The Physics of Toppling and Regaining Balance during a Pirouette.

One of the most common movements in dance is a turn around a vertical axis with one supporting foot on the floor--a pirouette. If the pirouette is not performed with the body on balance, it is not considered successful. Dancers are often taught to perform successful pirouettes by beginning the movement on balance and then keeping the body in that configuration, as opposed to correcting for an imbalance with small adjustments during the turn. Many, even advanced, dancers have significant difficulty performing more than two or three turns in a pirouette before losing balance, despite continued trial and error efforts to improve. To describe the mechanics of toppling and control of toppling during a pirouette, a theoretical model of a dancer in standard pirouette position was created, and an experimental study of real dancers performing pirouettes was conducted. Body segment parameters for the model (mass, length, etc.) were based on anatomical data and adjusted for sex, total body mass, and height. The principal moments of inertia were determined for several hypothetical dancers, and rigid body equations of motion numerically solved to express topple angle vs. time. When dancers reach too large a topple angle, they are forced to compensate by either hopping on the supporting foot in an attempt to regain balance or terminating the turn. The angle at which dancers lose stability and feel inclined to hop (θmax) was determined experimentally through a video analysis of nine intermediate to advanced ballet dancers' pirouettes (8 female, 1 male; 16 ± 2.3 years of age). The dancers hopped on the supporting foot after the body reached an average angle of 9.3 ± 1.9° from the vertical. With an average spin rate of 1.7 rev/s, it was found that a "rigid body" dancer (male or female) would need to begin the pirouette displaced less than one degree from the vertical in order to perform more than a double pirouette before reaching θmax. The results of this study demonstrate the difficulty of achieving many rotations when the body is held rigidly, whereas dancers may have success in consistently performing more pirouettes if they are taught strategies for regaining balance while turning.