The Effect of Spin on the Flight of a Baseball

The effect of spin on the flight of a baseball ͒ Alan M. Nathana Department of Physics, University of Illinois, Urbana, Illinois 61801 ͑Received 27 June 2007; accepted 13 October 2007͒ Measurements are presented of the Magnus force on a spinning baseball. The experiment utilizes a pitching machine to project the baseball horizontally, a high-speed motion analysis system to determine the initial velocity and angular velocity and to track the trajectory over 5 m of flight, and a ruler to measure the total distance traversed. Speeds in the range v=50–110 mph and spin rates ␻ ͑topspin or backspin͒ in the range 1500–4500 rpm were utilized, corresponding to Reynolds numbers of Re=͑1.1–2.4͒ϫ105 and spin factors SϵR␻/v in the range 0.090–0.595. Least-squares fits were used to extract the initial parameters of the trajectory and to determine the lift coefficients. Comparison is made with previous measurements and parametrizations, and implications for the effect of spin on the flight of a baseball are discussed. The lift coefficient CL is found not to depend strongly on v at fixed values of S.©2008 American Association of Physics Teachers. ͓DOI: 10.1119/1.2805242͔ I. INTRODUCTION swing angle and undercut distance were varied to maximize the range. The study found the surprising result that an opti- It is well known to players, fans, and even physicists that mally hit curveball travels farther than an optimally hit fast- a spinning baseball curves. In the language of physics, a ball, despite the higher batted-ball speed of the fastball. The spinning baseball in flight experiences the Magnus force, physics underlying this result is that, in general, a baseball which contributes along with gravity to the deviation of a will travel farther if projected with backspin. It will also baseball from a straight-line trajectory. The earliest experi- travel farther if it is projected with higher speed. A fastball mental investigations of the Magnus force were conducted will be hit with a higher speed, and a curveball will be hit 13 by Isaac Newton,1 who studied the motion of spinning tennis with greater backspin. It then becomes a question as to balls. A recent review of the Magnus force on a variety of which effect wins. The calculations of Ref. 5 showed that the different sports balls ͑baseball, tennis, golf, cricket, volley- latter effect wins and the curveball travels farther. This con- ball, and soccer͒ has been given by Mehta and Pallis.2 Were clusion depends critically on the magnitude of the Magnus Newton alive today, he would surely know that the Magnus force on a spinning baseball. A particular model was used for force is responsible for much of the subtlety in the battle the Magnus force based largely on experimental data ͑see between pitcher and batter, such as the sideways break of a Sec. II͒. This model and conclusions have been criticized by 14 slider or cutter, the drop of a curveball with topspin, and the Adair, who claims that the effect of spin on the flight of a “hop” of a fastball with backspin. He would also understand baseball was overestimated in Ref. 5. that home run hitters typically undercut the baseball to put The question of whether a curveball can be hit farther than backspin on the struck ball because they instinctively know a fastball is not an issue of primary importance to a physicist. that the Magnus force on a ball with backspin is primarily The issue is the quantitative effect of spin on the flight of a vertically upward, so that balls hit on a home run trajectory baseball, regardless of whether the trajectory is that of a long stay in the air longer and travel farther. fly ball, a popup to the infield, or a pitched baseball. The It might surprise Newton that more than 300 years after goals of the present paper are to report new experimental his initial investigations, we still do not have a completely data for the Magnus force and to use these data to update and quantitative description of the Magnus force and its effect on extend the earlier investigations of the effect of spin on the the flight of a baseball, despite the many experimental and flight of pitched or batted baseballs. The present experiment, computational studies that have appeared in the literature. including the data reduction and analysis, is described in Sec. Computational studies of the effect of spin on the flight of hit III. The results are compared to previous determinations of baseballs have been reported by Rex,3 Watts and Baroni,4 the Magnus force in Sec. IV and the implications for the and Sawicki et al.5 These studies utilized models of the Mag- flight of a baseball are discussed. A summary of our conclu- nus force based on the available experimental information, sions is given in Sec. V. which will be reviewed in Sec. II. Additional computations are reported in the books of Watts and Bahill6 and Adair.7 Experimental data on the effect of spin on the flight of pitched baseballs have been reported in Refs. 8–11. To our II. PREVIOUS DETERMINATIONS knowledge, there are no experimental data on the effect of OF THE MAGNUS FORCE spin on the flight of hit baseballs. The most recent and extensive computational study was When a spinning baseball travels through the atmosphere, reported in Refs. 5 and 12 in which the optimum bat-swing it experiences the force of gravity in addition to the drag and parameters for producing the maximum range of a batted Magnus forces, FD and FM, as shown in Fig. 1. Convention- baseball were determined. They used models for the ball-bat ally the magnitudes of these forces are parametrized as collision and the aerodynamic forces on a baseball and 1 tracked the ball from collision to landing. For given initial F = C ␳Av2, ͑1͒ parameters of the pitch ͑speed, angle, and spin͒, the bat D 2 D 119 Am. J. Phys. 76 ͑2͒, February 2008 http://aapt.org/ajp © 2008 American Association of Physics Teachers 119 17 Briggs data, increasing the CL values by approximately 50% and bringing them more in line with the expectation 12 that CL is proportional to S. The corrected values are shown in Fig. 2. A more extensive set of data was taken by Watts and Ferrer18 by using strain gauges to measure the force on a spinning baseball in a wind tunnel. These data cover the range S=0.4–1.0, but only for speeds up to about 37 mph. These data are roughly consistent with CL =S, independent of Re, which is the parametrization used by Watts and Baroni.4,6 The data of Alaways8,9 were obtained using a high-speed motion analysis technique to measure the spin vector and track the trajectory of a pitched baseball. From these data CL was extracted for S in the range S=0.1–0.5 and for speeds up to approximately 75 mph. The data of Jinji11 were taken Fig. 1. Forces on a spinning baseball in flight. The drag force FD acts in the ␻ϫ −v direction, the Magnus force FM acts in the ˆ vˆ direction, and the force using standard video cameras to track the trajectory of a of gravity FG acts downward. pitched baseball and a high-speed camera to measure the spin and spin axis just after release. Data were taken using both live pitchers and a pitching machine. Only the pitching 1 machine data are shown in Fig. 2 because the live data dis- F = C ␳Av2, ͑2͒ M 2 L play too much scatter to be useful. The plotted data cover the narrow range S=0.2–0.3 and v=65–78 mph. where A is the cross sectional area of the ball, v is its speed, Not shown in Fig. 2 are the wind tunnel data of Refs. 19 ␳ ͑ / 3͒ is the air density 1.23 kg m , and CD and CL are the drag and 20 in which CL was measured over a broad range of S 15 and lift coefficients, respectively. We will focus only on CL. =0.2–0.9 and v=32–73 mph. For SϽ0.4, these data follow 8,9 Data on other spherical sports balls suggest that CL is mainly the general trend of the Alaways data, but fall well below a function of the spin factor S=R␻/v, although it may also the Watts and Ferrer data18 at higher S values. Interestingly, be a function of the Reynolds number Re=2␳Rv/␮.2,16 Here the data of Refs. 19 and 20 show little or no dependence on R is the radius of the ball, ␻ is the angular velocity, and ␮ is Re for fixed S in the range investigated. More interesting is ͑ Ͻ ͒ the dynamic viscosity of the air. For a standard ball and air at that a reverse Magnus effect that is, CL 0 was observed normal temperature and pressure, ␮=1.85ϫ10−5 N-s/m2,so for a smooth ball with the dimensions of a baseball. For ϫ −3␻/ ␻ Ͻ that Re=2180v and S=8.53 10 v with v in mph and example at v=55 mph, CL 0 occurs for S=0.15–0.55. A in rpm. reverse Magnus effect was also observed on a smooth ball by 17 Previous determinations of CL for a baseball are shown in Briggs. No such effect has been reported for a real base- Fig.

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