^{1,a)}

### Abstract

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 motionanalysis system to determine the initial velocity and angular velocity and to track the trajectory over of flight, and a ruler to measure the total distance traversed. Speeds in the range and spin rates (topspin or backspin) in the range were utilized, corresponding to Reynolds numbers of and spin factors 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 is found not to depend strongly on at fixed values of .

The author thanks Joe Hopkins for his help in the early stages of this work, especially with the motion capture measurements. The author also thanks Dr. Hank Kaczmarski for the loan of the motion capture equipment and Dr. Lance Chong for his expertise in the use of this equipment. Finally, the author wishes to thank Professor Mont Hubbard for many clarifying discussions and for a critical reading of the manuscript.

I. INTRODUCTION

II. PREVIOUS DETERMINATIONS OF THE MAGNUS FORCE

III. EXPERIMENT AND DATA REDUCTION

IV. RESULTS AND DISCUSSION

A. Results for

B. Implications for the flight of a baseball

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Kinematics
- 16.0
- Cameras
- 15.0
- Aerodynamics
- 5.0
- Physics of sports
- 5.0
- Velocity measurement
- 5.0

## Figures

Forces on a spinning baseball in flight. The drag force acts in the direction, the Magnus force acts in the direction, and the force of gravity acts downward.

Forces on a spinning baseball in flight. The drag force acts in the direction, the Magnus force acts in the direction, and the force of gravity acts downward.

Experimental results for . The closed circles are from the present experiment. Open circles are from Watts and Ferrer (Ref. 18), open triangles are from Briggs (Refs. 12 and 17), open diamonds and squares are from Alaways two- and four-seam (Ref. 23), respectively (Refs. 8 and 9), and closed triangles are from the pitching machine data of Jinji (Ref. 11). Also shown are the parametrizations of Ref. 5 (solid) and Eq. (3) (dashed), the latter calculated for a speed of .

Experimental results for . The closed circles are from the present experiment. Open circles are from Watts and Ferrer (Ref. 18), open triangles are from Briggs (Refs. 12 and 17), open diamonds and squares are from Alaways two- and four-seam (Ref. 23), respectively (Refs. 8 and 9), and closed triangles are from the pitching machine data of Jinji (Ref. 11). Also shown are the parametrizations of Ref. 5 (solid) and Eq. (3) (dashed), the latter calculated for a speed of .

Calculated ratio of the Magnus force to weight for . The solid and dashed curves utilize the parametrizations of Refs. 5 and 7, respectively, the latter essentially reproducing Fig. 2.2 of Adair (Ref. 7).

Calculated ratio of the Magnus force to weight for . The solid and dashed curves utilize the parametrizations of Refs. 5 and 7, respectively, the latter essentially reproducing Fig. 2.2 of Adair (Ref. 7).

Trajectory data (top) for one of the pitches, where and are the coordinates of the dot on the ball in the coordinate system shown in the inset. The ball is projected at a slight upward angle to the direction and is spinning clockwise (topspin) about an axis perpendicular to the plane. Solid curves are least-square fits to the data using Eq. (4b), resulting in and . The long dashed curve is the center-of-mass trajectory for the coordinate, which is consistent with a downward acceleration of due to the combined effects of gravity and the Magnus force. The short dashed curves are the center-of-mass coordinates for both and with both and set to zero, indicating that the data are very sensitive to but not to . The fit residuals for the (points) and (curve) coordinates are shown in the bottom plot.

Trajectory data (top) for one of the pitches, where and are the coordinates of the dot on the ball in the coordinate system shown in the inset. The ball is projected at a slight upward angle to the direction and is spinning clockwise (topspin) about an axis perpendicular to the plane. Solid curves are least-square fits to the data using Eq. (4b), resulting in and . The long dashed curve is the center-of-mass trajectory for the coordinate, which is consistent with a downward acceleration of due to the combined effects of gravity and the Magnus force. The short dashed curves are the center-of-mass coordinates for both and with both and set to zero, indicating that the data are very sensitive to but not to . The fit residuals for the (points) and (curve) coordinates are shown in the bottom plot.

Results for from the present motion capture experiment.

Results for from the present motion capture experiment.

Results for from the present experiment for in the range 0.15–0.25, demonstrating that does not depend strongly on (or Re) for fixed values of .

Results for from the present experiment for in the range 0.15–0.25, demonstrating that does not depend strongly on (or Re) for fixed values of .

Calculated trajectories of a hit baseball with an initial speed of , angle of 30°, height of , and backspin of (solid), (long-dashed), and (short-dashed). The points indicate the location of the ball in intervals. These calculations utilize the values of Adair (Ref. 7) and the values from the parametrization of Sawicki *et al.* (Ref. 5).

Calculated trajectories of a hit baseball with an initial speed of , angle of 30°, height of , and backspin of (solid), (long-dashed), and (short-dashed). The points indicate the location of the ball in intervals. These calculations utilize the values of Adair (Ref. 7) and the values from the parametrization of Sawicki *et al.* (Ref. 5).

Calculated range of a hit baseball with an initial speed of , height , and backspin of (solid), (short-dashed), and (long-dashed), as a function of the initial angle . These calculations utilize the values of Adair (Ref. 7) and the values from the parametrization of Sawicki *et al.* (Ref. 5).

Calculated range of a hit baseball with an initial speed of , height , and backspin of (solid), (short-dashed), and (long-dashed), as a function of the initial angle . These calculations utilize the values of Adair (Ref. 7) and the values from the parametrization of Sawicki *et al.* (Ref. 5).

## Tables

Calculated deflection of a pitched baseball thrown with an initial horizontal velocity , spin , and spin factor after traversing . These calculations utilize the values of Adair (Ref. 7) and the values from the parametrization of Sawicki *et al.* (Ref. 5). These deflections are in accord with experimental observations (Refs. 11 and 28).

Calculated deflection of a pitched baseball thrown with an initial horizontal velocity , spin , and spin factor after traversing . These calculations utilize the values of Adair (Ref. 7) and the values from the parametrization of Sawicki *et al.* (Ref. 5). These deflections are in accord with experimental observations (Refs. 11 and 28).

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