A First-Principles Model of Curling Stone Dynamics

Abstract

An asymmetric force arising from grit and ice debris transfer gives rise to a model of curling stone trajectories that is compatible with observations. There are (almost) no free parameters in this model.

Appendix

First, we show why we can assume that grit and debris particles initially encounter the RB (at angle $\phi _0$ in the calculations of Sect. 4) only for $0<\phi _0<\pi$. This angular limitation is geometrically obvious for Type I particles that are distributed over the ice prior to the arrival of the stone: of course such pebbles can only make initial contact with the leading half of the advancing RB. However some of our particles are Type II ice debris made as the stone passes over—the RB abrades any pebble it makes contact with, and these abrasions can be at any initial angle $0<\phi _0<2\pi$. Even for debris particles that arise from such abrasions, however, we can neglect the contribution they make to the torque if they are located in the trailing half $\pi<\phi _0<2\pi$. The reasons are twofold. We showed in Fig. 4a that pebbles in the ’mid-latitudes’ of the trailing half are more severely worn by the stone passing over them than are other pebbles and so are in less close contact with the RB. We expect that the contribution of these pebbles to the CM torque is reduced because the abrasion force will be less than for the other pebbles. At the very back of the RB are some pebbles that are indeed in close contact, as we see in Fig. 4a, but these contribute very little to the torque because they are close to the y-axis. Thus in our torque calculation it is reasonable for us to make the approximation $0<\phi _0<\pi$.

Second, we derive the equation for $p_\lambda (\gamma )$, the probability density function describing the persistence of ice debris particle adhering to the curling stone RB. A debris particle is attached to the RB for an angle $\gamma$, as the stone rotates. We will assume that the probability for this particle to detach from the RB over the interval $d\gamma$ is proportional to $d\gamma$.⁶ That is, we assume a particle attached at $\gamma$ becomes detached by the time it reaches $\gamma +d\gamma$ with probability $\lambda d\gamma$, for some constant $\lambda$. Thus if $n(\gamma )$ is the number of particles that have been attached for an angle $\gamma$ then the number that detach between $\gamma$ and $\gamma +d\gamma$ is $dn=-n\lambda d\gamma$. Thus $n=n_0 \exp (-\lambda \gamma )$ where $n_0$ is the number of particles that were attached to the RB at $\gamma =0$, and so the probability that a particle persists for angle $\gamma$ is $P\equiv \frac{n}{n_0}=\exp (-\lambda \gamma )$. The probability that a particle detaches at $\gamma$ or some larger angle is thus

$$\begin{aligned} P=1-\int \limits_{\gamma }^{\infty } ds\,p_\lambda (s) \end{aligned}$$

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and so $p_\lambda (\gamma )=-\frac{dP}{d\gamma } =\lambda e^{-\lambda \gamma }$, as claimed in Sect. 4.