physics ballet

Physics of Ballet Dancing Center of Gravity and Balance Dance consists of movements of the bodyhttp://www.geocities.com/CapeCanaveral/Han interspersed with motionless poses. Often thesegar/4421/ poses demonstrate balance of the body over a small area of support on the floor. Physically, a by Sheila Dodge condition of balance exists if the dancer remains motionless above the area of support and does not Dancing is one of the most difficult, disciplined, fall. The balance condition will be achieved if andand beautiful art forms. Ballet is truly the only if the center of gravity lies on a vertical linefoundation of all dance. Ballet teaches self- passing through the area of support at the floor.discipline, correct body placement, strength, and If the dancer is motionless, the sum of all thegracefulness. Who would guess that ballerinas are forces and torques acting on the body must beactually physicists in disguise. Not too many zero. The force of gravity vertically downward willpeople, but in a way they are because they are be balanced by the force upward from the floor toshowing us many fundamentals of this science, the foot (the Normal force), acting on the samewhich allow them to produce breathtaking, dance vertical line.masterpieces. They are performing physics at its This concept is derived from Newton's Thirdbest as they dance, leap and turn across the stage. Law of Motion (the action-reaction law) which states: Whenever one body exerts a force on a Dancing is a unique art form, which lets one second body, the second body exerts an oppositelyexpress their feelings and use their body's directed force of equal magnitude on the first body.capabilities to full extent. In order to use their bodyand withhold their own kinesphere, the If the center of gravity is not in line with thesefundamentals of mechanical physics are needed to other equilibrium state forces, the dancer willensure that the art form is definitely an amazing be unbalanced and experience an angularone. acceleration towards the ground (in other words the dancer will fall to one side. The concepts of the following: • center of gravity and balance A dancer seldom achieves a true balance • rotational mechanics and the pirouette condition. If the center of gravity is close to that • friction and the glissade “balance area” that is if the forces are almost but • projectile motion and the grande jete not perfectly vertical, the acceleration away from vertical is initially quite small and the dancerall are part of physics that underlies the appears to be balanced.movements of ballet. Rotational Mechanics and the Pirouette This article discusses how steps in Ballet arereliant and dependent on mechanical physics. Turning movements are common in all forms ofLooking at dance from the mechanical physics dance. One of the most common turns is thepoint of view brings the art form of dance to a pirouette. The pirouette, in general, is a rotationdeeper level and shows the enormous amount of where the dancer supports herself on one leg whilediscipline needed to attain a beautiful, talented the other leg is in retire position. Any pirouetteballerina. must commence with some form of preparation position followed by a torque exerted against the floor. This torque of the floor against the dancer causes the angular acceleration that produces turning motion. The torque to initiate a turn can be exerted against the floor by two feet with some distance l. In Ballet, pirouettes are commonly done from one of two positions, fourth position or fifth position. A turn from fifth position, which has a smaller distance (I), will require more force to produce the same torque than a turn from fourth position, which has a larger distance (I). This is because Torque (T) is equal to the force (F) multiplied by the distance of the lever arm (l). Another way to describe the torque used in a pirouette is to take into account the dancer's Moment of Inertia. The moment of inertia depends on the mass of a body and its distribution relative

to the axis of rotation. A body of given mass will perpendicular to the floor (also called the normalhave a larger moment of inertia if the mass is far force) will give the force of friction which actsfrom the axis of rotation than if it is close. An against the motion of the dancer.arabesque position has a larger moment of inertiathan a retire position. Friction is very important in dancing. To increase friction many ballet dancers use a An arabesque turn is different than a pirouette substance called Rosin. Rosin has a highbecause the leg is now stretched out behind the coefficient of kinetic friction but a significantlydancer (larger R) rather than close to the dancer at smaller dynamic coefficient. This means that if thethe knee (smaller R). The moment of inertia for the foot is stationary on the floor a large horizontaldancer in arabesque or pirouette position with the frictional force is possible, but if the foot is movingaxis of rotation going vertically through the center that force is much smaller.of the body is I=1/3 MR2. Therefore, the larger theR, the larger the moment of inertia. This difference is very useful for a dancer performing glissades, a step which needs this If we can determine the angular acceleration (a) horizontal force only when the foot is not movingof an arabesque turn or pirouette, then we can against the floor. A glissade is a gliding motiondetermine the torque used to produce it. If we from fifth position to an open stretched positionknow angular velocity (w) then we can take the and then back to a fifth position.change in the angular velocity divided by the timeit takes to do the pirouette to find the angular Many dance movements involve sliding on theacceleration. floor or rotating against it. Whether a dancer needs a change in linear horizontal motion as in a Using the equation: T=Ia which is analogous glissade, or an angular acceleration as in a to Newton's second law of motion (F=ma), we pirouette, the floor must be able to supply the can calculate the T used for the turn. horizontal forces that provide for such accelerations. Too much friction may prevent theseConservation of Angular Momentum movements. A powerful concept for dance analysis involves Projectile Motion and the Grand Jeteangular momentum. Analogous to linear All jumps involve vertical accelerations andmomentum (p=mv), angular momentum is given by: L=Iw. It is a conserved quantity and is a constant forces. Because gravity acts vertically downward onif there are no torques acting on a body. In other our bodies at all times, we can only remainwords, the total angular momentum L of a body motionless if there is a vertical supporting forcecan change only if there is an external torque equal to our weight (the normal force). In order toacting on it. There is no way that changes in body jump off the ground we need to exert a forceposition alone, representing changes in the downward against the floor greater than our weightconfiguration of mass within the rotating system, (mg=mass * acceleration due to gravity for longcan change the total magnitude of L. enough to achieve the vertical upward velocity desired. It is sometimes claimed that \"spotting\" thehead one more time can squeeze and extra turn The height of a jump depends on the downwardout of a multiple pirouette. But rotating the head vertical distance through which that force isrelative to the rest of the body does not change the exerted. In order to obtain enough force to jump offtotal angular momentum of the body as a whole, the ground a dancer is required to bend theirand therefore cannot contribute any extra turning knees, a position called plie.motion. Jumps combined with horizontal motions such The body may descend from turning position as the grand jete (a leap in which the weight of thebefore completion of the turn, but the additional body is thrown from one foot to the other) producerotation of the head gives the appearance of a full trajectories or have projectile motion. Once theadditional turn. But once the initial torque against body loses contact with the floor, the center ofthe floor has been exerted, and the dancer rises on gravity will follow a parabolic trajectory that isthe supporting foot, the angular momentum L is totally determined by the initial conditions startingconstant, decreasing only gradually because of on the ground.friction with the floor. The trajectory will follow the path of a normalFriction and the Glissade projectile- it will begin with an initial velocity Friction involves the properties of surfaces, (v0), which will decrease as the dancer rises into the air until it reaches zero. This is at theboth chemical and mechanical. Chemical adhesion maximum height during the leap. Following thisand roughness contribute to a frictional force that maximum is the descent to the floor which isacts in a direction tangential to the interface an accelerating downward motion until thebetween the surfaces. Each type of surface has its dancer reaches the ground again.own coefficient of friction. This coefficient offriction multiplied by the vertical force

In order to calculate the different parameters ofthis parabolic trajectory, we can employ two-dimensional kinematics. Both the “x“ and the “y”component must be considered in this case. Alsoas in all projectile motions, the initial velocity inthe x direction is equal to the final velocity in the xdirection This means there is no horizontalacceleration. Finally the acceleration in the y orvertical direction is always due to gravity (-9.8m/s2). In conclusion dance is truly dependent onphysical concepts such as momentum, Newton'slaws of motion, rotational mechanics, and linearkinematics. Analysis of Ballet and correctTechnique employs physics to maintaingracefulness, strength, and an overall sense ofspace required for all beautiful dancers.List of definitionshttp://www.geocities.com/CapeCanaveral/Hangar/4421/definitions.html© 1997 Sheila Dodge Boston University