A 19.96 kg granite stone slides on pebbled ice. Friction depends on speed, sweeping lowers it, and a rotating stone drifts sideways. Change the physics and watch the curl.
The science
Pebbled-ice friction
Water droplets ("pebble") freeze on the sheet so the stone rides on tiny asperities. Measured kinetic friction is tiny (mu about 0.006-0.02) and falls as the stone speeds up. The model uses
mu(u) = mu1 · u^(-p) (u > 0.1 m/s), held constant below
with mu1 = 0.0128 and p = 0.5, so a stone released at 2.4 m/s at the hog line stops near the button (28.3 m later).
Stopping distance
Sliding straight (no spin) the energy balance integrates exactly:
dd = v dv / (mu g (1-s)) ⇒ d = v0^(2+p) / ((2+p) mu1 g (1-s))
where s is the fractional friction reduction while sweeping. Sweeping at s = 0.2 therefore adds 25 % to the distance (1/(1-s)). Experiment 1 checks this against the numerical integration.
Contact ring and curl
The stone touches the ice only on a running band of radius R (about 6.5 cm). Each of 24 contact points has local velocity u = v + omega × rho
and a friction force F = -mu(|u|) w N u/|u|, with a load weight w that may be heavier at the rear (epsilon < 0) or the front (epsilon > 0). For small spin the sideways force is
a_side = g mu omega R epsilon / (2V)
Friction always acts along the velocity, so this only turns the heading: dpsi/dt = a_side/V and the offset grows as x = integral of V psi dt. A rear-loaded ring (epsilon < 0) makes a counter-clockwise stone curl left, as in real curling; a front-loaded ring (epsilon > 0) would curl it the other way, the intuitive "wheel" answer that observation rejects. Because mu rises as the stone slows, most of the curl accumulates in the last few metres.
Spin decay
The friction torque slowly removes rotation: d omega/dt = -mu m g R^2 omega / (2 V I), with I = 0.14 kg m^2.
Collisions
Stones are equal masses; along the line of centres the impulse uses restitution e = 0.88.
Simplifications
The true cause of curl is still debated (pivot-slide, scratch-guide and asymmetric-friction models disagree). Here it is a simple ring model: it gets the direction, the late-run build-up and the effect of sweeping right, but its curl is proportional to spin at low spin, whereas real stones curl about the same whatever the turn rate. Ice is flat, air drag and stone-ice thermal effects are ignored, and sweeping is a uniform friction scale factor.