for values of*S*above 0.25.- For lower values of
*S*, the 4-seam orientation has higher lift than the 2-seam orientation. *C*&asymp_{L}*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 *C _{L}*
is approximately independent of Re for all ball velocities studied.
Briggs' data and the Watts/Ferrer data also confirm this.

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 &Omega*r*. In order for
*C _{L}* to be independent of

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.

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 C_{L} on *S*, Re, and κ.
Experimental work completed since the 1950s shows that

- For values of
*S*larger than 0.25,*C*rose linearly with_{L}*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. - 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. *C*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._{L}- As κ is decreased,
*C*changes sign, and develops a linear variation with Re for low values of_{L}*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].

© 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.

[1] D. Fleitz, "Candy Cummings", in *The
Baseball Biography Project.* Accessed 22 October 2007.

[2] I. Newton, Phil. Trans. R. Soc. **7**, 3078
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[3] L. Rayleigh, *Messenger of Mathematics*
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[4] R. Mehta, Ann. Rev. Fluid Mech.
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[6] R. Watts and E. Sawyer, Am. J. Phys. **43**,
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[7] J. Drury, "The Hell It Don't Curve," from *The
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[8] R. Granger, *Fluid Mechanics*
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[9] P. Bearman and J. Harvey, Aeronaut. Q. **27**,
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[10] R. Watts and R. Ferrer, Am. J. Phys. **55**,
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[11] L. Alaways, *Aerodynamics of the Curve-Ball:
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Trajectories*, Ph.D. Thesis, University of California, Davis
(1998).

[12] K. Borg, L. Soderholm, and H. Essen, *Phys.
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