In Chapter 1 we found that biomechanics provides tools that are needed to analyze human motion, improve performance, and reduce the risk of injury. In order to facilitate the use of these biomechanical tools, this text will emphasize the qualitative understanding of mechanical concepts. Many chapters, however, will include some quantitative examples using the algebraic definitions of the mechanical variables being discussed. Mathematical formulas are a precise language and are most helpful in showing the importance, interactions, and relationships between biomechanical variables. While more rigorous calculus forms of these equations provide the most accurate answers commonly used by scientists (Beer & Johnson, 1984; Hamill & Knutzen, 1995; Zatsiorsky, 1998, 2002), the majority of kinesiology majors will benefit most from a qualitative understanding of these mechanical concepts. So this chapter begins with key mechanical variables and terminology essential for introducing other bio-mechanical concepts. This chapter will emphasize the conceptual understanding of these mechanical variables and leave more detailed development and quantitative examples for later in the text. Next, nine general principles of biomechanics are introduced that will be developed throughout the rest of the text. These principles use less technical language and are the tools for applying biomechanical knowledge in the qualitative analysis of human movement. The chapter concludes by summarizing a model of qualitative analysis that is used in the application section of the book.

Before we can begin to understand how humans move, there are several mechanical terms and concepts that must be clarified. Mechanics is the branch of physics that studies the motion of objects and the forces that cause that motion. The science of mechanics is divided into many areas, but the three main areas most relevant to biome-chanics are: rigid-body, deformable-body, and fluids.

In rigid-body mechanics, the object being analyzed is assumed to be rigid and the deformations in its shape so small they can be ignored. While this almost never happens in any material, this assumption is quite reasonable for most biomechanical studies of the major segments of the body. The rigid-body assumption in studies saves considerable mathematical and modeling work without great loss of accuracy. Some biomechanists, however, use deformable-body mechanics to study how biological materials respond to external forces that are applied to them. Deformable-body mechanics studies how forces are distributed within a material, and can be focused at many levels (cellular to tissues/organs/ system) to examine how forces stimulate growth or cause damage. Fluid mechanics is concerned with the forces in fluids (liquids and gasses). A biomechanist would use fluid mechanics to study heart valves, swimming, or adapting sports equipment to minimize air resistance.

Most sports biomechanics studies are based on rigid-body models of the skeletal system. Rigid-body mechanics is divided into statics and dynamics (Figure 2.1). Statics is the study of objects at rest or in uniform (constant) motion. Dynamics is the study of objects being accelerated by the actions of forces. Most importantly, dynamics is divided into two branches: kinematics and kinetics. Kinematics is motion description. In kinematics the motions of objects are usually measured in linear (meters, feet, etc.) or angular (radians, degrees, etc.) terms. Examples of the kinematics of running could be the speed of the athlete, the length of the stride, or the angular velocity of hip extension. Most angular mechanical variables have the adjective "angular" before them. Kinetics is concerned with determining the causes of motion. Examples of kinetic variables in running are the forces between the feet and the ground or the forces of air resistance. Understanding these variables gives the track coach knowledge of the causes of running performance. Kinetic information is often more powerful in improving human motion because the causes of poor performance have been identified. For example, knowing that the timing and size of hip extensor action is weak in the takeoff phase for a long jumper may be more useful in improving performance than knowing that the jump was shorter than expected.

The language of science is mathematics. Biomechanics often uses some of the most complex kinds of mathematical calculations, especially in deformable-body mechanics. Fortunately, most of the concepts and laws in classical (Newtonian) rigid-body mechanics can be understood in qualitative terms. A conceptual understanding of bio-mechanics is the focus of this book, but algebraic definitions of mechanical variables will be presented and will make your understanding of mechanical variables and their relationships deeper and more powerful.

First, let's look at how even concepts seemingly as simple as numbers can differ in their complexity. Scalars are variables that can be completely represented by a number and the units of measurement. The number and units of measurement (10 kg, 100 m) must be reported to completely identify a scalar quantity. It makes no sense for a track athlete to call home and say, "Hey mom, I did 16 and 0"; they need to say, "I made 16 feet with 0 fouls." The number given a scalar quantity represents the magnitude or size of that variable.

Vectors are more complicated quantities, where size, units, and direction must be specified. Figure 2.2 shows several scalars and the associated vectors common in bio-mechanics. For example, mass is the scalar quantity that represents the quantity of

matter for an object. That same object's weight is the gravitational force of attraction between the earth and the object. The difference between mass and weight is dramatically illustrated with pictures of astronauts in orbit about the earth. Their masses are essentially unchanged, but their weights are virtually zero because of the microgravity when far from earth.

Biomechanics commonly uses directions at right angles (horizontal/vertical, longitudinal/transverse) to mathematically handle vectors. Calculations of velocity vectors in a two-dimensional (2D) analysis of a long jump are usually done in one direction (e.g., horizontal) and then the other (vertical). The directions chosen depend on the needs of the analysis. Symbols representing vector quantities like velocity (v) in this text will be identified with bold letters. Physics and mechanics books also use underlining or an arrow over the symbol to identify vector quantities. These and other rules for vector calculations will be summarized in chapter 6. These rules are important because when adding vectors, one plus one is often not two because the directions of the vectors were different. When adding scalars with the same units, one plus one is always equal to two. Another important point related to vectors is that the sign (+ or -) corresponds to directions. A -10 lb force is not less than a +10 lb force; they are the same size but in opposite directions. The addition of vectors to determine their net effect is called the resultant and requires right-angle trigonometry. In chapter 6 we will also subtract or break apart a vector into right-angle components, to take advantage of these trigonometry relationships to solve problems and to "see" other important pushes/pulls of a force.

There are two important vector quantities at the root of kinetics: force and torque. A force is a straight-line push or pull, usually expressed in pounds (lbs) or Newtons

(N). The symbol for force is F. Remember that this push or pull is an interactional effect between two bodies. Sometimes this "push" appears obvious as in a ball hitting a bat, while other times the objects are quite distant as with the "pull" of magnetic or gravitational forces. Forces are vectors, and vectors can be physically represented or drawn as arrows (Figure 2.3). The important characteristics of vectors (size and direction) are directly apparent on the figure. The length of the arrow represents the size or magnitude (500 N or 112 lbs) and the orientation in space represents its direction (15 degrees above horizontal).

The corresponding angular variable to force is a moment of force or torque. A moment is the rotating effect of a force and will be symbolized by an M for moment of force or T for torque. This book will use the term "torque" synonymously with "moment of force." This is a common English meaning for torque, although there is a more specific mechanics-of-materials meaning (a torsion or twisting moment) that leads some scientists to prefer the term "moment of force." When a force is applied to an object that is not on line with the center of the object, the force will create a torque that tends to rotate the object. In Figure 2.3 the impact force acts below the center of the ball and would create a torque that causes the soccer ball to acquire backspin. We will see later that the units of torque are pound-feet (lb •ft) and Newton-meters (N-m).

Let's look at an example of how kinematic and kinetic variables are used in a typical biomechanical measurement of isometric muscular strength. "Isometric" is a muscle research term referring to muscle actions performed in constant (iso) length (metric) conditions. The example of a spring is important for learning how mathematics and graphs can be used to understand the relationship between variables. This example will also help to understand how muscles, tendons, and ligaments can be said to

Figure 2.3. Vector representation of the force applied by a foot to a soccer ball. The magnitude and direction properties of a vector are both apparent on the diagram: the length of the arrow represents 500 Newtons of force, while the orientation and tip of the arrow represent the direction (15° above horizontal) of the force.

Figure 2.3. Vector representation of the force applied by a foot to a soccer ball. The magnitude and direction properties of a vector are both apparent on the diagram: the length of the arrow represents 500 Newtons of force, while the orientation and tip of the arrow represent the direction (15° above horizontal) of the force.

have spring-like behavior. Figure 2.4 illustrates the force-displacement graph for the spring in a handgrip dynamometer. A dynamometer is a force-measuring device. As a positive force (F) pulls on the spring, the spring is stretched a positive linear distance (displacement = d). Displacement is a kinematic variable; force is a kinetic variable.

Therapists often measure a person's grip strength in essentially isometric conditions because the springs in hand dynamometers are very stiff and only elongate very small distances. The force-displacement graph in Figure 2.4 shows a very simple (predictable) and linear relationship between the force in the spring (F) and the resulting elongation (d). In other words, there is a uniform increase (constant slope

of the line) in force with increasing spring stretch. We will see later on in chapter 4 that biological tissues have much more complex (curved) mechanical behaviors when loaded by forces, but there will be linear regions of their load-deformation graphs that are representative of their elastic properties.

Let's extend our example and see how another mechanical variable can be derived from force and displacement. Many simple force measuring devices (e.g., bathroom and fishing scales) take advantage of the elastic behavior of metal springs that are stretched or compressed short distances. This relationship is essentially the mathematical equation (F = k • d) of the calibration line illustrated in Figure 2.4, and is called Hooke's Law. Hooke's Law is valid for small deformations of highly elastic materials like springs. The stiffness (elasticity) of the spring is symbolized as k, which represents the slope of the line. In chapter 4 we will look at the stiffness of biological tissues as the slope of the linear region of a graph like this. If we plug in the largest force and displacement (700 = k • 0.01), we can solve for the stiffness of the spring, and find it to be 70,000 N/m. This says that the spring force will increase 70,000 Newtons every meter it is stretched. This is about 15,730 pounds of tension if the spring were stretched to about 1.1 yards! Sounds pretty impressive, but remember that the springs are rarely elongated that much, and you might be surprised how stiff muscle-tendon units can get when strongly activated.

Engineers measure the stiffness or elasticity of a material with special machines that simultaneously record the force and deformation of the material. The slope of the load-deformation graph (force/length) in the linear region of loading is used to define stiffness. Stiffness is the measure of elasticity of the material, but this definition often conflicts with most people's common understanding of elasticity. People often incorrectly think elasticity means an object that is easily deformed with a low force, which is really compliance (length/force), the opposite of stiffness. An engineer would say that there was less stiffness or greater compliance in the second spring illustrated as a dashed line.

Can you find the stiffness (spring constant, k) that corresponds to the dashed calibration line in Figure 2.4? Remember that the stiffness, k, corresponds to the slope of the line illustrated in the figure and represents the change in force for a given change in length. The slope or rate of change of a variable or graph will be an important concept repeated again and again in biome-chanics. Remember that forces and displacements are vectors, so directions are indicated by the sign (+ or -) attached to the number. What do you think the graph would look like if the force were reversed, i.e., to push and compress the spring rather than stretching it? What would happen to the sign of F and d?

It is also important to know that the previous example could also be measured using angular rather than linear measurements. There are isokinetic dynamometers

Activity: Elasticity

Take a rubber band and loop it between the index fingers of your hands. Slowly stretch the rubber band by moving one hand away from the other.The tension in the rubber band creates a torque that tends to abduct the metacarpophalangeal joints of your index finger. Does the tension your fingers sense resisting the torque from the rubber band uniformly increase as the band is stretched? Does a slightly faster stretch feel different? According to Hooke's Law, elastic materials like springs and rubber bands create forces directly proportional to the deformation of the material, but the timing of the stretch does not significantly affect the resistance. Chapter 4 will deal with the mechanical responses of biological tissues, which are not perfectly elastic, so the rate of stretch affects the mechanical response of the tissue.

that simultaneously measure the torque (T) and rotation (Figure 1.5). These angular measurements have been used to describe the muscular strength of muscle groups at various positions in the range of motion.

There are many other mechanical variables that help us understand how human movement is created. These variables (e.g., impulse, angular momentum, kinetic energy) often have special units of measurement. What all these mechanical variables and units have in common is that they can be expressed as combinations of only four base units. These base units are length, mass, and time. In the International System (SI) these units are the second (s), kilogram (kg), meter (m), and radian (rad). Scientific research commonly uses SI units because they are base 10, are used throughout the world, and move smoothly between traditional sciences. A Joule of mechanical energy is the same as a Joule of chemical energy stored in food. When this book uses mathematics to teach a conceptual understanding of mechanics in human movement (like in Figure 2.4), the SI system will usually be used along with the corresponding English units for a better intuitive feel for many students. The symbols used are based on the recommendations of the International Society of Biomechanics (ISB, 1987).

These many biomechanical variables are vitally important to the science of bio-mechanics and the integration of biome-chanics with other kinesiological sciences. Application of biomechanics by kinesiolo-gy professionals does not have to involve quantitative biomechanical measurements. The next section will outline biomechanical principles based on the science and specialized terminology of biomechanics.

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