Study of curling mechanism by precision kinematic measurements of curling stone’s motion

Why do curling stones curl? That is a question physicists are often asked, yet no answer has been established. Stones rotating clockwise curl right, contrary to our naive expectations. After a century of debate between contradicting hypotheses, this paper provides a possible answer based on experimental evidence. A digital image analysis technique was used to perform precision kinematic measurements of a curling stone’s motion to identify the curling mechanism. We observed a significant left–right asymmetric friction due to velocity dependence on the friction coefficient. Combined with the discrete point-like nature of the friction between ice and stone, swinging around slow-side friction points has been concluded as the dominant origin of the curling. Many new angular momentum transfer phenomena have been found, supporting this conclusion.


Results
, and 3 show the trajectories, the time sequences of the kinematic variables for the same four typical shots, and parameter configurations. P is defined in Fig. 3, from which we can approximately determine the swinging center. Indeed, P exactly acts as the swinging center for a case of pure rotation around a fixed friction point on the left side position. We can determine P uniquely, which is defined as the intersection of two straight lines. In cases with no static friction points, P can be interpreted as the virtual swinging center of the stone, which area is extended outside the actual stone volume. Especially, R rot ≡ CP → ∞ for pure translation with no rotation cases. Therefore, we can use R rot as a quantity representing how static the left side friction is.
The most symbolic phenomenon among the obtained results was the strong swinging observed before stopping, as shown in Fig. 1IV "ride-on swinging. " The stone swung around an almost static left position L ∼ = P on the radius of R edge as a simple orbital rotation (Fig. 1d). Here R edge was set to the inner radius of the "running band" (contacting ring) of the stone's bottom (Fig. 1a).
Similar relatively strong curling was observed, as shown in Fig. 1II-IV "gear-on" phase. In this phase, positions of P were not static but drifted while maintaining R rot ∼ = R edge positions (Fig. 1c). In Fig. 1III and IV, the velocities of L (i.e., v l ) were almost zero, as appeared as "kinks" of the label A, B's trajectories. In fact, Fig. 2IIIa and IVa show v l ∼ = 0 during R rot ∼ = R edge gear-on phase ( Fig. 2IIId and IVd). As shown in Fig. 1II and III, P moved from far distances to the R edge position during the "gear-off " phase ( Fig.1b) and then remained there stably after this "phase transition" to gear-on phase.  www.nature.com/scientificreports/ The minimum rotation shot also revealed interesting features. As shown in Fig. 2Ia, the deceleration rate of v(t) seems to be increasing at around the end ( t > 4 s ), suggesting the existence of velocity dependence of the friction coefficient µ . In fact, a(t) showed a large value just before the end, implying the strong-braking, as shown in Fig. 2Ia and Id. At the end of this section, we will discuss the significance of this behavior of µ(v) in a statistical analysis. In Fig.2Ib and Id, a(t) and ω(t) were shown by rebinning the time sequence combining 8 t to suppress fluctuations, which resolutions were σ 8�t a = 17 mm/s 2 and σ 8�t ω = 7.1 mrad/s 2 . |ω(t)| was small, but it can be highlighted that the rotation direction was significantly transitioned from CW (clockwise) to CCW (counter-clockwise) at t ∼ = 1.4 s occasionally. This transition timing coincided with the timing of the "inflectionpoint", which can be noticed if we carefully observe the compressed image of the trajectory shown in the inlet figure of Fig. 1I. It shows a transition of rightward to leftward curling. This result implies that ω was not simply    Fig. 1, of the kinematic variables; C's velocities v(t), C's accelerations a(t), angular velocities ω(t) , kinetic energies K.E., and R rot (t) . The square roots of K.E. were plotted for the translational components mv 2 /2 , the rotational components Iω 2 /2 , and their sums K.E. sum . Vertical error bars were drawn if they were not negligible. Gear-on, phase-transition, strong-braking, rotationacceleration, inflection-point in Fig. 1I, and constant ω are shown where they appeared.
i is the same position as L i on the stone's local frame, but that in t = t i+1 on the ice's frame. The "swinging" arm length is defined as www.nature.com/scientificreports/ decelerating but sometimes accelerating and that the changing of ω correlated with the curling. This "rotationacceleration" phenomenon can also be found in Fig. 2IIb for the slow rotation shot. The transitioning phenomena were also found for the translational motion. In Fig. 2IIIa "phase transition, " the deceleration rate of v suddenly decreased after t ∼ = 2 s . It was at the point that the gear-on phase began. At the same time, ω started a rapid deceleration (Fig. 2IIIb). This correlation can be well understood by checking the kinetic energies, K.E., as shown in Fig. 2IIIc. Their square roots were plotted to see quantities proportional to velocities. The translational and rotational components exhibited the transition, but their sum did not. This disappearance of the transition is particularly intriguing. It means that the total K.E. of a stone was conserved, except for the frictional loss while transferring the energy between translational ( 1 2 mv 2 ) and rotational ( 1 2 Iω 2 ) motions as But the total frictional loss rate in the system did not have the sudden change at the phase-transition timing. It is interesting to see the common features of strong-braking in the curves showing this velocity-dependent frictional deceleration at the timing just before the end of v in Fig. 2Ia, IIa, IIIa and IVa, ω in Fig. 2IIIb and IVb, and √ E sum in Fig. 2Ic, IIc, IIIc and IVc. The gear-off phase was also interesting, representing a situation in the actual curling games. ω was almost constant during the gear-off phase, as shown in Fig. 2IIIb. The transfer of translational and rotational energies helps to explain this situation. The rotational energy was fed by the translational energy, preventing deceleration due to rotational friction loss (Fig. 2IIIc). Therefore, we should not simply interpret the observed stability of ω during the gear-off phase as a result of minimal rotational friction coefficient. In contrast, the rotational energy fed the translational energy, as shown in the deceleration relaxation of v shown in Fig. 2IIIa during the gear-on phase.
All shots we measured were analyzed, not only for the typical shots shown in Figs. 1 and 2. R rot distributions at initial and final states are plotted in Fig. 4a, with Poisson errors = sample number . It can be confirmed that the convergence R rot → R edge was the common feature of all shots, independent of the initial conditions (v 0 , ω 0 ) , except for the minimum rotation cases. The observed peak of R peak rot = 58 ± 1.3 mm for the final states was compared with the running band regions of R band = 65 ± 5 mm . Then, their difference should be interpreted as the azimuthal angular distribution of the actual swinging center positions, which might be spread out at approximately ±cos −1 (R peak rot /R band ) ∼ = ±27 • around φ = 90 • . As shown in Fig. 2Id, µ seems strongly dependent on v. To confirm this, µ(v) might be estimated using the correlation between v(t) and a(t) = µg for the minimum rotation shots. However, ω of the minimum rotation shots was not precisely zero. Therefore, we should have considered the deceleration of v caused by energy leakage from translation to rotation. ṽ = √ 2K.E. sum /m and ã = dṽ/dt were used after rebinning the time sequence combining 8 t to suppress statistical fluctuations. Then, µ(ṽ) =ã/g was obtained as shown in Fig. 4b, using (1) d dt Iω 2 + frictional loss rate = 0,. www.nature.com/scientificreports/ the local value of the gravitational acceleration of g = 9.796 m/s 2 . Here, mean values and their standard errors (standard deviation/ sample number) of many ã samples from 38 minimum rotation shots, in the same ṽ bins, were used to obtain µ(ṽ) and their errors. The ṽ bin width of 25 mm/s was sufficiently larger than σ v , and only data with ṽ > 10 mm/s were used. The remarkable velocity dependence of µ was confirmed, showing a rapid increase before stopping. It meant that the strong-braking found in Fig. 2Ia and Id was confirmed as a common feature over all the samples in this statistical analysis. This is crucial to understand the curling mechanism, directly implying a friction enhancement on the slower side. This characteristic itself has been well known at least for a century 1 , as a result of ice melting. However, a reliable data set covering a very slow region at v < 0.1 m/s was newly obtained in this study. It owed to the high precision which enabled us to extend the sensitivity well down to v ∼ σ v . Figure 4b also shows the fitting result, which may be useful for future model calculations attempting to predict the curling trajectories as well as microscopic studies on the physics of friction. The static friction coefficient µ 0 measured using a spring scale was also shown, which was not used for the fitting. By utilizing this µ(v) data with m and I, one may be able to build a simulation tool to predict a curling stone's motion just by inputting v 0 and ω 0 .

Discussion
Let us now attempt to understand the obtained results shown above. First, the strong curling, as shown in Fig. 1IV "ride-on swinging, " was a clear indication of the existence of strong point-like frictions. It should be caused by pivoting due to relatively large pebbles on ice 12 or dust or scratching by the rock's rough bottom 15 or their accidental coincidences. Therefore, these phenomena occurred by chance, with less than 50% of our rotating shots exhibiting it.
The energy/momentum transfer between translational and rotational motions was found. This was observed as the accelerating rotation in Fig. 2Ib and IIb, as the deceleration relaxation of v in the gear-on phase in Fig. 2IIIa, and as the constant ω in Fig. 2IIIb. It also meant the transfer between the orbital-angular momentum (for revolution around a fixed position) L and the spin-angular momentum (for self-rotation) S 31 , conserving the total angular momentum J = L + S , if there is no external torque. For L = mvb around a friction point and S = Iω (b is the impact parameter, i.e., the perpendicular distance between the path of an incident particle and the center of force. See Fig. 1d, the L ↔ S transfer requires offset impact, i.e., b = 0 . Therefore, the observed angular momentum transfer must be caused by point-like impacts at a non-zero net offset position b . Any forward-backward asymmetric friction cannot produce such angular momentum transfer because �b� = 0 . Angular momentum transfer due to an offset collision to a fixed point cannot avoid swinging. The swinging leads to the leftward curling if the impact point is at 90 • φ < 180 • . This can be confirmed as a coincidence of the curling inflection and the ω change in the minimal rotation shot as shown in Figs. 1I and 2Ib. The converging R rot → R edge (Fig. 4a) can be understood as the frictional force at L being always opposite to the v l direction, suppressing |v l | . The observed converging v l → 0 (Fig. 2IIIa) also indicates that the friction was strongest at L around the φ direction in the running band. This backward friction at L assisted rotation when v l > 0 via the L → S transfer, preventing deceleration of ω during the gear-off phase. The stability of v l ∼ = 0 and R rot ∼ = R edge during the gear-on phase (Fig. 2IIIa,IVa,IIId,IVd) must be due to the large local static friction µ 0 at L . It prevented |v l | from enlargement by sequentially switching the engaging points by next-to-next, similar to engaging gears (Fig. 1c). The frictioning points for the gear-off phase were not static but dragged while scratching the ice, similar to slipping tires (Fig. 1b), which must be caused by the relatively large µ at small v l .
The observed µ(v) (Fig. 4b) indicated that the probability of having discrete impacts was greater at L than at R because continuum friction is not a fundamental concept but only a result of an artificial coarse-graining (averaging) treatment for many real microscopic impacts. As a result, it should be concluded that the combination of 1. swinging around a discrete frictioning point on the ice (pivoting/scratching) 10-15 and 2. the probability of the discrete frictioning is greater at the slow-side than at the fast-side because the velocity dependence of µ 1,10 should be the dominant curling mechanism. The convergence v l → 0 meant the existence of force to generate strong local static frictioning µ → µ 0 at the slow-side, which worked as the "adhesive friction" requested in the pivoting models [10][11][12] .
S provided the left-right asymmetry of the swinging probability but was not typically the primary momentum source of the swinging. That was L, which was transferred from a straight motion with the impact parameter b to an orbital rotational motion with the arm length b, resulting in swinging for v l > 0 , i.e., the slow rotating gear-off cases. S can directly contribute to the swinging via S → L transfer, but it was effective only for the fast rotation cases satisfying v l ≤ 0 . It should provide an answer to the known question 10,23 of why the total amount of curl is not proportional to, and only weakly depends on, ω 0 , part of which can be seen by comparing Fig. 1II and III. On the other hand, ω 0 dependence was becoming visible in the faster rotation cases due to the contribution from S 20 . In addition, deceleration relaxation of v due to S → L transfer during the gear-on phase (Fig. 2IIIa,IVa) should be the direct answer to why extremely fast rotating stones tend to travel further 10 . The stored intrinsic rotational energy should have been used to help the translational motion overcome friction.
In a real curling game, the brush-sweeping is effective not only for extending the stopping range by reducing µ but also for controlling the curling. Indeed, sweeping on the forward-right region leads to leftward curling. It is because of the reduction of the discrete frictioning on the right side. The unpredictable motion of the minimum rotation case is analogous to "knuckleball" in baseball, implying that a slight rotation should be preferred for a stable control to avoid random angular momentum transfer. The players should also remember that the occasional ride-on swinging phenomenon randomly affects the final stopping position.
Finally, the forward-backward asymmetry was examined. Although the inhomogeneous distribution of µ cannot be measured directly, we can estimate it because the discrete frictioning probability is proportional to µ . A useful tool was comparing the lengths R rot and R ′ rot (See Fig. 3 www.nature.com/scientificreports/ The swinging center positions were estimated. For example, A P = −1 for pure rotation around P without drifting, and A C = O(�θ 2 ) ∼ 0 for rotation around C with forward drifting. For rotation around F (B) with forward driting, A F(B) > (<) − 1 , which means that a large value of A implies swinging around F rather than B. Details of the estimation of R ′ rot to obtain A for (P, C, F, B) are shown in Fig. 5. Mean values and their standard errors of A for all frames in 73 completely stopped shots, except for minimum rotation shots, were obtained as �A� = 0.149 ± 0.020 (gear-on), = 0.056 ± 0.013 (gear-off), showing positive A . Figure 4c shows the A distributions with Poisson errors, which were dominated by A C . The realistic left side swinging with forward drifting should be distributed in A P < A < A C , but it was not visible. This must be because the probability of causing A = A P is relatively negligible to A = A C . A slight asymmetry enhancing right side with respect to A = 0 can be noticed, indicating that A = A F was preferred to A = A B . Indeed, asymmetries of the integration of Fig. 4c between A > 0 and A < 0 were obtained with Poisson errors as (2.03 ± 0.14) × 10 −1 (gear-on), = (3.13 ± 0.14) × 10 −1 (gear-off). This asymmetry corresponds the positive A . The dominant A C contribution should have had symmetric shapes to A = 0 , and A P must have enhanced A < 0 . Thus, the observed positive A meant that the A F -A B asymmetry was significantly positive.
This result indicated that µ(front) > µ(back) , which is naively acceptable but may lead to cause opposite curling. Therefore, this effect should suppress the major curling, which can provide another possible reason for the weak dependence of the total curl on ω 0 . However, considering that the sensitivity of using A on probing frictioning points is not quantitatively known, we should interpret the observed statistically significant positive A F -A B asymmetry as that, we found no evidence for the unnaturally large negative asymmetry indicating µ(front) ≪ µ(back) requested by the previously proposed forward-backward asymmetry models [5][6][7][8][16][17][18][19][20][21][22] in this method.

Conclusion
In conclusion, it has been found that swinging around the discrete left-right asymmetric frictioning points is the dominant curling mechanism. Each part of this conclusion is not perfectly new, repeatedly suggested as attractive hypotheses in previous works 1, [10][11][12][13][14][15] . Except for µ(v) , most of the featured rich results supporting the conclusion presented here are new, indicating the angular momentum transfer and the point-like nature of friction. Future model calculations must reproduce not only curling trajectories but also the phase transition and other angular momentum transfer phenomena. This work does not propose a hypothesis but presents the principle of curling based on the model-independent experimental evidence to solve this "mystery of the century. "

Data availability
The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request. Especially, the trajectory data will be useful to compare with simulation studies.