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Modeling the trajectory and measuring the Magnus coefficient and force of a spinning ping pong ball Cho, Raymond; Leutheusser, Samual Apr 30, 2013

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Modeling the Trajectory and Measuring the Magnus Coefficient and Force of a Spinning Ping Pong Ball Raymond Cho, Samuel Leutheusser Science One Program The University of British Columbia Vancouver, British Columbia April 2013  (I) Abstract Several sports including tennis, baseball, cricket and ping pong take advantage of the Magnus Effect, a forced caused by the spin of a ball travelling at a translational velocity. Because sports place a great emphasis on this phenomenon, there is an interest in modelling the trajectory of a ball involving this variable. To explore this concept, a ping pong ball was launched from a track with varying trajectories and an exposure-controlled strobe light photo was taken. This was analysed to model the path of the ball and determine the Magnus coefficient and its subsequent force. For the ping pong ball, the Magnus Force was determined to be: 𝐹! = (0.000207   ± 0.000053  𝑘𝑔)(𝛚×𝑣) Where ω is angular velocity and v is velocity. Table of Symbols   Fm   v g S0  FD  A   ω m C a ρ CD  Magnus force translational velocity gravitational acceleration Magnus coefficient drag force cross sectional area of ball  angular velocity mass of the ball drag coefficient  angle between v and horizontal density of air dimensionless drag coefficient  (II) Introduction The curved paths of tennis balls, ping pong balls and soccer balls are well-documented as an intuitive strategy in their individual fields of play. In higher levels of ping pong, topspin, backspin and sidespin are all tactics used to keep opponents off-balance by altering the ball’s trajectories and preventing return shots. Athletes inducing these spins on the ball maintain an intuitive sense of that force, known as Magnus Force, named after its discoverer, German physicist G. Magnus [1]. The simpler explanation for this phenomenon is that a rotating ball creates a current of air around itself, pushing the fluid forwards and backwards relative to the position of the ball. This effect is coupled with the translational movement of the ball and will slow down the movement of the air on one side while speeding it up on the other side. By Bernoulli’s principle, greater kinetic energy decreases the pressure of air and vice versa; therefore, the differential pressure on opposite sides of the ball will create a force, specifically, the Magnus Force [2]. Topspin slows down the movement of air above the ball while simultaneously speeding it up below it, causing a greater relative pressure above it and a consequent force in the downwards direction seen in 	
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    Figure 1. Backspin has the opposite effect and a force is induced in the upwards direction (Figure 2).  Figure 1: Topspin Causing Down Magnus Force in a Flowing Fluid (figure reproduced without permission from [5])  Figure 2: Backspin Causing Down Magnus Force in Flowing Fluid (figure reproduced without permission from [5])  A more exact justification for this phenomenon uses the concept of boundary layers and its separation point from the ball. The boundary layer is the stratum of fluid in the immediate surroundings of the ball in which the fluid’s viscosity has a significant effect on its motion. As seen in Figure 3, the boundary layer conforms to the shape of the ball but eventually reaches the separation 3: Boundary Layer, Separation Point and Wake (figure point, the start of the wake where there is a Figure reproduced without permission from [6]) very low flow of dense air. When a spin is induced on the ball, the separation point occurs further along the original one on the side of the ball that is moving with the flow of air. On the opposite side of the ball spinning against the flow of air, the separation point occurs before the point shown on Figure 3. Simply put, this shifts the wake towards the side spinning against the air flow. Since the wake region has higher pressure air than the flowing air around it, the ball will be deflected by this Magnus Force in the opposite direction [2]. This experiment differs from several other experiments based on Magnus Force in that a very smooth and light ping pong ball is used as opposed to rougher objects including baseballs and soccer balls [3]. We expect that the Magnus Coefficient for a ping pong ball is much lower than that of other objects as ping pong balls have no prominent external stitching or ‘grip’ for the fluid air. 	
    (III) Methods 	
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    We first made a measurement of the drag coefficient. A Ping-Pong ball of mass 2.5 g and radius 0.018 m was dropped from a height of approximately 1.25 m. Three videos were taken at a rate of 30 frames per second, and then analyzed in Logger Pro. A drag model was fitted to these data using a least squares test. The average of the drag coefficients was used as a drag coefficient for the drag and Magnus force model developed. The Ping-Pong ball was marked with three lines encircling the ball, each contained within one of three perpendicular planes, effectively assigning a Cartesian coordinate system to the ball. An entry track (Figure 4) was aligned with a rotating tire of a remote controlled car, such that upon rolling down the track, the ball collided with the spinning tire, inducing the desired spin (Figure 5). The ball was then launched into the air with induced topspin. In the case of backspin, the ball was hit down the track, so as to minimize the initial forward spin, and then collided with the tire to induce backspin.  Figure 4: Side, Front and Top View of the Apparatus set-up.  The initial translational and rotational velocities of the ping pong ball were quantified via exposure controlled strobe light photography. A strobe light was turned on and set to approximately 1350 flashes per minute. Once a stable flash rate was reached, the ball was released down the entry track and a camera with exposure length of 2.5 s was initiated. The room was kept entirely dark, thus, flashes of the strobe light steadily controlled the frames recorded by the camera. These photos were then analyzed in Logger Pro. The angular velocity of the ball was slow enough that the lines could clearly be seen and thus, these measurements were reliable.  (IV) Mathematical Modelling In Logger Pro, the horizontal (x) and vertical (y) positions of the ball at times determined by the flash rate of the strobe light were measured using the photo analysis tool. This data was subsequently coupled with measurements of the angular position of the ball. The angular position of the ball was quantified by determining the angle at which one of the lines contained within a plane was rotated from the horizontal using Screen Scales. A linear regression of the angular velocity of the ball over time revealed that the angular velocity of the ball remained relatively 	
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    constant. The average angular velocity and drag coefficient from the three videos, along with the initial conditions of each trial, were then used to create a model of each trajectory. Ten trials were run for each of topspin and backspin, of which seven for backspin and eight for topspin had little enough side spin to be analyzed. 90°-­‐a	
    The model involved differential equations describing the evolution of the horizontal and vertical positions as a result of gravity, drag force and the Magnus force. These equations were solved numerically using Euler’s method. The differential equations were derived on the basis of the forces shown in Figure 5.  Figure 5. Free body diagram of forces on a ball travelling at velocity, v, and spinning at angular velocity, ω.  The force on the ball is found to be,     F = −mgjˆ − C | v |2 vˆ + S0 (ω × v)  (2)	
    To split this force into x and y components, each force was split into its specific components. The gravitational force is solely in the vertical direction, thus the magnitude of this force in each direction is: Fgy = -mg and Fgx = 0. The drag force is always opposite the velocity,  thus this force can be defined as FD = -C| v |2 vˆ . The velocity, v, can be written, by Pythagoras’s Theorem, as |v| = vx2 + vy2 . From Figure 5, it can be seen that for any velocity, v,  vy 2 x  v +v  2 y  = sin a = cos(90 o − a) and similarly,  vx 2 x  v +v  2 y  = cos a = sin(90 o − a) . From analyzing the 2  components of the drag force we see that FDx = -C vx2 + vy2 cos(a) = -Cvx vx2 + vy2 and FDy = -Cvy vx2 + vy2 . Since the Magnus force is a cross product, it is perpendicular to the velocity and the angular velocity pseudovector, which are perpendicular to each other. This means the Magnus force can be written as FM = S0|ω||v|sin(90°) = S0|ω||v|. As seen in Figure 5, the Magnus force can be broken into FMx = S0|ω| vx2 + vy2 cos(90°- a) = S0|ω|vy and FMy = S0|ω|vx. The combination of gravity, drag and the Magnus force yields:   Fnetx = −Cvx vx2 + vy2 − S0 ω vy and   Fnety = −mg − Cvy vx2 + vy2 + S0 ω vx  	
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    (3) (4)  Differential equations to describe the acceleration of the ball, at time steps determined by the flash rate of the strobe light, were formatted on the basis on these forces, using Newton’s second law. These equations were evaluated using Euler’s method and cross referenced with the raw data to determine the Magnus coefficient that best described the observed trajectories.  (V) Results The drag coefficient was measured to be 0.001027 ± 0.000209 kg/m (p=0.05). As a ρ ACD v 2 dimensionless quantity and using the drag force model F = our dimensionless drag 2 coefficient is found to be CD = 0.406 ± 0.104. Through modeling of the trajectories observed in the stroboscopic photos, the Magnus coefficient was found to be 0.000207 ± 0.000053 kg (p=0.05). As seen in Figure 6, the observed trajectory (a) of a ball undergoing topspin agrees well with our drag model (b). Agreement between observed (a) and model (b) was also seen with backspin (see Figure 7). The variance in the backspin trials was greater than in the topspin trials. Also, the Magnus coefficients measured were on average smaller in the backspin trials as compared to the topspin. The initial translational and rotational velocities of the ball were also on average faster in the topspin trials. 	
    	
    	
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   Example Topspin Trajectory. (a) Stroboscopic Photo for Topspin. (b) Trajectory from Stroboscopic Photo. (c) Trajectory Calculated by Topspin Model.  Figure 7:  	
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    (VI) Statistical Analysis: 	
   Statistical analysis was run to quantitatively examine the accuracy of the model relative to the data attained through examination of stroboscopic motion photography. Principally, the chi-squared test was used to compare the model and the data: n  χ2 =  ∑ ( f (x ) − y ) i  2  i  i =1  n  (5)  Essentially, as the average of the differences squared is the chi-squared value, the lowest value gives the most accurate model for each photo taken of the trajectories. After the determination of the drag coefficient, 0.001027 kg/m, all the variables pertaining to the model except Magnus coefficient were kept constant. Thus, with both the topspin and backspin data, using Microsoft Excel Spreadsheet, the Magnus coefficients were randomly manipulated until the lowest possible chi-squared or least-squared value was attained. The chi-squared value typically ranged within the range of 10-4 and 10-6, suggesting that the data closely agreed with the model. The Magnus coefficients for top and backspin were verified separately to discern between different sets of data. After some trial and error for each photo to attain the lowest chi-squared value, the Magnus coefficient was determined to be 0.000259 kg. The same process was performed for backspin and the coefficient was 0.000147 kg. Using our force model, the Magnus coefficient must have the unit kg. A weighted average was run as 8 photos were used for topspin and 7 for backspin as follows:  8 ⋅ 0.000259 + 7 ⋅ 0.000147 = 0.000207kg 15 Therefore, using this coefficient, we can determine the Magnus Force contributing to the trajectory of the ping pong ball (refer to the introduction, where S0 is the Magnus coefficient): 𝐹! = (0.000207   ± 0.000053  𝑘𝑔)(𝛚×𝑣)  (6)  (VII) Errors There are a few prominent potential sources of errors that may have influenced our results. Firstly, there is potential error associated with the sidespin of the ball. The model assumed that all spin was directed perfectly forwards or backwards; however, in reality some trials were influenced by sidespin. Due to the presence of the track, sidespin was nearly 	
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    completely eliminated for topspin trials. The method of initializing motion of the ball for backspin allowed for some sidespin, increasing the error, and measured range of Magnus coefficients, for these trials. Since the topspin trials gained rotational velocity while rolling down the track, these initial rotational velocities were found to be higher. Due to the higher initial translational velocities, the higher Magnus coefficients measured in the topspin trials may be explained by potential velocity dependence of the Magnus coefficient [1]. Previous studies on the trajectories of spinning spheres have shown that the Magnus coefficient may be velocity dependent, although with the small range of velocities used in this experiment (0.5 – 3 m/s) it is unclear whether this would have a significant effect. Nonetheless, the difference in topspin and backspin Magnus coefficients may be explained by this effect. Finally, the susceptibility of a projectile to lateral forces whilst travelling in air is much greater for a ball spinning at a low angular velocity [4]. The lower angular velocity of the backspin balls likely made these trials more susceptible to lateral forces that were not accounted for in the model. This effect may explain the greater variance of the backspin trials as compared to the topspin trials. (VIII) Concluding Remarks: As we now see the effect that Magnus Force has on the ping pong ball, we can understand why sports place such a high emphasis on spin. Further studies on the Magnus Force and the trajectory of a smooth ball should include investigation of velocity dependence. Using this experiment’s procedure the initial velocities can be controlled and thus, the velocity dependence could be examined. Attaining more precise data could be accomplished by using wind tunnels and fluid imaging. This same experimental procedure can be used to examine the Magnus Effect on rougher balls that are roughly symmetric to broaden our understanding of how Magnus coefficients may be related to properties of the projectile. (IIX) References: [1] Kray, T., Franke, J., & Frank, W. (2012). Magnus effect on a rotating sphere at high Reynolds numbers. Journal of Wind Engineering and Industrial Aerodynamics, 110, 1-9. doi: 10.1016/j.jweia.2012.07.005 [2] Robert G. Watts, & Ricardo Ferrer. (1987). The lateral force on a spinning sphere: Aerodynamics of a curveball. American Journal of Physics, 55(1), 40. doi: 10.1119/1.14969 [3] Samad, A., & Garrett, S. J. (2010). On the laminar boundary-layer flow over rotating spheroids. International Journal of Engineering Science, 48(12), 2015-2027. doi: 10.1016/j.ijengsci.2010.05.001 [4] Mehta, R. D. (1985). Aerodynamics of sports balls. Annual Review of Fluid Mechanics, 17(1), 151-189. doi: 10.1146/annurev.fl.17.010185.001055 [5] The Magnus Force. (n.d.). Aviation for Kids. Retrieved March 13, 2013, from http://www.aviation-for-kids.com/the-magnus-force.html [6] Fitzpatrick, R. (2012, April 27). Boundary Layer Separation. Richard Fitzpatrick . Retrieved March 10, 2013, from http://farside.ph.utexas.edu/teaching/  	
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