Tuesday, April 23, 2013

Vector Diagrams & Analysis

Vector Diagrams & Analysis

An Introduction to visualizing force, frictions, acceleration, and displacement

Through out our study of the Human Body and the many directions we are capable of moving, we will us vector analysis to visualize the Physical aspects of every type of movement.  Before you are able to diagram the physics of any situation, you must be able to differentiate between scalar and vector quantities. Note that in science, we use the International System of units, abbreviated SI Units, i.e. meter not yards, kilograms not pounds, etc. 


In Newtonian Physics, A physical quantity that is not changed by coordination system rotations or reflections is known as a scalar.  (The Coordinate system is best represented by the unit circle pictured above in the Mathematical Movement banner.) Basically scalars have no directions; however, it is more than a number because physical quantities are an expression of numerical value and a physical unit.


Vectors have many uses from cross products in three dimensional space to multivariate random variables in statistics; as well as, applications in computer science. More commonly referred to as a Vector this geometric entity has many names: Euclidean, Geometric, & Spatial vectors.  A vector is a physical quantity represented by an arrow that has a magnitude and a direction.  Magnitude is the numerical value in physical units represented by the tail of the arrow.  The larger the magnitude the longer the tail. The direction  is at times given or easily interpretative   In physics, you should be careful with using intuition.  The best way to determine direction is to look at the signs (positive or negative) in which positive quantities move up or right and negative quantities move down or left.


When faced with any problem to solve, I encourage my students to always list the constants and variables (both known and unknown) first.  In physics, I require this first step. Second, you label each item as a vector or scalar quantity. Then we exercise the right side of the brain by drawing the object in the given environment and diagramming the physical quantities involved.  If you are not interested in exercising your artistic side, simply draw a square or circle to represent the object is acceptable. Below is a simple example. We will study difficult analysis as we investigate the human body in motion.

e.g. A ball with a mass of 0.5 kg is dropped 2 meter in 3 seconds with negligible air resistance.  
  • mass (m) = 0.5 kg ---> Scalar
  • distance (d) = 2m ---> Scalar
  • drag (Fair) = + .0000634 N ---> Vector
  • time (t) = 3 s ---> Scalar
  • weight (Fgrav) = (0.5) * (-9.8m/s^2) = - 4.9 N ---> Vector
By using a vector diagram it is easy to see how different forces affect the given object.  In the above example, the force of gravity is stronger than the force of air resistance. we can predict the motion of the ball by comparing the tails of each vector. The longer vector represents the stronger force.  Gravity will overcome the air resistance and the ball will fall until contacting the grounds. Vector diagrams can be used to observe the direction and magnitude of any vector quantity including velocity, acceleration, gravity, friction, applied force, etc. 


Now the you understand a basic vector analysis, you are ready to explore the various vectors that create and resist movement.  In this blog, I will break down many dance moves using vector diagrams. By looking at our activities analytically, you will better understand what makes a move difficult.  In more advanced diagrams, we will see how angles affect a given object which will help us visualize the proper body mechanics of movement from walking to dancing.   

Monday, April 22, 2013

Absorbing the Shock




Jumping, my favorite aspect of dance, is one of the most exciting moves we are capable of making. For me it is the sensation of flight that I enjoy. For animals it is a means of transportation  and often a tool for athletes to score, dive, catch, flip, dodge, etc. Be it a Wide Receiver jumping to catch a football or children jumping at the sight/sound of an Ice Cream Truck, It is exciting to watch and when we are overcome with excitement jumping is a natural outlet for Expression.    

Professional Athlete, or not, it is important to understand the mechanics of Jumps to prevent injury and sustain a long health life. The danger in jumping is landing; which is actually a collision with the floor. There are many physical factors that help our bodies cope with this collision and adsorb the shock.  
Let's Compare the two Images Below: 
Image A: High Speed Head on Collision 
Image B: Fender Bender in a Parking Lot 

It is obvious there is much more damage in Image A than Image B. A big factor in car accidents is the velocity at which the impact occurs.  Mechanical Engineers do there best to design a body that will resist the impact and protect the passengers  but our best way to stay safe is abiding by the speed limits. Another variable is mass the larger vehicles had more damage than the smaller. And the final differential is direction and displacement of Impact. Image A the vehicles hit straight on moved the vehicles off the road.  Image B there was little displacement and the contact is at an angle.


Do you get the picture? we observed how theses variables relate to car crashes, but what does that have to doe with Jumping.  Below we will explore how the same variables above relate to landing of a jump on the basketball court or descending from a grand jete or simply not slipping when you jump to reach for an object off a high shelf. Below we will examine the physics of collisions and analyze the mechanics of Jumping. 
  • Mass is an amount in kilograms of an object relative to the amount of gravitational force applied. Our ability to jump well is related to our weight, or the amount of gravitational force applied to the collision with the ground in landing.   Yes, it is hard to put math in to words that is why we rely on Newton's Second Law of Gravity 
    • Force (weight) = mass * g, where g=9.8m/s^2
    • Healthy eating and exercise go together because our weight effects our Joint's ability to support our bodies during vigorous activity.  Good Nutrition (not starvation) is important to providing the energy for Jumping
    • Our joints are like the suspension system on a vehicle.  Vehicles have there towing limits just as our joints have limits for the weight they carry. 
  • Hooke's Law is a principal of Physics that explains the force need for displacement (extension or compression) of a spring proportional to some distance.  For every material, there is a constant characteristic that represents the Stiffness or elasticity.  
    • Force = kx, where x is the distance and k is the constant characteristic
    • Oppose to an object, our bodies are constantly changing; therefore, our joint's level of elasticity is constantly changing.  Our Skin is the most elastic part of our bodies and is the first line of defense when we collide with the earth.  Second is our joints, a series of springs, that cushion the descent of a jump.
    • A positive change could be managing a healthy weight; have you ever notice after regular exercise you have more spring in your step?  A negative change is from compensating muscles to cope with an injury. 
    • Improper body mechanics will deplete the joint of collagen and may cause serious injury. In physics distance is a vector quality represented by a straight line in a given directions.  Our goal as movers is to keep each joint in a straight line. I like to visualize three parallel: two from ankle to shoulder and one up the spine.  Imaging the middle line longer than the other two, and if it isn't too foreign see it moving infinitely down & up to provide a strong central axis.  
  • Are you familiar with the phrase what goes up must come down? Or maybe you know Newton's says for every action there is an equal and opposite reaction. The same force to accelerate upward must be equal to the force of resistance when colliding with the floor.
    • This concept is from Newton's Third law of motion:  F1 = - F2  
    • Bending and straightening our legs (or Plie), Forces our body upward, slowing down at a rate of 9.85 m/s^2, untile our body reaches a velocity of zero before it changes directions, accelerating downward at the same rate of 9.85 m/s^2. The rate of acceleration due to gravity and our bodies mass is constant; therefore, the force is equal and the direction varies. 
    • While suspended at the peck of the jump, there is a sensation of flight. It is an amazing feeling, yet can be destructive and a distraction.  Eager to fly, I tend to apply as much force as possible, and when finding that suspension I want to linger which results in a rough collision with the floor.  
    • To prevent a high-speed head-on crash with the dance floor, My teachers always coached me that the beginning and end of a jump should have as deep of a plie
Each of these variable and constant in physics can be related to the body's function to jump.  If you struggle with one particular aspect of jumps, the body naturally compensates.  The above description is an ideal demonstration and takes diligent effort to achieve. When practicing your long jump, slam dunk, or jete always remember the landing is a collision. Take care of your body to sustain our natural ability to absorb shock. Practice is not always accomplished by action. The best way to protect our bodies and absorb shock  is from conditioning to strengthen and stretch the muscles around the joints.