The spinning ball spiral

We discuss the trajectory of a fast revolving solid ball moving in a fluid of comparable density. As the ball slows down owing to drag, its trajectory follows an exponential spiral as long as the rotation speed remains constant: at the characteristic distance where the ball speed is significantly affected by the drag, the bending of the trajectory increases, surprisingly. Later, the rotation speed decreases, which makes the ball follow a second kind of spiral, also described in the paper. Finally, the use of these highly curved trajectories is shown to be relevant to sports.


Introduction
Since Galileo, spheres have been used by physicists to probe movement and friction [13,14,27]. In the context of hydrodynamics, in particular, the motion of a solid sphere (radius R, velocity U 0 , density ρ s ) in a quiescent liquid (viscosity η, density ρ) is the paradigm for characterizing the laws of friction at low and high Reynolds numbers.
For low Reynolds number (Re = ρU 0 R/η 1), Stokes [30] established that the drag force experienced by the solid during its motion is F = 6πηU 0 R. This very classical result was then verified by several authors in the range Re < 1 [8,24]. For high Reynolds numbers, Newton [22] was probably the first to propose an heuristic expression for the drag: F = 1/2C D ρU 2 0 π R 2 , where C D is a coefficient provided by the experiments. According to Eiffel [12], C D is of the order of 0.4, a value later confirmed in the range 10 3 < Re < 2 × 10 5 [28]. For intermediate Reynolds numbers (1 < Re < 10 3 ), the asymptotic expansion method proposed by Oseen [23] led to lots of theoretical developments [4]. Beyond Re ≈ 2 × 10 5 , the resistance crisis experienced by the sphere once the boundary layer becomes turbulent has also been studied in depth [1,17,29].
For spinning spheres, according to Barkla and Auchterloniet [2], the work seems to go back to Robins [25] and then Magnus [18], who got the credit for the associated lift force. Besides these academic studies, the widespread use of balls in sports also motivated many studies, in baseball [21] and golf [9] in particular, a review of which can be found in [20]. Most of these studies consider a lift force F L = 1/2C L ρU 2 0 π R 2 , where the lift coefficient C L is known to increase with the spin parameter S = Rω 0 /U 0 .
Here, we study the trajectory of spinning spheres in water and try to understand their surprisingly curved trajectory, an example of which is presented in figures 1 and 2. In figure 1, the trajectory is decomposed into eight images, whereas in figure 2, the same sequence is presented within a single image by superimposing the successive positions of the ball. Both figures reveal a spiral trajectory. With solid friction and rotation, similar curved trajectories can be obtained, for example in French billard [6] and in lawn bowls [7]. In figure 2, we also observe the formation of an air cavity behind the sphere, a consequence of the high speed of penetration [5,10,11,15,16,19,32]. The bending of the trajectory starts as soon as the ball enters the bath, as reported in [31] for vertical impacts. Our aim here is to focus on the spiral trajectory and to discuss its relevance to sports.

Experimental facts
The projectiles used in this study are balls made of either polypropylene (ρ s 920 kg m −3 ) or polyacetal (ρ s 1400 kg m −3 ), a few millimetres in size. Spin and high velocities (20-50 m s −1 ) are achieved using a slingshot, consisting of a forked stick attached to a pocket by two rubber strips. The velocity can be varied by tuning the average tension applied to the rubbers, whereas the spin is controlled by the difference in tension between them: the motion is a pure translation when the extension of the two arms is symmetric, and spinning occurs when one strip is more stretched than the other. Both translational and spin velocities at the moment of impact U 0 and ω 0 are measured on the images recorded with a high speed video camera. Rotation is made visible by drawing a line on the equator and illuminating the spheres.
The effect of spin is illustrated in figure 3. Without rotation (ω 0 = 0), the ball goes straight ( figure 3(a)). With a bottom spin (ω 0 > 0), figure 3(b) shows that the ball deviates upwards;  it is even able to escape from the bath (last two images). Finally, for top spin (ω 0 < 0), the ball deviates downwards (figure 3(c)). We focus now on the bottom spin case and show in figure 4(a) the trajectory of a polypropylene ball (ρ s /ρ = 0.92) of radius R = 3.5 mm thrown in a water bath at a velocity U 0 = 27 m s −1 , with a spin rate ω 0 = 1000 rad s −1 and an impact angle θ 0 = 70 • (defined from the vertical). In this trajectory, the constant time step between two data is t = 384 µs. Clearly, the velocity of the ball decreases as it moves through water ( figure 4(a)). The evolution of the ball velocity is reported in figure 4(b) as a function of the curvilinear location s (s = 0 at impact). The semi-log presentation stresses that the velocity decreases exponentially with s. The characteristic length of the decrease is here 5.5 cm. Despite The trajectory bends only if spin is present and the sign of its curvature changes with the sign of the spin.  a strong variation in the velocity, the spin rate ω remains almost constant as the ball moves through water, as demonstrated in figure 4(c). This difference is discussed in the following section and is shown to be the key fact to account for the spiral trajectory.

Drag
The motion of the sphere of mass M is described in the Serret-Frenet coordinate system (t, n) introduced in figure 2. We first focus on the direction t. The Reynolds number Re = ρU 0 R/η is of the order of 10 4 , which implies a drag F ≈ 1/2ρU 2 π R 2 · C D , with C D ≈ 0.4 [28]. The equation of motion along t thus is written as In this equation, C M stands for the added mass coefficient, which, for a sphere, is of the order of 1/2, independent of the speed U [3]. Using the condition U (s = 0) = U 0 , equation (1) can be integrated as The velocity thus decreases exponentially in water, with a characteristic penetration length L ≈ 7ρ R ≈ 10R. This behaviour agrees with the results displayed in figure 4(b). We deduce from such measurements the value of L for different systems. Our data are presented in figure 5 as a function of the lengthρ R and compared to the results obtained by May [19] and Truscott and Techet [31]. All the data collapse in the same curve, L ≈ 7ρ R, in good agreement with equation (3). This comparison underlines that the entrained air cavity visible in figures 1-3 does not significantly affect the drag on the sphere. The time variation of the velocity can finally be deduced from equation (1), which classically yields U (t) = U 0 /(1 + t/τ ), where τ = L/U 0 is the characteristic slowing time of the ball. In equation (1) and in the above discussion, we neglected the effect of gravity. This assumption remains valid as long as the drag F is large compared to the Archimedean force 4π/3(ρ s − ρ)g R 3 . Using the expression for the drag, we conclude that the low gravity regime is achieved as long as U U * , where This condition is always fulfilled in the iso-density case but it fails otherwise at the 'end' of the trajectory, when the velocity of the ball vanishes. In this paper, we focus on the hydrodynamic effects and do not address the classical gravitational problem. Our conclusions thus hold above the critical speed U * , which in our case (ρ s /ρ ≈ 1.2, R ≈ 4 mm) is approximately 10 cm s −1 , much smaller than the impact speeds (≈10 m s −1 ).

Lift
Along the direction n, the equation of motion can be written as  [19] and Truscott and Techet [31], respectively. The symbol is used for our results. Equation (3), that is L ≈ 7ρ R, is represented by the solid line. (b) Initial curvature (dθ/ds) 0 of the ball trajectory as a function of ω 0 /U 0 · 1/ρ. The symbol is used for the data of Truscott and Techet [31] and the symbol for our results. The solid line shows the fit of (dθ/ds) 0 = 0.14ω 0 /U 0 · 1/ρ.
where F L = ρ U RC n is the lift force resulting from the circulation = 2π R 2 ω. In the limit of low Reynolds numbers (Re < 1), Rubinow and Keller [26] have shown that we have C n = 1/2. For large Reynolds numbers (Re ≈ 10 5 ), Nathan [21] collected the data obtained by several authors on the lift force experienced by spinning balls in air. From these results, we deduce C n ≈ 0.13.
Since our experiments are done in water with an entrained air cavity, we found it useful to measure C n . For this purpose, we focused on the impact region (s < L), where the dynamical parameters (U, ω) are constant, so that equation (4) predicts a constant curvature for the ball trajectory, This initial curvature is presented in figure 5(b) as a function of the inverse length ω 0 /ρU 0 ( ).
In the same figure, we also report the data extracted from Truscott and Techet [31] ( ). Equation (5) nicely predicts the initial curvature of the trajectory, and the best fit on both sets of data suggests C n ≈ 0.1, a value comparable to the one deduced from Nathan [21]. There again, the air cavity behind the ball does not affect the evaluation of the lift, a consequence of the entrainment of a water boundary layer around the projectile.  (6)) is drawn with a solid thin line (y = 0 is the surface of the water bath).

The ideal spiral
The next step in the derivation of the ball trajectory is to assume that the circulation remains (almost) constant during the motion, that is, over timescale τ . This assumption is suggested by figure 4(c) and we discuss it further in section 3.4. Then, for = 2π R 2 ω 0 , equation (4) together with (2) implies where = 4C n /C D ≈ 1 and S = ω 0 R/U 0 . The deviation of the ball from its initial orientation θ 0 thus increases exponentially with the curvilinear coordinate s, which defines the spinning ball (ideal) spiral. The characteristic length L for which the spiral coils up precisely is the penetration length expressed by equation (3). We compare in figure 6 the observed trajectory ( ) to equation (6) (solid line). The comparison is made in the plane (x, y) using the geometrical relations dx/ds = sin θ , dy/ds = −cos θ and for S ≈ 0.09, the value of the spin parameter in this experiment. The theoretical prediction is in close agreement with the experimental path up to the point where the ball escapes from the bath, whose surface is defined by y = 0.
An ideal spiral would converge to a centre C (figure 6) located at a distance D from the impact point. Since C is approached when θ(s) − θ 0 is of the order of π (corresponding to a U-turn of the ball), we obtain from equation (6) D ≈ L ln[1 + π/ S]. The distance D is a linear function of L and slowly diverges (as ln 1/S) when the spin number goes to 0. Conversely, for large spin numbers, the spiral centre is expected to converge towards the impact location (D ≈ L/S).

Variation in the rotation speed
The 'ideal' spiral derived in equation (6) is a good approximation for the ball trajectory as long as the rotation speed ω remains close to its initial value ω 0 . This assumption is valid for the 'shallow' spiral presented in figures 4 and 6, but cannot be used for the 'deep' spiral displayed in figures 1, 2 and 7. In the latter figure, we first show the ball trajectory (figure 7(a)) and observe that the 'ideal' spiral (thin solid line) only captures the data ( ) in the first part of the trajectory (s < L). At larger distances, figure 7(b) makes it clear that the rotation rate of the ball decreases, from ω 0 to 0.3 ω 0 at the end of the movement. This decrease of the rotation speed obviously lowers the lift, so that we expect the actual trajectory to be less curved, as observed in figure 7(a). In order to model the decrease in ω, we assume that the angular momentum MR 2 ω changes due to the torque of viscous forces acting on the surface of the ball, and we introduced ν = η/ρ for the kinematic viscosity. The viscous torque does not exist without rotation (see figure 3(a)) and originates from the difference in velocity Rω between the two sides of the ball. The boundary layer thickness associated with the rotation scales as √ ν/ω (typically of the order of 30 µm), from which we deduce a viscous stress η Rω/ √ ν/ω. Since the stress is applied over the surface area R 2 , we obtain the following angular momentum equation, ignoring the numerical coefficients, Equation (7) can be integrated, which leads to the time evolution of ω, The spin velocity of the ball thus decreases over a characteristic time τ ω (where β is a numerical constant, whose value is discussed below). The trajectory remains close to the ideal spiral as long as the travelling time t (s) = τ (e s/L − 1) is smaller than τ ω (we recall that τ = L/U 0 ). One thus expects to leave the ideal spiral when s becomes larger than L ln[1 + U 0 / √ ω 0 ν ], that is, a few times L, as observed in figure 7. Conversely, if τ ω < τ , ω decreases quicker than needed to make the spiral, and the corresponding lift force vanishes before curving the trajectory. One thus expects a linear propagation of the ball in this limit, which corresponds to impacts in viscous liquids (ν > U 2 0 /ω 0 ). For the parameters in our experiments (U 0 ≈ 30 m s −1 ; ω 0 ≈ 1000 rad s −1 ), this limit corresponds to oils at least 1000 times more viscous than water.
In the limit τ/τ ω 1, the evolution of the local angle reduces to the ideal spiral (6). However, even if τ/τ ω is small, its product with the exponential term e s/L can lead to an observable effect of the spin decrease. For β = 3.1, the trajectory obtained with equation (10) is drawn with a thick solid line in figure 7(a), showing fair agreement with the data. At long distances, equation (10) predicts that the ball follows a straight line, deviating from the impact direction by an angle Sτ ω /τ proportional to the spin number S.

Application to sports
The physical origin of the spinning ball spiral lies in the difference in velocity dependences of lift and drag, which are linear and quadratic, respectively. This behaviour is specific to high Reynolds number flows around spinning spheres. In our experiments, we used water to minimize the effect of gravity and to reduce the spatial scale of the spiral L ∼ρ R. However, the spinning ball spiral should also exist in air, and we discuss here its influence in ball games. For different sports, table 1 shows the ball size, the density ratio, the maximum ball velocity, the characteristic spin parameter and the size of the field, L. In the special case of baseball, L represents the distance between the pitcher and the batter. Using these data, we also display the penetration length L = 7ρ R and the length scale U 2 0 /g on which gravity acts. By comparing L and U 2 0 /g, one can identify sports dominated by aerodynamics (table tennis, golf and tennis) and sports dominated by gravity (basketball and handball). In between, we Table 1. Specifications for different sports. The first three sports are dominated by aerodynamic effects (L U 2 0 /g). For the last two sports, gravity dominates aerodynamics (L U 2 0 /g). In between, we identify sports for which both gravity and aerodynamics can be used to control the ball's trajectory. In this table, L is the size of the field except for baseball, where it stands for the distance between the pitcher and the batter. find sports where both gravity and aerodynamics play a comparable role (soccer, volleyball and baseball). Indeed, in the first category of sports, the spin is systematically used, while it is not relevant in the second category, and it only appears occasionally in the third one, in order to produce surprising trajectories. Focusing on sports where aerodynamics plays a role, we observe that the penetration length, which is also the characteristic length of the spiral, is generally larger than the size of the field. Since the spin parameter is smaller than one, the spiral centre (section 3.3) will lie outside the field. This suggests that the ball trajectory (6) can be expanded for s/L 1. In this limit, the spiral reduces to a circle of curvature (5), and we can evaluate the length d by which the ball deviated from its initial direction by its own size R: d ≈ √ 2LR/ S. This distance is shown in the last column of table 1. It is found to be systematically smaller than L, the field size, which makes relevant the use of spin effects to control the trajectory of the ball.
The case of soccer, where L is twice as small as L, is worth commenting on. The ball trajectory can deviate significantly from a circle, provided that the shot is long enough. Then the trajectory becomes surprising and somehow unpredictable for a goalkeeper. This is the way we interpret a famous goal by the Brazilian player Roberto Carlos against France in 1997 (http:// www.youtube.com/watch?v=crSkWaJqx-Y). This free kick was shot from a distance of approximately 35 m, that is, comparable to the distance L for which we expect this kind of unexpected trajectory. Provided that the shot is powerful enough, another characteristic of Roberto Carlos' abilities, the ball trajectory brutally bends towards the net, at a velocity still large enough to surprise the keeper.

Conclusions
We have studied the motion of spinning spheres at high Reynolds number and in the limit of low gravity. In this regime, we showed that the curvature of the ball trajectory changes as it moves, following law (9), rewritten here as dθ ds ∼ ω(s) U (s) · 1 ρ .
We have identified the characteristic length L ∼ 7ρ R over which the ball slows down and coils. Using this length, we have classified different phases in the ball trajectory. (i) In the initial phase (s/L 1), neither the velocity nor the spin varies, and the ball follows a circular path whose curvature C 0 can be deduced from (11): ω 0 /ρU 0 . (ii) As s approaches L, the velocity is changed but the spin is only weakly affected. This difference in behaviour is all the larger since dimensionless number √ ω 0 ν/U 0 is small. In this phase (s/L ≈ 1), the spinning ball coils up and forms a spiral. (iii) The last phase of the flight is reached when both the velocity and the spin decrease (s/L > 1). The trajectory then deviates from the spiral and tends to a straight line as the ball stops.