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Dependence of the Magnus Force on Velocity and Spin of a Smooth Ball Jacob Bringewatt ABSTRACT The dependence of the Magnus effect, the force that causes a spinning ball to curve, on the velocity and spin of a smooth ball was investigated. Two previously proposed models – one that shows the Magnus force to depend on the product of the spin rate and velocity, and the other that indicates that the force depends on the spin rates times the velocity squared – were compared. A subsonic wind tunnel was built for the experiment, and a pool ball on an axle was hung rigidly from two force sensors in the test section of the wind tunnel. Force data was taken for a variety of spins and velocities, ranging from 5.9 to 44.2 rad/s and 5.9 to 11.6 m/s, respectively. Within this range the Magnus force seemed to be jointly proportional to the spin rate and the velocity squared. Due to noise in the data, further work will be necessary to strengthen the conclusions.
Introduction The Magnus force, first noted in published literature by Newton, is the force that causes spinning balls to curve. (Newton 1671). Some papers report an ωV dependence, while others demonstrate that the force varies with ωV^2. Even after much experimentation, the quantitative description of the Magnus force is unknown; papers discern between an ωV (Watts and Ferrer 1987) and a ωV2(Briggs 1959) relationship, where ω is the spin rate and V it the linear velocity of the ball. The difference is encapsulated in the lift coefficient, as can be seen in the general equation for the Magnus force (Eq. 1).
Briggs dropped spinning baseballs through a 1.8m wind tunnel with a horizontal wind of known velocity. The balls were coated lightly with lampblack-containing lubricant so that when they hit a piece of cardboard attached to the tunnel floor their point of impact was recorded. To determine the lateral deflection two measurements were made, one with the ball spinning clockwise and the other with the ball spinning counterclockwise. The lateral deflection was determined to be one half the distance between the two marks. As shown in Fig. 1, Briggs’s data show that for translational baseball velocities in the range of 20m/s to 40m/s and angular velocities of 125 rad/s to 188 ras/s the Magnus-induced deflection is directly proportional to the speed of the ball squared. He also determined that the Magnus force is directly proportional to the spin rate.
where A is the cross-sectional area of the ball, ρ is the density of the air, and CD is the lift coefficient. The issue is made even more complicated by the experimentally observed reverse Magnus effect where a smooth, spinning ball will sometimes curve in the opposite direction as compared to the normal Magnus effect (Briggs 1959). New data focuses on spins and velocities ranging from 5.9 to 44.2 rad/s and 5.9 to 11.6m/s. The experiment also seeks to verify the mechanism Cross proposes for the reverse Magnus effect.
Previous Experiments Some of the first significant experiments on the aerodynamics of spinning balls were done by Briggs (Briggs 1959). His experiments were notable not only for their originality; he also was the first to experimentally observe the reverse Magnus effect for a smooth spinning ball. Taking this unexpected observation into account is an essential component of any description of the Magnus force.
Fig. 1 Graph showing that the ratio of deflections depends on the velocity squared. The ratio of deflections plotted on the y axis is directly proportional to the Magnus force.(Briggs 1959). Volume 3 | 2013-2014 | 81
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This early data on the Magnus force on a baseball was not Briggs’s only contribution. His paper also includes evidence for a reverse Magnus effect for a smooth spinning ball. Using the same apparatus, Briggs recorded the deflections for both a smooth rubber ball, and a smooth Bakelite sphere. He found that at moderate velocity and spin rates that these smooth balls curved opposite the usual direction. From these results, Briggs concluded that this negative Magnus effect occurred due to the boundary layer flow around the ball remaining laminar on the side of the ball moving with the wind while becoming turbulent on the side moving against the wind stream (Briggs 1959). Another significant set of data was taken by Watts and Ferrer (Watts and Ferrer 1987). For their experiment Watts and Ferrer impaled baseballs with shafts in three different seam orientations. The strain on the support was measured using a calibrated microstrain indicator. The effects of drag on the strain were removed by conducting experiments with the ball spinning both clockwise and counterclockwise. The difference of these two results divided by two was taken to be the results from the lift (Magnus) force alone (Watts and Ferrer 1987). The data obtained by Watts and Ferrer are shown in Fig. 2. The open shapes are Briggs’s data and the dotted line lines are a set of data obtained by Sikorsky, who had concluded that seams had a significant effect on the flight of a baseball. Sikorsky apparently never published his data in scientific literature (Watts and Ferrer 1987).
Fig. 2 Lift data from Watts and Ferrer experiments. (Watts and Ferrer 1987). Watts and Ferrer’s data was contradictory to Briggs’. Their data and subsequent analysis of other sources pointed to the Magnus force being proportional to ωV rather than ωV2 as Briggs concluded. Watts and Ferrer applied the Kutta-Zhukoviskii theorem that states whenever a two dimensional object moves through an inviscid fluid, and there is a net circulation of fluid around the object, a net 82 | 2013-2014 | Volume 3
force arises perpendicular to both the velocity and vorticity vectors with a magnitude of ωV (Watts and Ferrer 1987). The pair cite references claiming that this appears to apply to rotating cylinders and spheres, as well. Thus they argue that the lift coefficient follows the relationship in Eq. 2.
where D is diameter of the ball, ω is the angular velocity, and V is the mean wind stream velocity. Dω /V is known as the spin factor S. For Briggs’s linear dependence of the lift force on ωV2 to hold, the lift coefficient must be the product of the relationship above and the Reynolds number. This is shown in Eq. 3.
where v is the kinematic viscosity of air. This equation disagreed with the data of Watts and Ferrer and several others who showed that the Magnus force has at most a weak dependence on the Reynolds number for R > 0.5×105 (Watts and Ferrer 1987).
The Boundary Layer The Magnus force, regardless of direction, speed, or spin of the ball, is believed to be caused by uneven boundary layer separation. Essentially the boundary layer is a thin layer of fluid very close to the solid wall of an object moving relative to the fluid. For a fluid flowing past a surface, the fluid layer immediately adjacent to the surface “sticks” to the surface, due to the fluid’s viscosity; thus, its velocity relative to the surface is zero. Each successive layer rubs against the layers next to it, creating shearing forces between layers. These shearing forces mean that the faster moving layers drag the slower layers along, so that the velocity of the fluid relative to the surface increases in the direction perpendicular to the surface. After some distance, the fluid is unaffected by the surface; the velocity is referred to as the mean wind stream velocity. The region between this layer and the surface makes up the boundary layer, the thickness of which is typically between 3 and 30mm (Watts and Bahill 1990). In a direction perpendicular to the surface, the velocity of the air relative to the surface increases, while as air moves along the surface it is slowed by friction. At the point where both u=0 and , a phenomenon called boundary layer separation occurs and the boundary layer disappears rearward of this point. u is the component of the velocity along the surface and y is the direction perpendicular to the surface (Cross 2012). For a smooth, non-spinning sphere, separation usually occurs halfway between the front and rear of the ball. For a ball moving horizontally that is viewed side-on, this separation occurs near the top and bottom of the ball.
Physics and CompSci Research This boundary layer is the origin of the typical description of the Magnus force. Essentially, the rotation imparted to the boundary layer by a spinning ball affects the points of boundary layer separation in a way that causes the Magnus force. The standard explanation using this idea is as follows. For a spinning ball, separation is delayed on the side where the ball’s surface is moving in the same direction as the free stream velocity and separation occurs prematurely on the side where the ball’s surface is moving against the free stream velocity. As a result, the wake of a horizontally moving ball projected with backspin is deflected downwards. This is because at the point of separation, the air separates approximately tangentially to the ball’s surface. By Newton’s Third Law, this deflection of air downwards imparts an upward force on the ball (Watts and Ferrer 1987). For a ball with topspin, the air is directed upwards and the Magnus force is directed downwards (Mehta 1985).
(V − 0.5ωD) D
< 1.0 × 10
(V + 0.5ωD) D
On the side of the ball with a greater velocity relative to the air, the turbulence pulls in higher speed air from outside the boundary layer. This means that the boundary layer will separate later on this side, causing a reverse Magnus effect. This is depicted in Fig. 3. However, at relatively low mean flow speeds, flow in the boundary is laminar on both sides and at high speeds flow is turbulent on both sides, so that the Magnus effect occurs normally (Cross 2012).
The Reverse Magnus Effect Thus far, we have discussed the qualitative origin of the regular Magnus force. The proposed explanation for the reverse Magnus force depends on turbulence (Briggs 1959, and is further elaborated upon in a recent paper by Cross (Cross 2012). For a non-spinning, smooth ball at relatively low speed (i.e. Reynolds numbers below ) air flow in the boundary layer is laminar. The wake is turbulent, but within the boundary layer the flow is smooth, causing the boundary layer separates near the top and bottom of the sphere, when viewed side-on. However, if flow in the boundary layer on one side of the ball becomes turbulent, say from roughness or a raised seam, then separation will occur later on this side. This results in a deflection of the ball opposite the direction of the wake. A turbulent boundary layer induces later separation by pulling in high speed air at the edge of the boundary layer, increasing the average air speed near the ball’s surface. Greater air speed means it takes friction on the ball’s surface longer to slow the air within the boundary layer, and thus the result is later separation. Similar to the Magnus effect, this uneven boundary layer separation results in a force that deflects the ball from its normal trajectory. At high ball speeds, airflow becomes turbulent on both sides regardless of asymmetrical roughness, leading to delayed separation on both sides, resulting in no lateral force on the ball. (Assuming it has no spin.) Similarly, turbulence can be used to explain the reverse Magnus force. Essentially, for a ball projected with backspin at certain speeds and rotation rates, transition to turbulence in the boundary layer will occur on the bottom, but not the top of the ball, as the relative velocity of the air past the ball is greater on the bottom than the top. Given that boundary layer turbulence occurs when Re > , this will occur when the velocity and spin rate are such that,
Figure 3: A negative Magnus force can arise as shown here if the air flow is laminar on the upper side of the ball and turbulent on the lower side. In this example, the tangential speed of the ball relative to the air due to spin is 4.4 m/s, and the center of mass speed is 10 m/s. Thus point A translates to the right at 5.6 m/s and point B translates at 14.4 m/s. The air flow near A is laminar, and the flow near B is turbulent. Diagram from Cross (Cross 2012).
Materials and Methods Equipment The research was conducted with the use of a wind tunnel designed specifically for this exploration. For the actual experimentation, we used an air compressor and hose, two Vernier Dual Range Force Sensors, a Kestrel 1000 anemometer (Kestrel), a standard international 5.715cm diameter pool ball, and a laptop to collect data using LoggerPro software (LoggerPro). Wind Tunnel Design and Construction The open wind tunnel consists of three basic parts: the bellmouth, where air enters the tunnel; the test section where experimentation occurs; and the diffuser that contains the fan that draws air through the tunnel. A 3-dimensional schematic is shown in Fig. 6. As will be discussed later, the tunnel was designed so that the airflow through Volume 3 | 2013-2014 | 83
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Figure 4: A schematic of an open design wind tunnel. Units are in centimeters. the tunnel was fast enough that the full range of boundary layer flow types on both sides of the spinning ball could be achieved (e.g. laminar on both sides, laminar on one side, while turbulent on the other, and turbulent on both sides). Since we want a high airflow rate, we want the cross sectional area to be smaller for a given volumetric flow rate. As the wind tunnel was built on a limited budget it had to have the smallest possible area such that the walls of the tunnel had a limited impact on the results. The maximum blockage for the test section is 7.5%, where blockage is the percent of the cross sectional area blocked by the model under experimentation (San Diego State). Thus for a 5.715cm pool ball the minimum cross sectional area for the test section of the wind tunnel is 342cm2. This corresponds to minimum dimensions of about 18.5cm by 18.5cm for the test section. The fan used is a 48 inch, 19,500cubic-feet/min, Q-Standard Belt-Drive drum fan, manufactured by Northern Tool and Equipment (Northern Tool and provides a terminal flow velocity of 6 m/s. Approximating the air to be incompressible yields the relationship below (Eq. 4)
where A is the cross sectional area and V is the air velocity. This approximation is valid at wind speeds significantly below the speed of sound (NASA). Choosing 60m/s as the desired (idealized) maximum speed in the test section, the necessary cross sectional area was calculated to be 1530cm2, which corresponds to a test section of dimensions 39.1cm by 39.1cm. This speed was chosen to give a significant margin of error to allow for air flow losses in the tunnel. Then the length of the test section was determined. There must be at least 0.5 diameters from the beginning of the test section to the front of the model in order for irregularities in the flow due to contraction to 84 | 2013-2014 | Volume 3
smooth out (Mehta and Bradshaw 1975). Thus the test section was determined to be 40cm in length The bellmouth must have a large cross sectional area and a short length, while still smoothly constricting flow into the test section. For a small, open wind tunnel the area contraction ratio should be 6 to 9 (Mehta and Bradshaw 1975).Thus the opening to the bellmouth was chosen to have a cross sectional area of 10000cm2 (100cm side lengths). This contracts down to the test section area in a smooth bell-like shape over 61cm. At the entrance to the bellmouth there is a wire mesh screen to help remove turbulence from the flow. The fan used was 48 inches in diameter, so the diffuser had to expand the tunnel from the test section to that size. While the length ratios arenâ€™t quite as important on this side of the test section, aluminum flashing was used to smoothly transition to the greater cross sectional area, in order to ensure low-turbulence air flow. The test section was built out of wood and acrylic sheet, and the bellmouth and diffuser were constructed from aluminum flashing on a wooden frame. Upon completion of construction, a smoke test was conducted to ensure that air flow in the tunnel was smooth. The results of the test were positive. Experimental Details To conduct the experiment, a standard American (5.175cm diameter) pool ball was mounted on an axle with countersunk ball bearings and then rigidly hung in the test section from two Vernier Dual-Range force sensors (Vernier). A schematic is shown in Fig. 5.
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Figure 5: A simple cross-section schematic of the experimental set up. The ball could be spun by blowing compressed air over the top of the ball. The same hole that allowed access for the hose nozzle was also large enough to put an anemometer (Kestrel) into the tunnel to measure the wind speed. To vary the spin rate, force data was simply taken from the instant the compressed air was shut off until the ball’s spin slowed to a stop. This interval was recorded using a 480fps high speed camera (Casio) so that video analysis during data processing could give the angular velocity at various points during this time interval. The linear velocity of the air past the ball was controlled by hooking up the fan to a 20A Variac (variable voltage transformer), which could then be adjusted to control the power (and thus volumetric flow rate) of the fan. After calibration of the force sensors, eleven trials were run using this method, with linear velocities ranging from 5.9 to 11.6 m/s.
Results and Analysis The raw force data were noisy due to what is likely a combination of vibration and the pool ball not being perfectly centered on the axle. In order to filter out this noise a large amount of data processing had to be done. First, the forces from each of the two sensors were summed to get the total force on the ball at a given instant. This total force is the result of the Magnus force, as the sensors were zeroed before data collection to account for the weight of the ball and axle, the only other force acting in the vertical plane (see Fig. 6). The sensors were also calibrated using default factory settings stored on the device (Vernier). At higher spin rates much of the noise in the data is most likely due to vibrations, so the force data was averaged over 0.2s intervals in order to reduce this noise. The angular velocity of the ball was also calculated over these same 0.2s intervals using video analysis techniques. At lower spin rates, the variation in the data is caused more by uncertainty in the angular velocity rather than by vibrations. Thus for the data with ωV2<2000 rad∙m2/s3, 0.05s averaging intervals were used.
Essentially, the LoggerPro software was used to track the coordinate location of a small marker placed on the pool ball at either 0.2 sec (96 frame) or 0.05 (24 frame) intervals. Then, the average angular velocity of the ball could be determined for the time interval simply by dividing the change in angular position by the time interval. Force data was collected every 0.02s, so a simple arithmetic mean was used to determine the average force for each time interval. To determine whether ωV and ωV2 gave the most accurate description of the Magnus effect the force was plotted versus each of these products and a linear regression was done. These results are shown in Fig. 8. Uncertainties in the force measurements were calculated by doing a standard deviation of the mean for each time interval.
Figure 6: A force diagram of the forces acting on a ball in flight where FM is the Magnus force, FD is the drag force, and mg is the gravitational force. The ball is projected from left to right with backspin.
Figure 7: Smoke test of baseball portraying boundary layer disruption . To determine whether ωV and ωV2 gave the most accurate description of the Magnus effect the force was plotted Volume 3 | 2013-2014 | 85
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Figure 8: This upper graph shows the vertical (Magnus) force plotted vs. Ď‰V and the corresponding linear regression. The lower shows the same for the Magnus force vs. Ď‰V2.
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Physics and CompSci Research versus each of these products and a linear regression was done. These results are shown in Fig. 8. Uncertainties in the force measurements were calculated by doing a standard deviation of the mean for each time interval. There is a correlation of 0.59 and 0.48 between the Magnus force and ωV2and the Magnus force and ωV, respectively. Also, the spin and velocity appears to have little effect on the Magnus force until about ωV2<2000 rad∙m2/s3. From this data it appears that the ωV2 relationship is the correct one, however, given the noise in the data one cannot put much confidence in these results. In the realm of about 2300 rad∙m2/s3 there are some positive forces, which possibly corresponds to the reverse Magnus effect. However, once again, given the noise in the data, one can not definitively state that this actually is a reverse Magnus effect. Briggs (Briggs 1959) and Cross (Cross 2012), who both present reverse Magnus force data, have vastly different spin factors at which the reverse effect occurs - Briggs’s spin factors are about 0.25, whereas Cross’s are around 0.84. Our new data show a potential reverse Magnus effect at spin factors of around 0.15. Further data may determine if these results really do correspond to a reverse Magnus effect.
Conclusion/Future Work The dependence of the Magnus effect, the force that causes a spinning ball to curve, on the velocity and spin of a smooth ball was determined. A subsonic wind tunnel was built for the experiment, and a pool ball on an axle was hung rigidly from two force sensors in the test section of the wind tunnel. Force data was taken for a variety of spins and velocities, ranging from 5.9 to 44.2 rad/s and 5.9 to 11.6 m/s, respectively. Within this range the Magnus force seems to depend on the product of the spin rate and the velocity squared; however, further research will be necessary to further support this conclusion. A possible reverse Magnus effect was also observed where this product was in the range of approximately 2100 to 2400 rad∙m2/s3. While the research suggests some preliminary results, due to noise in the data it is impossible to make any conclusions with a high degree of certainty. Future work would include designing a method to reduce vibration of the apparatus from which the ball is hung, in order to achieve results with more precision.
Acknowledgements I would like to acknowledge Dr. Jonathan Bennett and Dr. William McNairy of the North Carolina School of Science and Mathematics for their assistance in my research process.
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HarperCollins, New York, 2002  L. W. Alaways. “Aerodynamics of the curve ball: An investigation of the effects of angular velocity on baseball trajectories.” Ph.D. thesis, University of California, Davis, 1998.  L. W. Alaways and M. Hubbard. “Experimental determination of baseball spin and lift.” Journal of Sports Science 19 (2001): 349-358  G. K. Batchelor. An Introduction to Fluid Dynamics Cambridge University Press, London, 1967  L. J. Briggs. “Effects of spin and speed on the lateral deflection (curve) of a baseball and the Magnus effect for smooth spheres.” American Journal of Physics 27 (1959): 589-596  R. Cross. “Aerodynamics in the classroom and at the ballpark.” American Journal of Physics 80 (2012): 289297  R. Cross. Physics of Baseball & Softball Springer, New York, 2012  “Dual-Range Force Sensor.” Vernier Software & Technology. N.p., n.d. Web. 22 Sept. 2013.  C. Frohlich. “Aerodynamic drag crisis and its possible effect on the flight of baseballs.” American Journal of Physics 52 (1984): 325-334  T. Jinji and S. Sakurai. “Direction of spin axis and spin rate of the pitched baseball.” Sports Biomechanics 5 (2006): 197-214  “Kestrel 1000 Wind Meter | Pocket Wind Meter.” KestrelMeters.com. N.p., n.d. Web. 22 Sept. 2013.  “Logger Pro.” Vernier Software & Technology. N.p., n.d. Web. 22 Sept. 2013.  R. Mehta. “Aerodynamics of Sports Balls.” Annual Review Fluid Mechanics 17 (1985): 151-189  R. Mehta and P. Bradshaw. “Technical Notes: Design Rules for Low Speed Wind Tunnels.” The Aeronautical Journal of the Royal Aeronautical Society (1979): 443-449  A. Nathan. “The effect of spin on the flight of a baseball.” American Journal of Physics 76 (2008): 119124  I. Newton. “A new theory about light and colors.” American Journal of Physics (Reprint) 61 (1993): 108-112  “Q Standard Belt-Drive Drum Fan — 48in., 1 1/2 HP, 19,500 CFM Model# 10248.” Portable Generators, Pressure Washers, Power Tools, Welders. N.p., n.d. Web. 22 Sept. 2013.  L. Rayleigh. “On the irregular flight of a tennis ball.” Messenger of Mathematics 7 (1877): 14-16.  A.F. Rex. “The effect of spin on the flight of batted baseballs.” American Journal of Physics 53 (1985): 10731075  G. S. Sawicki, M. Hubbard, and W. Stronge. “How to hit home runs: Optimum baseball bat swing parameters for maximum range trajectories.” American Journal of Physics 71 (2003): 1152-1162  S. Sawicki, M. Hubbard, and W. Stronge. “Reply to Volume 3 | 2013-2014 | 87
Physics and CompSci Research ‘Comment on How to hit home runs: Optimum baseball bat swing parameters for maximum range trajectories.’” American Journal of Physics 73 (2005): 185-189  “Vehicle Aerodynamics.” Vehicle Aerodynamics. San Diego State, n.d. Web. 19 Apr. 2013.  R.G. Watts and A. Bahill. Keep Your Eye on the Ball: The Science and Folklore of Baseball, W.H. Freeman and Company, 1990  R.G. Watts and S. Baroni. “Baseball-bat collisions and the resulting trajectories of spinning balls.” American Journal of Physics 57 (1989): 40-45  R.G. Watts and R. Ferrer. “The lateral force on a spinning sphere: Aerodynamics of a curve ball.” American Journal of Physics 55 (1987): 40-44  “Wind Tunnel Parts.” Wind Tunnel Parts. NASA, n.d. Web. 19 Apr. 2013. Company, 1990  R.G. Watts and S. Baroni. “Baseball-bat collisions and the resulting trajectories of spinning balls.” American Journal of Physics 57 (1989): 40-45  R.G. Watts and R. Ferrer. “The lateral force on a spinning sphere: Aerodynamics of a curve ball.” American Journal of Physics 55 (1987): 40-44  “Wind Tunnel Parts.” Wind Tunnel Parts. NASA, n.d. Web. 19 Apr. 2013.
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