Rikkyo University

In his research, Murata used an image-processing displacement sensor, a cutting-edge technology developed for experiments, to verify the law of universal gravitation on a micrometer scale to search for higher-dimensional spaces of four or more dimensions. Using this technology, he measured the movement of stones at the micrometer level to discover that leftward-curling is caused by rotation resulting from the stone’s gravity center swinging at a frictional fulcrum out of the center. He also found that the coefficient of kinetic friction, which is believed to be stable under most conditions, increases as the speed slows. According to Murata, this velocity dependence is the most likely reason why the center of rotation is asymmetric on the right and left sides, which have different speeds on the ice.

The results of his study were published in the Sept. 3, 2022, issue of Scientific Reports, a journal published by Springer-Nature.

• Title: Study of curling mechanism by precision kinematic measurements of curling stone’s motion
  
• Author： Jiro Murata
  
• Publication： Scientific Reports 12, 15047 (2022)
  
• URL： https://www.nature.com/articles/s41598-022-19303-4
  
• Supplementary URL including movies: https://rdcu.be/cUTFA

Watching curling matches at the Winter Olympics, not a few people might feel puzzled about how the stones turn. They instinctively think that stones rotating counterclockwise should curl to the right, in the direction of travel, since the friction between the stones and the ice is more robust in the front than in the back (Figure 1). This apparent mystery was reported in an academic journal for the first time in 1924. Since then, various hypotheses have become the focus of fierce debate in Nature and other journals. The first hypothesis discussed was the right-left asymmetry model: friction differs on the right and left sides because they move at different speeds (Figure 2). This model is based on the fact that a side moving faster has less friction because ice on the faster side melts more due to frictional heat.

Since then, many conflicting theories have been developed. One of them is the forward-backward asymmetry model. According to the model, although a curling stone touches the ice on its narrow, ring-shaped underside, friction becomes stronger at the back of the stone than at the front for some reason.

According to pivoting model, the stone curls because protrusions on its underside catch on the ice surface (Figure 3). This idea is straightforward from the point of view of physics; the only academic knowledge required to understand the phenomenon is the elementary dynamics that high school students and first-year university students learn. Nonetheless, solving the mystery has been too difficult, because not enough data to verify the hypotheses could be collected technologically.

In this study, Murata measured the behavior of curling stones at the micrometer level for the first time with an image-processing displacement sensor. This patented technology was developed for experiments to test the law of universal gravitation on a micrometer scale to search for higher-dimensional spaces of four or more dimensions. The experiments were carried out so the phenomenon could be analyzed through scientific data. Murata made the observations at the Karuizawa Ice Park of Karuizawa Kazakoshi Park, which was a venue of the 1998 Nagano Winter Olympics. The only devices used in the study were a digital camera and a tripod, but Murata was able to obtain accurate data on the behavior of stones in all 122 shots he took.

As a result, Murata observed a phenomenon in which the stone rotates with its underside, engaging with the ice like gears. He was also able to precisely measure the phenomenon whereby the coefficient of kinetic friction increases as the stone’s speed slows. This observation led to the discovery that the phenomenon makes the probability of forming a frictional fulcrum, which causes curling, different between the left and right sides of the stone. These discoveries experimentally proved that the center of rotation is asymmetrically formed by the velocity dependence of the dynamic friction coefficient--the most reasonable explanation for this century-old mystery. By studying the distribution of rotation centers, Murata was also able to confirm that friction is stronger at the front than at the back, as naively expected, contrary to what the forward-backward asymmetry model claims. Murata did not simply observe the motion of stones. He confirmed that the energy and angular momentum of their motion and rotation in the direction of travel are conserved, allowing him to observe the physical phenomenon correctly and reach a credible conclusion.

The pivoting model is a result of dynamics rather than a hypothesis. Though it has still not been determined how much it contributes to deflection, there is no room for discussion about its correctness on a qualitative basis. This theory is not based on the constant force of friction between even-toned surfaces, but on friction with discretized nature. If the left-right asymmetry, which has been discussed from the beginning, is considered with this model, it could be understood why curling stones turn to the left in the direction of travel.

Regarding the left-right asymmetry model, for many years many people have supported the claim that the trajectory of stones could not be bent because the model is based on the law of constant friction, as described in high school textbooks. A force of friction, however, should have been just a formula to simplify a problem and make it easier to be dealt with by averaging multiple collisions at the micro-level. It is a macro and statistical concept, with the amount of information, such as gas temperatures and pressures, intentionally reduced. At the micro-level, if a nub off the center of a moving stone digs into the ice and decreases the speed or creates a friction fulcrum, the center of gravity rotates around it. It is similar to when a runner swings and turns to the left after grabbing a pole standing vertically on the left side with his or her left hand. This phenomenon can be understood through the law of the conservation of energy, or the law of conservation of angular momentum. If the friction phenomenon is seen at a macro-level, micro-multiple crashes usually happen at the same frequency on both the left and right sides. Therefore, the force of friction as their average consequently has only the component of speed against the direction of travel after components on the left and right sides balance out. However, this should be considered correct only under ideal conditions. In reality, curling stones move with large clattering sounds and with vertical and horizontal vibrations. This means that friction happens discretely.

In this research, Murata could better understand this phenomenon by considering that the force of friction is not a fundamental concept but a coarse, secondary statistic. This phenomenon is one example showing that mechanical energy is not conserved in a phenomenon involving the force of friction, which is also known as non-conservative force. This way of thinking should be considered a reminder to avoid mistakes that are sometimes made in science, such as confusion caused by elevating a concept initially introduced for descriptive purposes to a principle, as in the case of curling stones. This is cognate with the trouble that Fleming’s left-hand rule violates the rule of parity symmetry. Murata thinks this is why the behavior of curling stones has not previously been solved.

Murata has come up with the idea that a mysterious phenomenon, which according to macro laws should not occur, could be understood by recalling micro basics after getting a cue from a so-called “ghost effect” known in the field of rarefied gas dynamics. This phenomenon is out of line with the Navier-Stokes equation, which is basic in fluid mechanics. Murata is not a specialist in sports science, but in physics concerning elementary particles, nucleus, and gravitation. He finds that the velocity dependence on the dynamic friction coefficient is very similar to the Bragg curve, a behavior of radiation within the material that is used in radiation therapy. Also, he found the transition of angular momentum with rotation by getting a cue from nuclear spin polarization, which occurs during experiments of nucleus collision with accelerators. By using nuclear spin polarization, Prof. Murata has recently been studying time-reversal symmetry at TRIUMF, a national particle accelerator center in Canada. Time-reversal symmetry is true in most cases under the laws of microphysics, such as reactions of particles. At the same time, it is generally believed that the arrow of time from the past to the future appears at a macro level due to the law of entropy. However, this explanation still doesn’t satisfy everyone. Since entropy is also statistic, Murata expects the universal lesson learned from the motion of curling stones to help clarify this issue in the future.

This research proves, based on data, that deviation of curling stones can be explained qualitatively from rotation around a frictional fulcrum formed not continuatively, but discretely, at asymmetrical ratios on the left and right sides. Anyone with physics knowledge up to the first-year university level and mathematics learned in high school can understand this research paper. The paper is available to the public and can provide any researcher and student who would like to try experiments on friction mechanics and simulations of the motion of curling stones with kinetic data that has been measured precisely for the first time. This paper is also expected to provide curling players with a fresh perspective on why brush sweeping can control the curling of stones, and how significant rotation, which could affect scores, occurs accidentally shortly before the stone stops.