Angular Momentum & Rotational Motion
Momentum: Inertia in motion. It's the product of the mass of an object times its velocity.
Angular Momentum: Rotational inertia times rotational velocity.
Impulse: It's the product of force and time interval. The shorter the time the stronger the force, and viceversa.
Rotational Motion
Rotational Speed: Is the number of rotations per unit of time.
Radial Distance: Distance from the axis of rotation.
Rotational Inertia: The property of objects to resist change in their rotational state of motion.
Torque: The force needed to change the state of motion of a rotation.
Center of Mass: The average position of all the mass that makes up the object. It's proportional to the center of gravity.
Centripetal Force: Any force directed toward a fixed center.
Applying the terms in:
- Balance:
For a dancer to have balance, the dancer must keep his or her center of mass/gravity above the base area to have stability. This means that for example, if a dancer was doing an attitude, (Figure 6) her center of gravity/mass must be directly on top of her right foot, which is the base, to give her stability and balance.
In the same way, the idea of this position is to make an arc with the leg that’s folded (the left leg), and feel like it is trying to get to the opposite shoulder. In this way, as all of the weight of her body is at one single foot the other foot needs to put some of the weight to the opposite side to keep her center of gravity directly on top of her foot and keep her balance.
Her hips, on the other hand, need to keep facing the front and form a perfect square in order to keep her balance, because a misplacement of the hips to either side would cause the center of gravity to shift and cause trouble in the dancer's stability.
Figure 6
- Turns:
Fouettes: To start the turn the dancers push on the floor, which pushes back with an equal and opposite reaction (Newton’s 3rd Law), to generate torque, which allows dancers to start the turn, but the friction between their foot and the floor cause them to lose speed, and momentum, which is why after every turn dancers pause, face the audience, and their foot flattens as they push on the floor again to generate more torque. In the same way, their arms open and close again which also increases their momentum. Their other leg also extends and folds after every turn to gain more momentum.
As their legs move from the front to the side it is storing momentum that is then used by dancers when the leg folds back in. This sometimes allows dancers to perform more than one turn out of every leg extension, because the longer the leg is extended, the more momentum it stores, and the more momentum is given to the body when that leg retracts, allowing them to turn more than once out of that single extension. This happens because of the turn’s angular momentum, which is measured by multiplying the rotational inertia, times the angular velocity. So the rotational inertia is the resistance of a dancers' body to rotational motion, this inertia is increased by increasing the radial distance, or the distance between dancers, or in this case their arms and the legs, form the axis of rotation. This is why after extending the arms and legs they retract them again quickly, decreasing the rotational inertia, so the velocity of the turn must increase in order to maintain the angular momentum. (Figure 7 & 8)
Figure 7
Figure 8
It is important for dancers to keep their center of gravity directly on top of the base, which is their foot, to maintain their balance. For this purpose, the dancers focus on a specific point and turn their heads as quickly as possible to maintain stability.
In this other turn (Figure 9), the dancer does not retract the leg for the first turns so all of the torque comes from the base foot that flattens and pushes on the floor again after each turn, and from the arms that open and retract. All of this is done to gain more rotational velocity, to maintain the angular momentum, because as the leg is extended throughout most of the turns, the rotational inertia is greater. The dancer retracts the leg slowly and extends it again once to gain more momentum and perform more turns, before continuing his turns a la seconde (with the leg extended).
Figure 9
