The physics of Ballet

During February (2017) I was lucky enough to attend a lecture/demonstration jointly hosted by the Institute of Physics and the Royal Ballet held at the Royal Opera House about the link between physics and ballet.

The presenters were:

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Dr James Shippen, who is a mechanical engineer at Coventry University, specialises in applying engineering theory to biological systems;

imageCopyright: ROH Brian Slater

Sian Murphy (above left), who is assistant ballet mistress and a former first artist of the Royal Ballet;

Giacome Rovero (above centre), who has just joined the Royal Ballet’s Aud Jebsen Young Dancers Programme. He is wearing the special suit that will be described later in the blog;

Brian Maloney, who is a ballet rehabilitation specialist and class teacher of the Royal Ballet. He was initially a dancer and retired from the Royal Ballet as a soloist in 2013.

Now I have to admit that I am not known as an arts lover. I was once called a philistine by a woman at the Royal Opera when I booed a production of Aida and I can go round an art gallery in 10 minutes but I do have a lot of time for ballet (although I don’t know one step from another). It amazes me that female dancers can balance on their toes (en pointe) and I am mesmerised by the dancing of a good Corps de Ballet

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http://pointe-shoes.moonfruit.com/freed-dancers/4581997636

Dr Shippen began the session by defining some of the terms that he would use in his part of the presentation.

The term Biomechanics is made up of Bio (biology – specifically human biology) and mechanics (which in this case is to do with the movement of the human body). So we are applying physics to the movement of the human body.

Essentially there are two types of motion, linear where the movement is along a line where the speed and/or direction are constant and angular motion (of great importance to ballet) where the movement is circular around a fixed point or axis (the tip of a pointe shoe for instance). Of course you can get a combination of the two motions.

In order for the human body to start moving, change its movement or change its direction you need a force. A force is basically a push or a pull that alters or tends to alter, the state of motion of a body. This could mean a change in speed or a change in direction. In ballet this could be the whole of a dancer’s body or just a part of it. Isaac Newton’s first two laws of motion state that:

An object will remain at rest or move at a constant speed in an unchanging direction (uniform motion) if no net force acts on it (Newton 1 also known as the law of inertia);

An object will experience an acceleration (a change in its movement i.e. speed or direction) if it experiences a net force (force = mass x acceleration or F = ma). The acceleration takes place in the direction in which the force acts. If the mass of the object is constant then the net force is proportional to the acceleration (Newton 2).

As our bodies have mass we can exert a force on external objects just by standing on them (or sitting etc.). This force is called weight and is equal to the value of our mass (kg) x gravitational field strength (known as “g” and equal to approximately 9.81 kg/m2). In ballet the point of contact of the weight is via the shoe to the floor but there is also another external force of the floor on the shoe. It should be noted that we are only aware of our weight because of the opposing force to our weight. Newton had a law about this too:

If a body A exerts a force on body B then body B will exert a force on body A, which is the same size but acts in the opposite direction. When it was initially translated from the original Latin the law came out as “For every action, there is an equal and opposite reaction” but as a physics teacher I have found that this is often misinterpreted by students (Newton 3).

A force is a vector so it has size as well as direction. The direction is very important when adding forces as it is the net force that counts.

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The net force in the above example is zero.

Another example of an external force besides weight (force due to gravity downwards) and support from the floor (upwards is friction from floor (sideways). Friction is the force that resists relative motion between two bodies in contact. Friction is almost always necessary in ballet, but the dancer must apply the correct amount of friction for her movement’s to be successful. As a dancer begins her leap she needs friction so that she can push against the ground, because this friction allows her to accelerate forwards and upwards, due to Newton’s third law. The friction of the dancer pushing against the ground in a backwards and downwards motion has an equal and opposite reaction, causing the dancer to move forwards and upwards as she leaps. If the dancer used no friction then there would be no force allowing her to accelerate forwards into her leap, so she would not be able to move forwards and properly perform her leap. Importantly if there was no friction at all the dancer would slip.

http://physicsofballet.weebly.com/friction.html

Besides the external forces there are also internal forces in the body too. It is the job of the muscles, attached across joints to produce movement of individual body parts.

If any movement involves twists/rotations then torques are involved. A torque is a measure of the force that can cause an object to rotate about an axis. Just as force is what causes an object to accelerate in linear motion, a torque is what causes an object to accelerate in a circle (motion in a circle involves an acceleration i.e. angular acceleration, even if the speed is constant because the direction of the body is constantly changing). A torque is a vector quantity. The direction of a torque vector depends on the direction of the force on the axis. If a ballet dancer is rotating his/her whole body then this axis would be the centre of mass (centre of gravity). The centre of mass is defined as the point in space (normally within the body) where all the mass of a body appears to be concentrated. You can balance the body at this point and mathematically, it is the point at which the torques from the mass elements sums to zero. In other words the net torque is causing the body to rotate but the centre of mass does not rotate because there is zero torque there.

If the dancer is not turning then in order for her to stay balanced the centre of mass/gravity must remain directly above the area of contact with the floor. Then gravity’s downward push and floor’s upward push will go through the dancer’s centre of gravity and there is no total force and no total torque. If this area is very small (the dancer is en pointe), it’s harder to balance.

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The size of the torque depends on the size of the force and/or the distance the force is from the axis or rotation/ pivot point. If a dancer is rotating his/her arm then the pivot point is the shoulder joint. Other pivot points are found in the back, lumber spin for example.

Examples of dance torques include arabesque, a body position in which a dancer stands on one leg (the supporting leg) with the other leg (the working leg) turned out and extended behind the body, with both legs held straight. In classical ballet, an arabesque can be executed with the supporting leg en pointe or demi pointe or with foot flat on the floor. The axis of rotation is the hip (if the dancer is rotating the leg) and the torque is generated by forces in the muscles off set from the hip. The more the leg raises the tighter the hamstrings and the force in the hip increases.

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In fouette turns, a dancer plies with her leg extended, then as the she relevé, her leg moves around (rond de jambe) out to second position before bringing her leg to passe. This leg movement creates torque and momentum for the dancer’s turn – the faster her leg whips around, the faster she turns. In a pirouette, a dancer begins from a deep plie in fourth position. Then the she springs from her plie into relevé and brings her arms to first position and her leg to her knee. This movement of her arms creates momentum. The most important part of a pirouette is the dancer making sure she has a deep enough plié to create enough momentum to get her around. If the plié isn’t deep enough there won’t be enough momentum. Another thing that helps with turning is spotting, which is the dancer whipping her head around, keeping her from getting dizzy whilst turning. It is also used to keep her head at the axis rotation and helps her keep her centre of balance. Rosin is used to stop dancers from slipping or sliding while dancing. It creates friction between the satin of the dancer’s shoe and the wood of the floor

An interesting aside is that Irish dancers can experience far worse forces than ballet dancers. Dr Shippen’s work showed that when they “rock” they can experience a force 14 times their body weight through their ankles. Chorography had to be changed to reduce this force to 8 times their body weight. It is important to understand these large forces in order to reduce the likelihood of injury.

Other physics terms involved in dance include energy which equals force x distance and, in the case of a dancer leaping upwards, equals the mass of the dancer x height reached x gravitational field strength (g). For a dancer the energy required for this movement is transferred via the muscles, which contract. Power which equals work done divided by time taken is the rate at which the dancer uses the energy.

Momentum is mass times velocity (velocity is a measure of how fast something is moving/speed in a given direction). If an object has a large momentum, it’s hard to stop it! To change an object’s momentum, you have to apply some kind of force for some time! So change in momentum = force x duration of time the force was applied. No force means no acceleration or change in momentum.

Rotational inertia is the inertia of a rotating object and indicates how difficult it is to start an object spinning (or to stop it, if it’s already spinning). It depends on the mass of the object, the greater the mass, the greater the rotational inertia. It also depends on how far the mass of the object is placed from the rotation axis. If this distance is doubled, the rotational inertia gets quadrupled!

Inertia in general is the tendency of an object to keep doing whatever it is doing; the greater the inertia the greater the difficulty of starting an object moving, stopping it moving or changing its direction.

Angular momentum is rotational inertia times angular velocity (L = I x ω). If an object has a large angular momentum, it’s hard to stop its spinning! I is a measure of where the dancers mass is. Torque is required to change angular momentum. Angular momentum is a conserved quantity and is a constant if there are no torques acting on a body. In other words, the total angular momentum L of a body can change only if there is an external torque acting on it. There is no way that changes in body position alone, representing changes in the configuration of mass within the rotating system, can change the total magnitude of L. In other words if no torque is applied, the angular momentum stays the same and since angular momentum = rotational inertia times angular velocity, the greater the rotational inertia, the smaller the angular velocity!

This explains why the angular velocity (ω) of a rotating ballet dancer increases if she brings her arms towards her body when the moment of inertia deceases.

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You can see Marianela Nunez showing this by clicking the link seen below,

A pirouette, in general, is a rotation where the dancer supports herself on one leg while the other leg is in releve’ position. Any pirouette 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. Increasing the number of pirouettes increases the amount of torque, momentum and velocity required.

A fouette involves:

1) The dancer starts to turn – her arms are brought together (r small, rotational inertia small, so large angular velocity)

2) The dancer stops for a moment by extending her arms and leg (r large, rotational internal large, so small angular velocity)

3) The dancer continues turning – her arms are brought together (r small, rotational inertia small, so large angular velocity)

Change in angular momentum = torque x duration of time the torque was applied

As mentioned before friction can also create torque. When the dancer pushes the floor one way, the friction between the leg and the floor creates the push the other way.

The laws of physics are essential for understanding what is going on during a dance and for improving energy use and preventing energy.

Dr Shippen’s work was initially developed in hospitals branching into sports science in the mid-1990s. It is now being increasingly used in dance to show how injury can be reduced.

The videos that can be found by clicking the links below summarise the processes involved.

http://www.danceuktv.com/biomechanical-analysis

Motion capture is used with optical tracking. Small white balls are attached to the dancer and cameras measure the location from monitoring them. Angles are measured and trigonometry is used. Computers are used to work out the 3 dimensional motions to the nearest mm.

imageCopyright: ROH Brian Slater

The above image shows Giacomo wearing a suit containing about 17 sensors. Once he was “calibrated” a simplified stick-like video of him was produced by a computer and displayed in real time (as can be seen behind him in the image). In this his centre of mass was shown as a small green sphere. The sensors include linear accelerometers monitoring three directions, gyroscopes and magnetometers. Together they tell where a dancer is in space and enables the computer to work out where the dancer goes.

Other images can show which muscles are under the most stress. The red areas in the following images show the most stressed muscles and the blue areas are the least stressed.

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The above images were kindly given to me by Dr Shippen

The tracking system can also display the trajectory of the moving dancer’s body as well as how parts of the body move, e.g. leaps such as the sissone. The tracks show the motion of the hands, feet and centre of mass.

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We were shown an animation of a dancer performing a sissone, a grand jeté to attitude, a double revoltade, a double saut de basque and a 540 jump. These showed him as a skeleton with key muscles at work – those bearing the most load were coloured red while those bearing a medium load were green and those bearing no load were coloured blue. The five leaps may look totally different but the same physics applies.

The force due to gravity plays a major role in jumps. The total effect of gravity is the same as if it were acting on the dancer’s centre of mass/gravity only. Gravity only affects vertical (not horizontal) motion and the gravitational force is proportional to a dancer’s mass.

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The vertical position of the dancer’s centre of mass against time describes a parabola. There is no force in the horizontal direction, so the horizontal position is a straight line if it is plotted against time.

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During a revoltade the pelvis doesn’t move but the centre of mass does, along a parabola.

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While the trajectory is always the same, some dancers can create an illusion of floating in space. This is done by raising the legs, arms and adjusting the head mid-leap, i.e. raising the arms and lowering the head or vice versa. The vertical motion of the head is smaller than the vertical motion of the centre of gravity. So changing the centre of mass in relation to the head (as shown above) fools the eye into thinking that the dancer is floating in air for a fraction of a second.

The dancer pushes off with the leg muscles coordinating with the arms to stay in the air for as long as possible

http://ed.fnal.gov/trc_new/demos/present/physofballet.pdf

http://iceskatingresources.org/physicsballet.pdf

https://prezi.com/lezj71_tope9/ballet-physics-in-motion/

At the end of the Dr Shippen’s part of the presentation questions were asked. Here are some answers:

The dance floor is sprung to reduce injuries. It is a sort of waffles structure made up of an absorbing material to absorb energy.

Choreographers tend not to understand the physics but have inherent knowledge.

Biomechanics research is important to help with treating dancer’s injuries, helping with rehabilitation and monitoring the dance to prevent injuries happening in the first place.

Giacomo thought the monitoring process would help improve his technique.

The tracking information enables movement, torques, forces in joints, forces between the feet and floor, energy and power to be measured.

Dame Monica Mason, former artistic director of the Royal Ballet, wanted to know what the green colour on the tracking images meant. It actually shows the medium stress that the muscles experience. I wondered if she thought it would be useful to see which dancers weren’t putting enough effort into their dancing.

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https://en.wikipedia.org/wiki/Monica_Mason

The technology could get rid of the need for ballet notation if there is just one dancer.

Holding a partner up at a constant height doesn’t involve a change in external energy but the muscles inside the body require energy.

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