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The Physics of Fitness: Part 1

By Chris Comfort, CFT 2 ,C.H.E.K. Practitioner

----For many of you, just the sound of the word "Physics" makes you cringe. It is one of those high school or college classes that we were required to take and we probably categorized it with the statement," This is a such a waste of time or I'm never going to use this in real life". I personally remember sitting in the back of my high school class, with several other jocks, all mumbling these exact words and maybe adding a few other expletives. Well, little did we realize or appreciate for that matter, that the Laws of Physics affects us on a daily basis. There is not a single moment during our lives that we are not being acted upon by some form of Physics. The irony is, that at least with mechanical physics, real life is exactly what it does apply to. Gravity, inertia, friction, and ground-reaction forces are just some of the qualities of nature that are relentlessly inflicting themselves upon us. Put us in the gym or on a playing field, and their effects are magnified.

----This article is the first of a three part series that is intended to shed some light on the subject of physics, mechanical physics to be exact. Part 1 will be a simplified introduction to some of the basic concepts, laws and formulas of mechanical physics. In Parts 2 and 3, we will apply these laws in the gym and in several sport-specific situations to further explain how this knowledge can be applied to better focus our training efforts.

----We will begin our discussion with an introduction to Sir Isaac Newton and his Laws of Motion. Despite their simplicity, these laws are the fundamental roots of all modern day theories of mechanical physics. Listed below are the laws:

1.) The Law of Inertia : A body in motion or at rest tends to continue in that state until acted upon by a force.

2.) The Law of Acceleration : The amount of change of motion of an object is proportional to the amount of force impressed upon it and is made in the direction of the straight line on which the force is impressed.

3.) The Law of Action-Reaction : For every action there is an equal and opposite reaction.

4.) The Law of Universal Gravitation : The gravitational pull of one object to another is equal to the product of the masses of the two objects divided by the square of the distance of them from one another. (For simplicities sake, we can think of the acceleration that gravity imposes on all objects here on Earth as: g = 32 feet/second/second or 9.81 meters/second/second.)


----Add to these the Law of the Conservation of Energy, which says:

----Energy can neither be created nor destroyed; only transformed from one form to another, but the total amount of energy never changes.

----Although often referred to as the "first law of thermodynamics ", it rings equally true with all forms of energy. It not only applies to the universe as a whole, but also to any closed system. In the collision of two objects, an example of a closed system, the total kinetic energy of the 2 objects (the sum of their momentum) just before the collision must equal the total energy just after the collision. Some of the kinetic energy after the collision will result in flying pieces, crumpled metal or flailing limbs.

----Energy can take on many forms. You are probably familiar with, or have at least heard of chemical, thermal and electro-magnetic energy. Since we are talking about mechanical physics, we will be focusing on mechanical energy, which in itself, comes in two forms. The first form is kinetic energy, which is the energy of motion and objects in motion, and is represented by the following formula:

(1) KE = 12mv2 ---- (Where m is the mass of the object and v is the velocity that the object is moving at.)

----The second, potential energy (PE), is the energy due to position. Potential energy can be further separated into gravitational potential energy , represented by:

(2) PE = mgh ---- { Mass (m) times the acceleration of gravity (g) times the height of the object off of the ground (h) }


----An apple falling out of a tree would be an example of gravitational potential energy being converted into kinetic energy. Strain energy (SE) is the energy of distortion or elasticity. An example of this would be the bending of a pole in pole vaulting or the stretch of an elastic band. Both have a tendency, or potential, to "spring back" into their original shape. This is represented mathematically as:

(3) SE = 12k"x© ---- (Where k is the stiffness or spring constant of the material and ."x is the change in length or deformation of the object from its undeformed position)

----Energy by itself doesn't mean a whole lot to most of us until it either acts on us or on the objects around us. That brings us to force.

----A force (F), in simplest terms, is a push or a pull, and is represented by:

(4) F = m(a) ---- (The mass of the object times the acceleration of that object)

----We will look at two categories of forces: internal (muscular contractions or momentum of a body) and external (anything else, including gravity, that externally affects the person or object in question). In human activity, especially in athletics, there are multiple forces acting on an object (person) in multiple directions at any given time. Colinear forces act with or against one a other along a single line of force. Concurrent forces act on one point at the same time along different lines of force. As we learned from Newton, a single force acts on an object in a straight line. Therefore, any time we talk about the force of an object, we must take the direction of that force into consideration. This is where speed and velocity differ.

----While speed measures how fast a distance is traveled, velocity gives us the speed and provides a sense of direction. The number value for velocity, like speed, is given as a distance divided by the time it takes to travel that distance, like miles-per-hour or meters-per-second. Lets look at the movements of a football running back, for instance. If he is running straight down the field at a steady speed of 10 mph, his velocity is 10 mph downfield. Now, if he is suddenly hit in such a way that he is thrown directly sideways, his velocity has changed. Even if he somehow happens to be moving sideways at the same speed as he was when he was running downfield, his forward velocity becomes zero and his lateral velocity goes from zero to 10 mph. This change in velocity is what is known as acceleration(a).

----Most of us know acceleration as an increase in speed. This is only partially true. Acceleration is, in fact, any change in velocity. And because it is directly related to velocity, it is also contingent on direction. So, acceleration is any increase in velocity, decrease in velocity, starting, stopping, or any change in direction, as well as any combination of these. It is represented with a value such as feet per second©. An example of this would be the acceleration of gravity, which again is 32 feet per second per second (or second©). This tells us that if something is in a freefall towards Earth, it's velocity increases by 32 feet/second for every second that it falls. So, if something falls for 3 seconds before it hits the ground, it's velocity just before impact (it's maximum velocity) would be:

(5) 32 ft./sec.© x 3 seconds = 96 ft./sec.

(Notice that the value for gravitational acceleration is a positive number. This means that the falling object increases in velocity as it falls. If the number were negative (which happens to not be possible with gravity, since it is a constant) we would have negative acceleration which is a decrease in velocity over time.)

----Looking at Newton's law of inertia, we can take our new understanding of acceleration and say that inertia of an object is its resistance to acceleration (again, any change in motion). An object in motion has momentum, which is inertia in motion. Newton's first three laws of motion can be combined together to give us a kind of "law of momentum". Once an object is in motion, it will continue to stay in motion until acted upon by another force that significant enough to cause a change. So, in a sense, the object is resisting change. Since any change would be considered acceleration, its momentum is inertia (again, the resistance to any acceleration) in motion. Then, Newton's second law tells us that the "momentum" of the object is directly proportional to the force that acted on it to cause its movement (or more accurately, its acceleration). This may all seem like a lot of double-talk for something that seems so obvious, but it will help us to better understand the interrelationships between these concepts.

----One thing that the second law doesn't directly point out, but is implied, is that the "amount" of force is a product of the magnitude (strength) and the amount of time that it is applied. Since that quantity given for a force is its magnitude, we can say mathematically that:

(6) Momentum = F(t) (Force times time)

----It is also important to understand that

(7) F(t) = m(v) (mass times velocity)

----This gives us what is called the impulse of the momentum. That is, the momentum of an object can be increased if either the force is increased or the amount of time that the force is applied is increased. This will be explored further in parts 2 & 3 of this series.

----A great example of inertia and its relation to the third law of action-reaction is seen when you try to push something heavy, like a bookcase. Provided you possess the strength necessary to push it to begin with and you have a good footing, the bookcase will eventually move. If, however, you are wearing sox and standing on a hard wood floor, your pushing force may reverse its effect and cause you to slide backwards. Of course, even if you move the bookcase, the forces that go through your wrists, elbows and shoulders, through your trunk, hips and knees and translate from your feet into the floor is exactly equal to the amount of force that you are pushing with. There is an exception to this when there are energy leakages in the body, which cause a redirection of a portion of your effort. We will also see more of this later in the series.

----Whenever a force causes a change in the position of an object, like when you moved the bookcase in the above example, work is being done to it. Work, in this sense, is the means by which energy is transferred from one object or system to another. (This is work in the quantitative scientific sense and has no relation to how much exertion that you put into the task, or how hard you are working.) The amount of work (U) is equal to the force (F) applied times the distance that the object was displaced (d):

(8) U = F(d)

----Power(P) is the work done divided by the amount of time it takes to do the work:

(9) P = U/t

----Power is one of the most important qualities in most sports and will be one of the main focuses in the continuation of parts 2 & 3.

----All of the forces described so far have been in a straight line, as any one force always is. What about motions that do not move in a straight line, like just about every joint motion in the human body? (Some exceptions would be the sheering translation that takes place in the Pubic Symphysis and in the knee and shoulder, in conjunction with their angular motions.) For the most part, gross joint movements are circular (angular) motions. We need to understand the components of angular motion and how they relate to human movement.

----A simple angular motion always goes around a central point, called the axis of rotation. (In more complex angular motions, there may be multiple axes of rotation or a single axis that moves.) The distance from the axis to the center of gravity of the object being moved is called the moment arm or lever arm (r). As any one force always moves in a straight line, we must further qualify (r) as the perpendicular distance between the line of action of the force and a line parallel to it that passes through the axis of rotation. So, if the force of the object is moving in a straight line (as would be the case if the force was gravity pulling towards Earth) the value for (r) changes with the change position. Just like objects moving in a straight line (linear motion), objects that move in angular motions have inertia (for angular motion, it is called a moment of inertia):

(10) I = mr2

----(Where I is the moment of inertia and m is the mass of the object)

----Once the object is moving its inertia becomes angular momentum (H) and is represented by:

(11) H = IÉ.

----(Where É is the angular velocity, typically measured in degrees per second)

----Angular motion can be thought of as an infinite number of linear forces being accelerated (change in direction) by a force that pulls the moving object towards the axis of rotation. This is known as a centripetal force. If you were to tie a small weight on the end of a string and spin it overhead, the centripetal force is the force from your hand, translated through the string, that keeps the weight from flying off. As every force has an equal and opposite reaction, the centripetal force is constantly counter acted by a centrifugal force, which pulls outward away from the axis. Since the centripetal and centrifugal forces of a closed system are always equal to each other, Rothman says the terms are practically interchangeable. You can feel the centrifugal force pull on the string as you spin it. The longer the string is, the more energy you must use to spin the weight at the same speed. Also, the faster you try to spin the weight with the same string length, the more energy you must use. In either situation, the centripetal (and centrifugal) force must increase proportionally. (Again, we see an example of action-reaction.) As you can see using formula (10), the weight's resistance to being moved(I) increases with the increase in the length of the string (r). And in formula (11), once the weight is moving, it's resistance to going faster (H) increases with an increase in velocity (É.). Your arm must, therefore, generate more energy in order to either accommodate for the increase in string length or to increase the velocity. Energy that has a tendency to rotate, turn or twist an object an axis or axes is called torque or moment(T), and is represented mathematically as:

(12)T = F(r)

----In the example above, the force causing the weight to spin overhead was coming from the axis of rotation, in this case your hand and arm. The movement of the joints of the human body is generated in a slightly different manner. Instead of the force of movement being generated at the actual point where the two bones join, the muscles, around the joint, that are associated with that movement, produce it. The joint, and the bones that it is composed of, are, in essence, being used to transform the linear force of the contracting muscle into a torque (moment). This is an example of a lever, a simple mechanical machine that can either increase the magnitude of a force (while sacrificing speed) or increase the speed and range of motion (while sacrificing magnitude). It consists of a lever arm (separated into two parts) a fulcrum (the axis of rotation) a force that provides the kinetic energy for the work and a resistance (object) that the work is being done to. There are three classes of simple levers :

First Class 
Second Class 
Third Class
= Kinetic Force
= The Fulcrum
= The Resistance

----Most of the joints in the human body are first class and third class (usually one at each joint set up to oppose each other). The factor that determines which effect the lever has on the force depends on the difference between the length of the force lever arm (the distance from the fulcrum to the point of the application of force) and the load lever arm (the distance from the fulcrum to the center of gravity of the load). Both the force and the load exert a torque on each other through their respective lever arms. In order for a lever to be in balance, both torques must be equal. This is represented mathematically as:

(13) TF = TL

----(For the torque of the force side and the torque of the load side)

----This becomes:

(13) {F(r)} F = {F(r)} L

----So, what does all of this really mean? Let's look at an example of a third class lever in the body, the biceps muscle working as an elbow flexor. If you were performing a dumbbell curl with a 20 lb dumbbell and you wanted to maintain an isometric contraction at 90 ° of elbow flexion, how much force must your muscles produce? Let's say the length of your arm from the elbow joint (the fulcrum) to the center of the palm of you and where you are holding the dumbbell (the load) is 1 ft (12 inches), and your biceps inserts into your forearm about 1/2 inch from the elbow joint. To make this simple, we will discount the weight of your arm itself and the action of the synergistic muscles. Combining formulas (4) and (14) and plugging in the above values, we get:

(Muscle) (Dumbbell)
(15) {F (1/2 inch)} F = {ma (12 inches)} L (since there is no movement a = 1)
F = (20 lb) (12inches) 2
F = 480 inch/lb.

----So, your biceps must generate a force equal to 480 inch/lb (or 40 ft/lb) of torque to maintain the isometric contraction at 90 ° with the 20 lb dumbbell. If you were to perform a full rep and measure the displacement (distance traveled) of the dumbbell and compare it with the change in length of the biceps from the bottom of the rep to the top, it would be obvious how the levers in our bodies modify the forces generated by our muscles. Our bodies are designed to sacrifice some of the force for the sake of speed and a greater range of motion. As athletes, maximizing our training efforts to increase our power output seems even more imperative.

----This brings us to the conclusion of part 1 of the Physics of Fitness. Again, we will explore this and many other relations between physics and fitness in the upcoming installments of this series.

Bibliography:

Hewitt, Paul G. (1997)Conceptual Physics, 8th, Edition . Boston, MA : Addison Wesley Longman, Inc.
Lindeburg , Michael R. (1997). Mechanical Engineering Reference Manual : For the PE Exam . Belmont, CA: Professional Publications, Inc.
McGinnis ,Peter M. ,Ph D. (1999) Biomechanics of Sport and Exercise. Champaign ,IL: Human Kinetics.
Newton, I. (1934) Principia (Vol. I-II ,Andrew Mott's translation revised by Florian Cajoari). Berkeley: University of California Press. (Original work published 1686 ,Motte.s English translation, 1729.)
Rothman, Tony ,Ph.D.(1995) Instant Physics ,From Aristotle to Einstein ,and Beyond . N e w York ,NY : Ballantine Books.

 

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