The Motion of Curling Rocks: An Analytical Approach to the Scratch-Guiding Mechanism

The motion of a curling rock on ice exhibits an interesting behavior. It curls towards the left (right) of its initial direction for an anticlockwise (clockwise) spin and the final lateral displacement (curl) is nearly independent on the initial angular velocity (in a normal playing regime). These features are compatible with a previously proposed scratch-guiding mechanism. Here, we present an analytical approach to the scratch-guiding quantitative model developed by Penner. We use a first-principle based friction coefficient that includes ploughing and lubricating contributions. We obtain analytical expressions for the kinematics of the rock in the limit where the rotational speed of a point of the rock in contact with the ice is small compared to its translational speed. We generalize Penner’s model to the small initial angular velocity regime and show that the final curl depends on the square of the initial angular velocity. We explore the effect of temperature, rock and ice wear on the rock trajectory.