Physics in Dance and Dance

Physics in Dance and Dance

to Represent PhysicAL Processes

CAPOCCHIANI Vilma, (I), LORENZI Marcella (I), MICHELINI Marisa, (I),

ROSSI Anna Maria, (I), STEFANEL Alberto (I)

Abstract. Physics and dance are, apparently, very distant topics. Nevertheless dance offers to physics teachers many opportunities to teach physics in a motivating way, showing how models in physics are constructed and how they are at the base of all human reasoning in interpreting phenomena and constructing theories. For instance, physics can be used to describe in a very simply way complex dance motions, such as pirouettes and artistic jumps. From another point of view dance offers the opportunity to interpret and represent physical processes, through the opportunity to introduce dynamic actions.

Our research based activities were developed in these two directions. On one hand we implemented an educational path based on the questions emerging while dance movements are observed, as starting points to study the dynamics of systems. On the other hand an interdisciplinary team developed a choreutic performance representing some physical processes, in particular “time” and the process of understanding what time is.

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Mathematics Subject Classification: Arts, music, language, architecture 97M80, Physics, astronomy, technology, engineering 97M50, Models of real-world systems 91B74, Teaching units and draft lessons 97D80

1 Introduction

Usually in Physics Education we take into consideration some processes that happen under controlled conditions, often apparently far from everyday life. In particular, in teaching mechanics, the tendency to simplify or emphasize the study of dynamics and kinematics using one-directional motion is very common. Problems in learning reported by the students of different ages seem to be strictly correlated to similar reductionist approaches [1-6] and to the disconnection of the scientific knowledge from everyday and experiential knowledge [7]. On the other hand, positive results in learning are well-documented in approaches starting from situations not far from young people’s experience, proposing the study of physics in contexts [8-14], using new technologies to realize active and explorative strategies [3,15-20].

Dance, which apparently is a context very far from physics, offers from this point of view very interesting hints to approach different areas of physics in inspiring but not trivial ways. As a matter of fact, it involves very complex and apparently surprising movements, as the acceleration of angular velocity of a dancer performing a pirouette or the apparent horizontal trajectory of the jumps of the dancers, that physics can explain by means of the construction of more and more complex models. Therefore dance offers to physics teachers many opportunities to explain in a motivating way, showing how models in physics are constructed and how they are at the base of all human reasoning in interpreting phenomena and constructing theories [21-23]. Moreover, it allows to create meaningful relations to other scientific disciplines, as mathematics and biology, or to other fields, such as humanities, the arts and physical education. From another point of view, dance offers the opportunity to interpret and represent the physical processes, introducing dynamic actions and exploiting the metaphorical character of the choreutic performance.

Our research-based activities were developed in these two directions. On one hand we implemented an educational path based on the questions emerging while dance movements are observed, as starting points to study the dynamics of systems. On the other hand an interdisciplinary team developed a choreutic performance representing some physical processes, in particular the concept of “time” and the process of understanding what time is.

A discussion about how these activities were carried out follows.

2 Models for interpretation of dance positions and movements

This sections presents the contexts and the nodes in which our teaching activities have been developed. The core of the proposal is based on the analysis of the external forces and their momenta exerting on the dancer’s body, given by the weight force and by the push of the floor on the feet of the dancer, and on the construction of simple physical models to represent the processes taken into account. These models are based mainly on the material point model and on the rigid body model, and do not take into account the interaction between the parts of the body. Only for the analysis of the jumps a simulation based on a mechanically linked two rigid poles model. The model analysis here presented has been conducted following an experimental approach with high school students, as follows.

2.1 The equilibrium of a ballerina

The movements of a ballet dancer, as every movement of a human body, starts from a situation of equilibrium that, at a certain point, is broken. However, before dealing with motion, one might ask: which is the state of equilibrium of a body?

In the case of dance, ballet style, there are few basic positions and several dance steps or moves, among which the ballerina balancing on one foot is particularly interesting. They maintain their balance by positioning their centre of mass over their base of support. That is, the body should assume a correct posture (which differs depending on each person and on the assumed position) in order to maintain the centre of mass over the supporting base. In the case of the first position, in which the heels are touched together and the toes are turned out at an ideal 180 degree angle, the ballerina is in a state of relative equilibrium with respect to lateral movements, while she is less stable for back and forth motions. In the fifth position, the feet come together, turned out in the different directions, the toes of each foot reach the heel of the other, touching against each other: this position is not as stable as the first one for lateral movements, but it is better for back and forth shifts.

In the case of a ballerina balancing on one foot (“en pointe”), every little movement is crucial, since the centre of mass can easily go out the base of support, determining a new situation in which a couple of forces (weight and vincular reaction of the supporting plane) take action on the body of the dancer, whose momentum is not null. If we assume that the centre of mass of the ballerina does not change remarkably the distance L from the base of support (the surface of the ballet shoe), the angular acceleration acquired by the body of a dancer in an unbalanced position could be evaluated applying the model of the rigid body:

L/2 Mg sen q = ML2/3 dw/dt (1)

in which M is the mass of the dancer, q is the angle between the vertical direction and the line between the centre of mass and the point around which the body rotates (Fig. 1).

Following the hypothesis that sinq ~ q, the angular acceleration of the ballerina is given by the equation:

d2 q /dt2 = (3/2)(g/L) q (2)

where one could obtain for the angular velocity w=dq/dt~ (3g/2L) 0.5 q°, that is with L=1.8 m, w ~ (2.9) q° t, with q° as initial angle. Assuming the model of the material point we obtain w ~ (3.3) q° t. An horizontal movement of the centre of mass of 5 cm implies an initial angle of q°=3° that would suddenly lead to a loss of balance of the ballerina and would thus require a correction of the position directly performed by the dancer

2.2 The motion of a dancer

How does a dancer start moving? He/she must accelerate, being thus subject to external forces. The only force that can act from outside should be exerted by the floor. There are two fundamental mechanisms. In the first case, from a position standing on two feet, one ahead the other, the dancer passes to a position in which the rear foot exerts a load on the floor. Following the third law of dynamics the floor will push the body forward. This mechanism is typical of walking. The second mechanism implies that, starting from joint feet, the dancer lets his/her body fall forward. The more the centre of mass is forward in respect to the feet, the bigger the push that it is possible to exert. This is the mechanism adopted by the sprinters in athletics. A third mechanism is typical of dance. It consists in lifting forward a part of the body, for example a leg as in a movement of degage. The throw produces a reaction of the other leg, that would have a tendency to rotate backwards, but, being fixed on the floor, it pushes it backwards. Once again, following the third law of dynamics, the dancer will advance thanks to the reaction of the floor. The movement on a circular trajectory implies a centripetal acceleration that determines the shift of direction of the velocity vector during time. This acceleration could only be given by the external force exerted by the floor on the support foot of the dancer. In order to receive this push the dancer bends his/her body in the same way the sprinters that face the stretches of bendy athletics track do. Using a simple model it is possible to establish that the angle of inclination q of the dancer in relation to the vertical is given by tgq= v2/(rg) where v is the linear velocity of the dancer, r the radius of curvature of the trajectory, g the gravity acceleration. It is thus recognized that the angle q is independent from the mass of the dancer and depends only upon his/her velocity. In the case v is 5 m/s (half the velocity of a sprinter) on a circular trajectory of radius 5 m, a dancer should lean of 26.6° in respect to the vertical.

2.3 The jumps

The vertical jump can be easily described in an analogous way to the bounce of a ball on the floor and can be analyzed correspondingly to the jump of a high jumper [21]. Also in this case the thrust Fs must be provided by the floor, as reaction to the push of the dancer to the floor. Given that Fs = R * P, with R an a-dimensional coefficient, R must be R>1 so that the dancer can leave the ground, or R=1 when the dancer is still. The thrust Fs has an effect on the dancer since his centre of mass is in the lowest position (height h1) to the moment in which he detaches his feet from the floor (height h2). If we use the material point model in order to describe the dancer, during the phase of thrust the work of the acting forces is given by:

W = Fs d - P d, with d = h2 - h1 (3)

The kinetic energy theorem provides the kinetic energy at the take-off: Fs d - P d = ½Mvo2, with vo velocity at take-off and M mass of the dancer.

In the phase of flight, the centre of mass goes from h2 to h3, that is the rise is H = h3 - h2, under the action of barely the weight force. The variation of kinetic energy during the phase of the flight should be equal to the variation of the potential energy of the dancer:

0 - ½ M vo 2 = - P H (4)

From equations (3) and (4) we obtain an expression for H depending only upon d and R:

H = d ( R-1) (5)

that is, the height of the jump H = h3 - h2 is equal to the height of thrust multiplied by the factor R-1. The heights reached for R=1 (thrust equal to P), 1.5 and 2 (thrust equal to 2P), are respectively H=0, H=d* 0.5 (half the height of thrust), H=d; a jump of 1 m with a bending of 0.5 m requires a push equal to R = H/d+1 = 1/0.5 +1 =3, that is Fs = 3P.

Since the only force that is acting on the dancer during the flight is the weight, the time of the flight depends only on the height H of the jump and is independent from both the mass and the typology of dancer. If we take into account that the motion during the flight is uniformly accelerated , it is simple to realizee that the flight time is given by t=(2H/g)0.5. Since from that relation it turns out that Dt/t=0.5 DH/H, a percentage variation in the time of flight Dt/t implies twice the same percentage variation in the height of the jump DH=H.

So how to jump higher? In order to effectively bring the center of mass at an higher level it is possible to fling the arms high at the same time of the phase of thrust, so to acquire quantities of motion during the phase of take-off. In order to increase in a simply way the distance of the foot from the ground, it is possible to gather the legs together during the flight. In the second case the effect is that of giving a perception of a higher jump, in comparison to a jump with stretched legs. The same effect is obtained by the ballerinas that perform a Grand jeté: since they splay their legs in flight, the visual effect is that of a fluctuation in the air.

Figure 2. Simulation realized with Interactive Physics evidencing the parabolic trajectory of the center of mass of the system and the near horizontal trajectory of the snodable constraint.

The same happens in the case of a vertical jump made by a ballerina, in which she is subject only to the weight force. The flight of the centre of mass occurs on a vertical plane defined by the vertical and the direction of the velocity at take-off and cannot occur on a curved horizontal path. The trajectory of the centre of mass is parabolic and it is determined by the bare initial velocity at the moment of the detachment. The motion of the body of the ballerina in relation to the centre of mass, that determines mainly the visual effect of the jump, satisfies the principle of conservation of angular momentum. Therefore during the jump the dancer looks like fluctuating in the air since, while her centre of mass describes a parabola, her trunk, that lowers in relation to the centre of mass, practically describes an horizontal trajectory. The same effect is obtained by the hurdlers in athletics. On the other hand, in the high jump following Fosbury flop technique, the jumper bends his body in a a certain way so that the centre of mass of the body does not clear the bar.