Objectives
After reading this chapter, the student should be able to:
In Chapters 8 and 9, we discussed linear and angular kinematics. Kinematics was defined as the description of motion with no regard to the cause of the motion. The motion described was translatory (linear), rotational (angular), or a combination of both linear and rotational (general). In this chapter, the concern is with the causes of motion. For example, how do we propel ourselves forward in running? Why does a runner lean on the curve of a track? What keeps an airplane in the air? Why do golf balls slice or hook? How does a pitcher curve a baseball? The search for understanding the causes of motion date to antiquity, and answers to some of these questions were
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suggested by such notables as Aristotle and Galileo. The culmination of these explanations was provided by the great scientist Sir Isaac Newton, who ranks among the greatest thinkers in human history for his theories on gravity and motion. In fact, the laws of motion described by Newton in his famous book Principia Mathematica (1687) form the cornerstone of the mechanics of human movement (14). The branch of mechanics that deals with the causes of motion is called kinetics. Kinetics is concerned with the forces that act on a system. If the motion is translatory, then linear kinetics is of concern. The basis for the understanding of the kinetics of linear motion is the concept of force.
Force
Force is a very difficult concept to define. In fact, we generally define the term force by describing what a force can do. According to Newton's principles, objects move when acted upon by a force greater than the resistance to movement provided by the object. A force involves the interaction of two objects and produces a change in the state of motion of an object by pushing or pulling it. The force may produce motion, stop motion, accelerate, or change the direction of the object. In each case, the acceleration of the object changes or is prevented from changing. A force, therefore, may be thought of as any interaction, a push or pull, between two objects that can cause an object to accelerate either positively or negatively. For example, a push on the ground generated by a forceful knee and hip extension may be sufficient to cause the body to accelerate upward and leave the ground–that is, jump.
Characteristics of A Force
Forces are vectors and as such have the characteristics of a vector, including magnitude and direction. Magnitude is the amount of force being applied. It is also necessary to state the direction of a force because the direction of a force may influence its effect, for example, on whether the force is pushing or pulling. Vectors, as described in Chapter 8, are usually represented by arrows, with the length of the arrow indicating the magnitude of the force and the arrowhead pointing in the direction in which the force is being applied. In the International System (SI) of measurement, the unit for force is the newton (N), although for comparison value, forces are often represented in the literature as a ratio of force to body weight (BW) or force to body mass. Sample peak force values for a variety of movements, expressed as a function of BW, are presented in Table 101.
Forces have two other equally important characteristics, the point of application and the line of action. The point of application of a force is the specific point at which the force is applied to an object. This is very important because the point of application most often determines whether the resulting motion is linear or angular or both. In many instances, a force is represented by a point of application at a specific point, although there may be many points of application. For example, the point of application of a muscular force is the center of the muscle's attachment to the bone, or the insertion of the muscle. In many cases, the muscle is not attached to a single point on the bone but is attached to many points, such as in the case of the fanshaped deltoid muscle. In solving mechanical problems, however, it is considered to be attached to a single point. Other points of application are the contact point between the foot and the ground for activities such as jumping, walking, and running; hand contact with the ball for a baseball throw; and the contact point between the racquet and the ball in tennis.
TABLE 101 Maximum Forces Acting on the Body 


The line of action of a force is a straight line of infinite length in the direction in which the force is acting. A force
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can be assumed to produce the same acceleration of the object if it acts anywhere along this line of action. Thus, if the force coming up from the ground in the last jump phase of a triple jumper has a line of action directed to 18° with respect to the horizontal, the jumper accelerates forward and upward in that direction. The orientation of the line of action is usually given with respect to an x, y coordinate system. The orientation of the line of action to this system is given as an angular position and is referred to as the angle of application. This angle is designated by the Greek letter theta (0). The four characteristics of a force– magnitude, direction, point of application, and line of action–are illustrated inFigure 101A for a muscular force and in Figure 101B for a highjump takeoff.
FIGURE 101 Characteristics of a force for an internal muscular force (A) and an external force generated on the ground in the high jump (B). 
Composition and Resolution of Forces
Forces are vector quantities that have both magnitude and direction. As presented in the discussion of kinematic vectors in Chapter 8, a single force vector may be resolved into perpendicular components, or several forces can be resolved into one vector. That is, a single force vector can be calculated or composed to represent the net effect of all of the forces in the system. Similarly, given the resultant force, the resultant force can be resolved into its horizontal and vertical components. To do either, the trigonometric principles presented in Appendix B are applied.
Several types of force systems must be defined to compose or resolve systems of multiple forces. Any system of forces acting in a single plane is referred to as coplanar, and if they act at a single point, they are called concurrent. Any set of concurrent coplanar forces may be substituted by a single force, or the resultant, producing the same effect as the multiple forces. The process of finding this single force is called composition of force vectors.
When force vectors act along a single line, the system is said to be collinear. In this case, vector addition is used to compose the forces. Consider the force system in Figure 102A The force vectors a, b, and c all act in the same direction and can be replaced by a single force, d, which is the sum of a, b, and c. Thus:
The force vector d would have the same effect as the other three force vectors. In Figure 102B, however, two of the force vectors, a and b, are acting in one direction, but the vector c is acting in the opposite direction. Thus, the force vector d is the algebraic sum of these three force vectors:
The force vector d still represents the net effect of these force vectors. In both of these examples, sets of collinear force vectors are present.
When the force vectors are not collinear but are coplanar, they may still be composed to determine the resultant force. Graphically, this can be done in exactly the same manner as described in Chapter 8 in the section on adding vectors. Consider Figure 102C. The force vectors a and b are not collinear, but they may be composed or added to determine their net effect. With the arrow of vector a placed at the tail of vector b, the resultant composed vector c is the distance between the tail of a and the arrow of b. This procedure is illustrated in Figure 102D with multiple vectors.
Multiple vectors can also be combined using trigonometric functions. First presented in Chapter 8, this involves first breaking each vector down into its components using resolution. After they are resolved into vertical and horizontal components, the orthogonal components for each vector are added, and the resultant vector is composed. To illustrate, the four vectors shown in Figure 102D will be assigned values of length 10 and 0 = 45° for vector A, length 6 and
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θ = 0° for vector B, length 5 and 9 = 30° for vector C, and length 7 and 9 = 270° for vector D. The first step is to resolve each vector into vertical and horizontal components.
To find the magnitude of the resultant vector, the horizontal and vertical components of each vector are added and resolved using the Pythagorean theorem:
FIGURE 102 Force vector addition. 


To find the angle of resultant vector, the trigonometric function tangent is used:
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The characteristics of the resultant vector (R = 16.60, 9 = 8.91°) can be clearly confirmed by examining the resultant obtained by combining vectors graphically using the headtotail method.
Refer to the walking ground reaction force (GRF) data in Appendix C: Using the peak anteroposterior (Fy) and vertical force (Fz) for frame 21, calculate the resultant force and the angle of force application at this point.
Laws of Motion
The publication of the Principia Mathematicain 1687 by Sir Isaac Newton (16421727) astounded the scientific community of the day (14). In this book, he introduced his three laws of motion that we use to explain a number of phenomena. Although these laws have been superseded by Einstein's theory of relativity, we can still use Newton's basic principles as the basis for most analyses of human movement in biomechanics. Newton's three laws of motion have demonstrated how and when a force creates a movement and how it applies to all of the different types of forces previously identified. His work has provided the link between cause and effect. To fully understand the underlying nature of motion, it is necessary to understand the cause of the movement, not merely the description of the outcome. That is, the forces that cause motion must be fully understood. The statements of these laws are taken from a translation of Newton's Principia Mathematica (14).
Law I: Law of Inertia
Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed on it (14).
The inertia of an object is used to describe its resistance to motion. Inertia is directly related to the mass of the object. Mass is a scalar and is the measure of the amount of matter that constitutes an object and is expressed in kilograms. An object's mass is constant, regardless of where it is measured, so that the mass is the same whether it is calculated on earth or on the moon. The greater the mass of an object, the greater its inertia and thus the greater the difficulty in moving it or changing its current motion.
For an object to move, the inertia of the object has to be overcome. Newton suggested that an object at rest– an object with zero velocity–would remain at rest. This seems obvious. A chair sitting in a room has zero velocity because it is not moving. Additionally, an object moving at a constant velocity would continue to do so in a straight line. This concept is not as obvious because the practical instances of individuals on the earth's surface rarely experience constant velocity motion. If it is noted that constant velocity results in zero acceleration just as zero velocity does, then it can be understood how this law holds for both cases. Therefore, the inertia of these objects would compel them to maintain their status at a constant velocity. Newton's first law of motion can be expressed as:
Note that 2F refers to the net force and takes into account all forces acting on the object.
Overcoming the inertia of such objects requires a net external force greater than the inertia of the object. For example, if a barbell has a mass of 70 kg, a force greater than 686.7 N, or the product of the acceleration due to gravity (9.81 m/s^{2}) and 70 kg, must be exerted to lift it. If an object is subjected to an external force that can overcome the inertia, the object will be accelerated. To get an object moving, the external force must positively accelerate the object. On the other hand, to stop the object from moving, the external force must negatively accelerate the object. Because body mass determines inertia, an individual with greater mass has to generate larger external forces to overcome inertia and accelerate.
Law II: Law of Acceleration
The change of motion is proportional to the force impressed and is made in the direction of the straight line in which that force is impressed (14).
Newton's second law generates an equation that relates all forces acting on an object, the mass of the object, and the acceleration of the object. This relationship is expressed as:
This equation can also be used to define the unit of force, the newton. By substituting the units for mass and acceleration in the righthand side, it can be seen that:
where kgm = kilogrammeters. In this equation, the force is the net force acting on the object in question, that is, the sum of all of the forces involved.
In adding up all forces acting on an object, it is necessary to take the direction of the forces into account. If the forces exactly counteract each other, the net force is zero. If the sum of the forces is zero, the acceleration will also be zero. This case is also described by Newton's first law. If the net force produces acceleration, the accelerated object will travel in a straight line along the line of action of the net force.
Rearranging the equation described by Newton's second law allows another important concept in biomechanics to be defined. Acceleration was previously defined as the time rate of change of velocity, or dv/dt. Substituting this expression into the equation of the second law:
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or
The product of mass and velocity in the numerator of the righthand side of this equation is known as the momentum of an object. Momentum is the quantity of motion of an object. It is generally represented by the letter p and has units of kilogrammeters per second. For example, if a football player has a mass of 83 kg and is running at 4.5 m/s, his momentum is:
Newton's second law can thus be restated:
That is, force is equal to the time rate of change of momentum. To change the momentum of an object, an external force must be applied to the object. The momentum may increase or decrease, but in either case, an external force is required.
Law III: Law of ActionReaction
To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal and directed to contrary parts.
This law illustrates that forces never act in isolation but always in pairs. When two objects interact, the force exerted by object A on object B is counteracted by a force equal and opposite exerted by object B on object A. These forces are equal in magnitude but opposite in direction. That is:
In addition, the force–the action–and the counterforcethe reaction–act on different objects. The result is that these two forces cannot cancel each other out because they act on and may have a different effect on the objects. For example, a person landing from a jump exerts a force on the earth, and the earth exerts an equal and opposite force on the person. Because the earth is more massive than the individual, the effect on the individual is greater than the effect on the earth. This example illustrates that although the force and the counterforce are equal, they may not necessarily have comparable results.
In human movements, an action force is generated on the ground or implement, and the reaction force generally produces the desired movement. As shown in Figure 103, the jumper makes contact with the ground and generates a large downward force because of the acceleration of the body combined with forces generated by body segments at contact, and a resulting reaction force upward controls the landing.
FIGURE 103 Vertical GRF during a landing from a jump. 
Types of Forces
The forces that exist in nature and affect the way humans move may be classified in a number of ways. The most common classification scheme is to describe forces as contact or noncontact forces (11). A contact force involves the actions, pushes or pulls, exerted by one object in direct contact with another object. These are the forces involved, for example, when a bat hits a baseball or the foot hits the floor. In contrast to contact forces are those that act at a distance. These are called noncontact forces. As implied by the name, these are forces that are exerted by objects that are not in direct contact with one another and may actually be separated by a considerable distance.
Noncontact Forces
In the investigation of human movement, the most familiar and important noncontact force is gravity. Any object released from a height will fall freely to the earth's surface, pulled by gravity. In Sir Isaac Newton's book, the Principia Mathematica (1687), he introduced his theory of gravity (14). With the law of gravitation, Newton identified gravity as the force that causes objects to fall to the earth, the moon to orbit the earth, and the planets to revolve about the sun. This law states: “The force of gravity is inversely proportional to the square of the distance between attracting objects and proportional to the product of their masses.”
In algebraic terms, the law is described by the following equation:
where
G = universal gravitational constant
m_{1} = mass of one object
m_{2} = mass of the other object
r = distance between the mass centers of the objects.
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The constant value G was estimated by Newton and determined accurately by Cavendish in 1798. The value of G is 6.67 ^{*} 10^{11} Nm^{2}/kg^{2}.
The gravitational attraction of one object of a relatively small size to another object of similar size is extremely small and therefore can be neglected. In biomechanics, the objects of most concern are the earth, the human body, and projectiles. In these cases, the earth's mass is considerable, and gravity is a very important force. The attractive force of the earth on an object is called the weight of the object. This is stated as:
The force of gravity causes an object to accelerate toward earth at a rate of 9.81 m/s^{2}. Newton determined through his theories of motion that:
where m is the mass of the individual and a is the acceleration due to gravity. Thus:
where g is the acceleration due to gravity. Body weight is thus the product of the individual's mass and the acceleration due to gravity. It is apparent, therefore, that an individual's mass and BW are not the same. Whereas body weight is a force and the appropriate unit for BW is the newton, body mass is a scalar with units of kilograms. To determine a person's BW, simply multiply mass times the acceleration due to gravity (9.81 m/s^{2}).
Because weight is a force, it has the attributes of a force. As a vector, it has a line of action and a point of application. An individual's total BW is considered to have a point of application at the center of mass and a line of action from the center of mass to the center of the earth. Because the earth is so large, this line of action is straight down toward the center of the earth.
The point of origin of the weight vector is called the center of gravity. This is a point about which all particles of the body are evenly distributed. Another term used interchangeably with center of gravity is center of mass, a point about which the mass of the segment or body is equally distributed. They differ in that the center of gravity refers only to the vertical direction because that is the direction in which gravity acts, but the center of mass does not depend on a vertical orientation. The computation of both the center of mass and the center of gravity is presented in Chapter 11.
The value for g, the acceleration due to gravity, depends on the square of the distance to the center of the earth. Because of the spin on its axis, the earth is not perfectly spherical. The earth is slightly flattened at the poles, resulting in shorter distances to the earth's center at the poles than at the equator. Thus, the points on the earth are not all equidistant from its center, and acceleration due to gravity, g, does not have the same value everywhere. The latitude–the position on the earth with respect to the equator–on which a long jump, for example, is performed can have a significant effect on the distance jumped. Another factor influencing the value of g is altitude. The higher the altitude, the lower the value for g. If one were weighing oneself and a minimal weight were desired, the optimum place for the weighin would be on the highest mountain at the earth's equator.
Contact Forces
Because contact forces are those resulting from a direct interaction of two objects, the number of such forces is considerably greater than the single noncontact force discussed. The following contact forces are considered paramount in human movement: ground reaction force, joint reaction force, friction, fluid resistance, inertial force, muscle force, and elastic force.
Ground Reaction Force
In almost all terrestrial human movement, the individual is acted upon by the GRF at some time. This is the reaction force provided by the surface upon which one is moving. The surface may be a sandy beach, a gymnasium floor, a concrete sidewalk, or a grass lawn. If the individual is swinging from a high bar, the surface of the bar provides a reaction force. All surfaces on which an individual interacts provide a reaction force. The individual pushes against the ground with force, and the ground pushes back against the individual with equal force in the opposite direction (Newton's law of actionreaction). These forces affect both parties–the ground and the individual–and do not cancel out even though they are equal in magnitude but opposite in direction. Also, the GRF changes in magnitude, direction, and point of application during the period that the individual is in contact with the surface.
Refer to the GRF data for walking in Appendix C. Calculate the resultant force using Fy (anteroposterior) and Fz (vertical) data for frames 18, 36, and 60. How does the direction of the force application change across the support phase?
As with all forces, the GRF is a vector and can be resolved into its components. For the purpose of analysis, it is commonly broken down into its components. These components are orthogonal to each other along a threedimensional coordinate system (Fig. 104). The components are usually labeled Fz vertical (updown), Fy anteroposterior (forwardbackward), and Fx, mediolateral (sidetoside). In the reporting of threedimensional (3D) data, however, some researchers label the axes as Fy (vertical), Fx (anteroposterior), and Fz (mediolateral). The former convention is used in this book because it is the most often used system. It is probably much clearer to represent the GRF components as F_{vertical}, F_{anteroposterior} and
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F_{mediolateral}. Regardless of the convention used, the anteroposterior and mediolateral components are referred to as shear components because they act parallel to the surface of the ground.
FIGURE 104 Ground reaction force components. The origin of the force platform coordinate system is at the center of the platform. 
Biomechanists record the GRF components using a force platform. A force platform is a sophisticated measuring scale usually imbedded in the ground, with its surface flush with the surface of the ground, on which the individual performs. A typical force platform experimental setup for the data collected in Appendix C is shown in Figure 105. This device measures the forces the foot on the performing surface or the force of an individual just standing on the platform. Force platforms have been used since the 1930s (28) but became more prominent in biomechanics research in the 1980s.
Although forces are measured in newtons, GRF data are generally scaled by dividing the force component by the individual's BW, resulting in units of times BW. In other instances, GRFs may be scaled by dividing the force by body mass, resulting in a unit of newtons per kilogram of body mass.
Refer to the GRF walking data in Appendix C. Locate the maximum vertical (Fz), anteroposterior (Fy), and mediolateral (Fx) forces (N). Report peak forces scaled to BW and to body mass (N/kg).
FIGURE 105 A typical laboratory force platform setup. 
GRF data have been used in many studies to investigate a variety of activities. Most studies, however, have dealt with the load or impact on the body during landings, either from jumps or during support phase of gait. For example, GRFs have been studied during the support phase of running (17,19), walking (36), and landings from jumps (27,53).
A vertical component curve of a single foot contact of an individual landing from a jump is presented in Figure 103. Only the vertical component is presented because it is of much greater magnitude than the other components and because the major interest in landings has been the effect of impacts on the human body. In the landing curve, the first peak represents the initial ground contact with the forefoot. The second peak is the contact of the heel on the surface. Generally, the second peak is greater than the first peak. Some individuals, however, land flatfooted and have only one impact peak. When the individual comes to rest on the surface, the vertical force curve settles at the individual's BW. Although the magnitude of the vertical component at impact in running is three to five BW, the vertical component in landing can be as much as 11 times BW, depending on the height from which the person drops (27,53).
If should be noted that the GRF is the sum of the effects of all masses of the segments times the acceleration due to gravity. That is, the sum of the product of the masses and accelerations of each segment. This sum reflects the center of mass of the individual. Consequently, the GRF acts at the center of mass of the total body (Fig. 106). Dividing a force by the mass, the result would be acceleration. For example, the equation relating the vertical GRF to acceleration is given by Newton's second law as:
Dividing both sides of the equation by body mass (m):
This value reflects the acceleration of the center of mass. Many researchers have related the vertical GRF component to the function of the foot during landings. Because the GRF acts on the center of mass, the relationship between a GRF component and foot function is tenuous at best.
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FIGURE 106 The ground reaction force vector acts through the center of mass of the body. 
Joint Reaction Force
FIGURE 107 The joint reaction force of the knee with its shear and compressive components. 
In many instances in biomechanical analyses, segments are examined, either singly or one at a time in a logical order. When a joint reaction force analysis is conducted, the segment is separated at the joints, and the forces acting across the joints must be considered. For example, if one is standing still, the thigh exerts a downward force on the leg across the knee joint. Similarly, the leg exerts an upward force of equal magnitude on the thigh (Fig. 107). This is the net force acting across the joint and is referred to as the joint reaction force. In most analyses, the magnitude of this force is unknown, but it can be calculated given the appropriate kinematic and kinetic data, in addition to anthropometric data describing the body dimensions.
Some confusion exists as to whether the joint reaction force is the force of the distal bony surface of one segment acting on the proximal bony surface of the contiguous segment. The joint reaction force does not, however, reflect this boneonbone force across a joint. The actual boneonbone force is the sum of the actively contracting muscle forces pulling the joint together and the joint reaction force. Because the force generated by the actively contracting muscles is not known, the boneonbone force is difficult to calculate, although sophisticated calculations have been done to estimate boneonbone forces (91).
Friction
Friction is a force that acts parallel to the interface of two surfaces that are in contact during the motion or impending motion of one surface as it moves over the other. For example, the weight of a block resting on a horizontal table pulls the block downward, pressing it against the table. The table exerts an upward force on the block that is perpendicular or normal to the surface. To move the block horizontally, a horizontal force on the block of sufficient magnitude must be exerted. If this force is too small, the block will not move. In this case, the table evidently exerts a horizontal force equal and opposite to the force on the block. This interaction, the frictional force, is due to the bonding of the molecules of the block and the table at the places where the surfaces are in very close contact. Figure 108 illustrates this example.
Although it appears that the area of contact influences the force of friction, this is not the case. The force of friction is proportional to the normal force between the surfaces, that is:
where µ is the coefficient of friction and N is the normal force or the force perpendicular to the surface. The coefficient of friction is calculated by:
FIGURE 108 The forces acting on a block being pulled across a table. 
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The coefficient of friction is a dimensionless number. The magnitude of this coefficient depends on the nature of the interfacing surfaces. The greater the magnitude of the coefficient of friction, the greater the interaction between the molecules of the interfacing surfaces.
Continuing the block and table example, if a steadily increasing force is applied to the block, the table also applies an increasing opposite force resisting the movement. At the point where the pulling force is at a maximum and no movement results, the resisting force is called the maximum static friction force (F_{MAX}). Before movement, it can be stated that:
where µ_{s} is the static coefficient of friction. At some point, however, the force is sufficiently large and the static friction force cannot prevent the movement of the block. This relationship simply means that if a block weighing 750 N is standing on a surface with µs of 0.5, it will take 50% of the 750 N normal force, or 375 N, of a horizontal force to cause motion between the block and the table. A µs of 0.1 would require a horizontal force of 75 N to cause motion and a µs of 0.8 would require a 600 N horizontal force. As can be seen, the smaller the coefficient of friction, the less horizontal force required to cause movement.
As the block slides along the surface of the table, molecular bonds are continually made and broken. Thus, after the two surfaces start moving relative to each other, it becomes somewhat easier to maintain the motion. The result is a force of sliding friction that opposes the motion. Sliding friction and rolling friction are types of kinetic friction. Kinetic friction is defined as:
where µ_{k} is the dynamic coefficient of friction or the coefficient of friction during movement. It has been found experimentally that µ_{k} is less than µs and that µ_{k} depends on the relative speed of the object. At speeds of one centimeter per second to several meters per second, however, µk is relatively constant. Figure 109 illustrates the frictionexternal force relationship.
Although translational friction is important in human movement, rotational friction must also be considered. Rotational friction is the resistance to rotational or twisting movements. For example, the soles of the shoes of a basketball player accomplishing a pivot interact with the playing surface to resist the turning of the foot. Obviously, the player must be able to accomplish this movement during a game, so the rotational friction must allow this motion without influencing the other frictional characteristics of the shoe. A basketball player executing a 180° pivot in a conventional basketball shoe on a wooden floor would have a rotational friction value 4.3 times greater than if the individual completed the same movement in gym socks (78). The measurement of rotational friction does not yield a coefficient of friction. The values used to compare rotational friction are based on the value of the resistance to rotation, usually measured on a force platform. For example, rotational friction values for a tennis shoe on artificial turf or artificial grass have been shown to range from 15.8 to 21.2 Nm (newtonmeters) and 17.1 to 21.2 Nm, respectively (59). Translational and rotational friction are not independent of each other.
FIGURE 109 A theoretical representation of friction force as a function of the applied force. The applied force increases with the friction force until motion occurs. 
Friction is a complicated but important influence on human movement. Just to walk across a room requires an appropriate coefficient of friction between the shoe outsole and the surface of the floor. In everyday activities, one may try either to increase or to decrease the coefficient of friction, depending on the activity. For example, skaters prefer fresh ice because it has a low coefficient of friction. On the other hand, a golfer wears a glove to increase the coefficient of friction and get a better grip on the club.
Many types of athletes wear cleated shoes to increase the coefficient of friction and get better traction on the playing surface. Valiant (76) suggested that µ equaling 0.8 provides sufficient traction for athletic movements and any greater coefficient of friction may be unsafe. In certain situations, cleated shoes may, in fact, result in too much translational and/or rotational friction. This appears to be the case with artificial turf. Many injuries, such as turf toe and anterior cruciate ligament tears, have been related to too much friction force on artificial turf.
When the coefficient of friction is too small, a slip hazard results, but when the coefficient of friction is too great, a trip hazard occurs. In the workplace, slips and falls are numerous and often cause serious injury. Cohen and Compton (20) reported that 50% of 120, 682 workers compensation cases in the state of New York from 1966 to 1970 were caused by slips. In England, Buck (12) cited 1982 statistics in which 14% of accidents in the manufacturing industry resulted from slips and trips. Thus, the coefficient of friction is a very important criterion in the design of any surface on which people perform, whether in the workplace or in athletics.
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FIGURE 1010 A towed sled device that measures applied force. The known weight constitutes the normal force, and the spring gauge measures the resistant horizontal force on the known weight. The coefficient of friction is computed by the ratio of the spring gauge force to the known weight. 
The static coefficient of friction of materials on different surfaces has been successfully measured using a towed sled device (2). This device, illustrated in Figure 1010, involves an object of a known mass and a forcemeasuring gauge. The moving surface, usually some type of footwear, is placed beneath the mass. The gauge pulls the mass until it moves, and the force is measured by the gauge at the instant of movement. The coefficient of friction can then be calculated using the known mass and the force measured from the gauge. This type of measurement is known as a materials test measure; it does not involve human subjects.
It is difficult to measure the static or kinetic coefficient of friction accurately without sophisticated equipment. Both, however, may be measured with a force platform. The shear components Fy and Fx are, in fact, the frictional forces in the anteroposterior and mediolateral directions, respectively. If the normal force is known, the dynamic coefficient of friction may be estimated. Generally, this is done using the vertical (Fz) component as the normal force. Thus, the coefficient of friction can be determined by:
Several researchers have devised instruments to measure both translational and rotational friction. A device developed in the Nike Sports Science Laboratory is one such instrument (Fig. 1011). It has been used to measure the friction characteristics of many types of athletic shoes.
In general, the magnitude of the coefficient of friction depends on the types of materials constituting the surfaces in contact and the nature of those surfaces. For example, a rubbersoled shoe would have a higher coefficient of friction on a wood gymnasium floor than a leathersoled shoe. Jogging shoes on an artificial track register static and dynamic frictional coefficients in the range of 0.7 to 1.1 and 0.7 to 1.0, respectively (59). This is compared with
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football shoes on artificial turf, for which the static and dynamic coefficients of friction range from 1.1 to 1.6 and 1.0 to 1.5, respectively (59). The relative roughness or smoothness of the contacting surfaces also affects the coefficient of friction. Intuitively, a rough surface has a higher coefficient of friction than a smooth surface. The addition of lubricants, moisture, or dust to a surface also greatly affect frictional characteristics. To determine the coefficient of friction, all of these factors must be considered.
FIGURE 1011 Device for mechanically evaluating translational and rotational friction characteristics of athletic footwear outsoles. This device was developed at the Nike Sports Research Laboratory. 
Fluid Resistance
In many activities, human motion is affected by the fluid in which the activities are performed. Both air (a gas) and water (a liquid) are considered fluids. Thus, the motion of a runner is affected by the movement of air and that of a swimmer by the water or the airwater interface. Projectiles, whether humans or objects, are also affected by air. For example, anyone who has ever driven a golf ball into the wind will understand the effects of air on the golf ball.
Density and Viscosity
The two properties of a fluid that most affect objects as they pass through it are the fluid's density and viscosity. Density is defined as the mass per unit volume. Generally, the more dense the fluid, the greater the resistance it presents to the object. The density of air is particularly affected by humidity, temperature, and pressure. Viscosity is a measure of the fluid's resistance to flow. For example, water is more viscous than air, with the result that water resistance is greater than air resistance. Gases such as air become more viscous as the air temperature rises.
As an object passes through a fluid, it disturbs the fluid. This is true for both air and water. The degree of disturbance depends on the density and viscosity of the fluid. The greater the disturbance of the fluid, the greater the energy that is transmitted from the object to the fluid. This transfer of energy from the moving object to the fluid is called fluid resistance. The resultant fluid resistance force can be resolved into two components, lift and drag (Fig. 1012).
Drag Force Component
Drag is a component of fluid resistance that always acts to oppose the motion. The direction of drag is always directly opposite to the direction of the velocity vector and acts to retard the motion of the object through the fluid. In most instances, drag is synonymous with air resistance. The magnitude of the drag component may be determined by:
where C_{d} is a constant, the coefficient of drag, A is the projected frontal area of the object; the Greek letter rho (p) is the fluid viscosity; and v is the relative velocity of the object, that is, the velocity of the object relative to the fluid. The magnitude of the drag component is a function of the nature of the fluid, the nature and shape of the object, and the velocity of the object through the fluid.
Two types of drag must be considered. Drag as a result of friction between the object's surface and the fluid is referred to as surface drag or viscous drag. When an object moves through a fluid, the fluid interacts with the surface of the object, literally sticking to its surface. The resulting fluid layer is called the boundary layer. The fluid in the boundary layer is slowed down relative to the object as it passes it by. It results in the object pushing on the fluid and the fluid pushing on the object in the opposite direction. This interaction causes friction between the fluid in the boundary layer and the object's surface. This fluid friction opposes the motion of the object through the fluid. A fluid with high viscosity will generate a high drag component. In addition, the size of the object becomes more important if more surface is exposed to the fluid.
From the formula for drag force, it can be seen that drag force increases as a function of the velocity squared. The relative velocity of the fluid as it passes by the object actually determines how the object will interact with the fluid. At lower movement velocities of the object, the fluid passes the object in uniform layers of differing speed, with the slowestmoving layers closest to the surface of the object. This is
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called laminar flow (Fig. 1013A). Laminar flow occurs when the object is small and smooth and the velocity is small. The drag force consists almost entirely of surface or friction drag. On every object, however, there are points, called the points of separation, at which the fluid separates from the object. That is, the fluid does not completely follow the contours of the shape of the object. As the object moves through the fluid faster, the fluid moves by the object faster. When this occurs, the points of separation move forward on the object and the fluid separates from the contours of the object closer to the front of the object. Thus, at relatively high velocities, the fluid does not maintain a laminar flow; rather, separated flow occurs (Fig. 1013B). Separated flow is also referred to as partially turbulent flow. In this instance, the fluid is unable to contour to the shape of the object, and the boundary layer separates from the surface. This produces turbulence behind the object. In power boating, for example, the turbulence behind the boat is called the wake. The wake of an object moving through a fluid is a lowpressure region. Separated flow occurs at a small relative velocity if the object is large and has a rough surface or under any conditions in which the fluid does not stick to the surface of the object.
FIGURE 1012 Fluid resistance vector with its lift and drag components. 
FIGURE 1013 Flow about a sphere. A. Laminar flow. B. Separated flow. C. Turbulent flow. 
When the fluid contacts the front of the object, an area of relatively high pressure is formed. The turbulent area behind the object is an area of pressure lower than the pressure at the front of the object. The greater the turbulence, the lower the pressure is behind the object. The net pressure differential between the front and back of the object retards the object's movement through the fluid. With increasing flow velocity, the point at which the boundary layer separates from the surface of the object moves farther to the front of the object, resulting in an even greater pressure differential and greater resistance. Drag resulting from this pressure differential is called form drag. In partially turbulent flow, both form and friction drag occur. As the wake increases, form drag dominates.
As the relative velocity of the object and the fluid increases, the whole boundary layer becomes turbulent. This type of fluid flow is called turbulent flow (Fig. 1013C). Interestingly, the turbulence in the boundary layer actually moves the point of separation toward the back of the object, reducing the ability of the boundary layer to separate from the object. The net result is a reduction in the drag force.
Moving from partially turbulent flow to fully turbulent flow, thus decreasing the drag force, can be accomplished by streamlining and smoothing the surface of the object. Contrary to what one might expect, putting dimples on a golf ball or seams on a baseball, thereby roughening the surface, actually helps in this transition.
The shift from laminar to partially turbulent flow can be forestalled, hence drag minimized, with a streamlined shape or smooth surface or both. Athletes such as sprinters, cyclists, swimmers, and skiers generally wear smooth suits during their events (Fig. 1014). A significant 10% reduction of drag occurs when a speed skater wears a smooth body suit (80). Wearing smooth clothing also prevents such things as long hair, laces, and loosefitting clothing from increasing drag (47). Kyle (46) reported that loose clothing or thick long hair could raise the total drag 2% to 8%. He calculated that a 6% decrease in air resistance can increase the distance of a long jump by 3 to 5 cm.
Streamlining the shape of the object involves decreasing the projected frontal area. The projected frontal area of the object is the area of the surface that might come in contact with the fluid flow. Athletes in sports in which air resistance must be minimized manipulate this frontal area constantly. For example, a speed skater can assume any of a number of body positions during a race. A skater who has the arms hanging down in front presents a greater frontal area than one in an arms back racing position. The frontal areas in these speed skating positions are 42.21 m^{2} and 38.71 m^{2}, respectively (84). Similarly, a ski racer assumes a tuck position to minimize the frontal area rather than the posture of a recreational skier. To decrease the drag component, a more streamlined position must be assumed. Streamlining helps minimize the pressure differential and thus the form drag on the object. For example, whereas the drag coefficient for a standing human figure is 0.92, it is 0.8 for a runner and 0.7 for a skier in a low crouch (46).
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FIGURE 1014 Examples of clothing used by athletes to reduce drag (Photos provided by Nike, Inc.) 
Much equipment has been designed to minimize fluid resistance. New bicycle designs; solid rear wheels on racing bicycles; clothing for skiers, swimmers, runners, and cyclists; bent poles for downhill skiers; new helmet designs; and so on have all contributed to help these athletes in their events. Research on streamlining body positions has also greatly aided athletes in many sports, such as cycling, speed skating, and sprint running (84).
Although it may seem counterintuitive, drag may also have a propulsive effect in some activities, particularly in swimming. This concept, called propulsive drag, was proposed by the famous Indiana University swimming coach, “Doc” Counsilman (21). This concept uses Newton's third law of actionreaction and states that, as the swimmer moves his or her hand through the water against the direction of motion, the reaction force of the water helps to propel the swimmer though the water. In addition, by changing the orientation of the hand as it moves through the water, a drag force is created in the direction opposite to the movement of the hand, thus further propelling the swimmer forward.
Lift Force Component
Lift is the component of fluid resistance that acts perpendicular to drag. Thus, it also acts perpendicular to the direction of motion. Although there is always a drag force component, the lift component occurs only under special circumstances. That is, lift occurs only if the object is spinning or is not perfectly symmetrical. The lift force component is one of the most significant forces in aerodynamics. This is the force, for example, that helps airplanes fly and makes a javelin and a discus go farther. Contrary to what the name suggests, this force component does not always oppose gravity.
Lift force is produced by any break in the symmetry of the airflow about an object. This can be shown in an object having an asymmetrical shape, a flat object being tilted to the airflow, or a spinning object. The effect makes the air flowing over one side of the object follow a different path than the air flowing over the other side. The result of this differential airflow is lower air pressure on one side of the object and higher air pressure on the other side. This pressure differential causes the object to move toward the side that has the lower pressure. Figure 1015 shows an airfoil (the crosssection of an airplane wing). The air flowing over the top of the airfoil moves at a higher speed than the air flowing beneath it. A principle
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first stated by Daniel Bernoulli in 1738 relates the speed of airflow to the pressure exerted by the fluid. Bernoulli's principle states that pressure is inversely proportional to the velocity of the fluid; it is expressed mathematically as:
FIGURE 1015 Bernoulli's principle in creating lift on an airfoil. Molecules of air A and B move from A_{1} to A_{2} and B_{1} to B_{2}, respectively, in the same amount of time, but the distance from A_{1} to A_{2} is greater. Thus, the velocity of A is greater than the velocity of B, which causes lower pressure on top of the airfoil than underneath. 
where P is the pressure and v is the velocity of the fluid. Thus, when the velocity of a moving fluid increases, the pressure exerted by the fluid decreases and vice versa. The result is that the airfoil develops lift force in the direction of the lower pressure. The lift force concept is used, for example, on the wings of airplanes and on the spoilers on cars.
Lift force also contributes to the curved flight of a spinning ball that is critical in baseball and, most maddeningly, in golf. The spin on the ball results in the air flowing faster on one side of the ball and slower on the other side, creating a pressure differential. Consider the spinning ball in Figure 1016. Side A of the ball is spinning against the airflow, causing the boundary layer to slow down on that side. On side B, however, because it is moving in the same direction as the airflow, the boundary layer speeds up. By Bernoulli's principle, this results in a pressure differential. This is comparable to the pressure differential about the airfoil. The ball, therefore, is deflected laterally toward the direction of the spin or the side on which there is a lower pressure area. This effect was first described by Gustav Magnus in 1852 and is known as the Magnus effect.
Baseball pitchers have mastered the art of putting just enough spin on the ball to curve its path successfully. Soccer players spin their kicks to induce a curved path on the ball (Fig. 1017). In this example, the defenders set up to prevent the ball from traveling in a straight line into the net. The player taking the free kick curves the ball around the defensive wall. Many golfers try not to put a sideways spin on the ball to avoid slicing or hooking the ball. They do, however, try to put backspin on the golf ball. The backspin, because of the Magnus effect, creates a pressure differential between the top and the bottom of the golf ball, with the lower pressure on the top. The golf ball gains lift and thus distance.
Just as there is a propulsive drag force in swimming, we can also discuss the concept of propulsive lift (21). A swimmer holds his or her hands so that they resemble airfoils. Pitching or changing the orientation of the hands puts the lift component in the desired direction of movement (5,69). Thus, lift can contribute to the forward motion of the swimmer.
Inertial Force
In many instances in human movement, one segment can exert a force on another segment, causing a movement in that segment that is not due to muscle action. When this occurs, an inertial force has been generated. Generally, a more proximal segment exerts an inertial force on a more distal segment. For example, during the swing phase of running, the ankle is plantarflexed at takeoff and slightly dorsiflexed at touchdown. The ankle is relaxed during the swing phase, and in fact, the muscle activity at this joint is very limited. The leg also swings through, however, and exerts an inertial force on the foot segment, causing the foot to move to the dorsiflexed position. Similarly, the thigh segment exerts an inertial force on the leg.
Muscle Force
When a force was defined, it was noted that a force constituted a push or pull that results in a change in velocity. A muscle can generate only a pulling or tensile force and therefore has only unidirectional capability. The biceps brachii, for example, pulls on its insertion on the forearm to flex the elbow. To extend the elbow, the triceps brachii must pull on its insertion on the forearm. Thus, the movements at any joint must be accomplished by opposing pairs of muscles. Gravity also assists in the motion of segments.
In most biomechanical analyses, it is assumed that a muscle force acting across a joint is a net force. That is, the
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force of individual muscles acting across a joint cannot be taken into consideration. Generally, a number of muscles act across a joint. Each of these muscles constitutes an unknown value. Mathematically, the number of unknown values must have a comparable number of equations. Because there is no comparable number of equations, there cannot be a solution for each individual muscle force. If a solution were attempted, it would result mathematically in an indeterminate solution, that is, no solution. Thus, we can only calculate the net effect of all the muscles that cross the joint.
FIGURE 1016 The Magnus effect on a spinning ball. Because of this effect, the ball will curve in the direction of B. 
FIGURE 1017 A soccer kick curves a free kick around a wall of defenders. 
It is also assumed that the muscle force acts at a single point. This assumption is again not completely correct because the insertions of muscles are rarely, if ever, single points. Each muscle can be represented as a single force vector that is the resultant of all forces generated by the individual muscle fibers (Fig. 1018). The force vector can be resolved into its components, one component (Fy) acting to cause a rotation at the joint and the other (Fx) acting toward the joint center. If the angle 9 is considered, it can be seen that as 9 gets large, such as when the joint is flexing, the rotational component increases while the component acting toward the joint center decreases. This can be assumed because:
and
At 9 = 90°, the rotational component is a maximum because the sin 90° = 1, but the component acting toward the joint center is zero because cos 0 = 0.
The actual muscle force in vivo is very difficult to measure. To do so requires either a mathematical model or the placement of a measuring device called a force transducer on the tendon of a muscle. Many researchers have developed mathematical models to approximate individual muscle forces (41,71). To do this, however, it is necessary to make a number of assumptions, including the direction
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of the muscle force, point of application, and whether cocontraction occurs.
FIGURE 1018 A muscle force vector, the angle of pull (0), and its vertical and horizontal components. 
For example, a simple model to determine the peak Achilles tendon force during the support phase of a running stride would use the peak vertical GRF and the point of application of that force. It must be assumed that the line of this force is a specific distance anterior to the ankle joint and the Achilles tendon is located a specific distance posterior to the ankle joint. In addition, the center of mass of the foot and its location relative to the ankle joint center must be determined. In this rather simple example, the number of anatomical assumptions that must be made is evident.
The second technique, placing a forcemeasuring device on the tendon of a muscle, requires a surgical procedure. Komi et al. (45) and Komi (43) have placed force transducers on individuals' Achilles tendons. Komi (43) reported peak Achilles tendon forces corresponding to 12.5 BW while the subject ran at 6 m/s. Although the technique to measure in vivo muscle forces is not well developed for longterm studies, it can be used to analyze several important parameters in muscle mechanics.
Elastic Force
When a force is applied to a material, the material undergoes a change in its length. The algebraic statement that reflects this relationship is:
where k is a constant of proportionality and As is the change in length. The constant k represents stiffness, or the ability of the material to be compressed or stretched. A stiffer material requires a greater force to compress or stretch it. This relationship is often applied to biological materials and represented in stressstrain relationships. The stressstrain relationship is explained in detailed in Chapter 2.
The effect of elastic force can be visualized in an example of a diver on a springboard. The diver uses BW as the force to deflect the springboard. The deflected springboard stores an elastic force that is returned as the springboard rebounds to its original state. The result is that the diver is flung upward. A considerable amount of work has been conducted to determine the elasticity of diving springboards used in competition (9,73).
In most situations, the biological tissue–muscles, tendons, and ligaments–do not exceed their elastic limit. Within this limit, these tissues can store force when they are stretched, much as a rubber band does. When the loading force is removed, the elastic force may be returned and, with the muscle force, contribute to the total force of the action. For example, using a prestretch before a movement increases the force output by inducing the elastic force potential of the surrounding tissues. There is, however, a time constraint on how long this elastic force can be stored. Attempts to measure the effect of stored elastic force have illustrated that using this force can affect oxygen consumption (3). Further estimates of stored elastic force in vertical jumping have been investigated by Komi and Bosco (44), who reported higher jumps using stored elastic force. It has been suggested by Alexander (1) that elastic force storage is important in the locomotion of humans and many animals, such as kangaroos and ostriches.
Representation of Forces Acting on a System
When one undertakes an analysis of any human movement, one must take into account a number of forces acting on the system. To simplify the problem for better understanding, a free body diagram is often used. A free body diagram is a stick figure drawing of the system showing the vector representations of the external forces acting on the system. In biomechanics, the system refers to the total human body or parts of the human body and any other objects that may be important in the analysis. It is critically important to define the system correctly; otherwise, extraneous variables may confound the analysis. External forces are those exerted outside the system rather than from inside the system. Thus, internal forces are not represented on a free body diagram.
After the system has been defined, the external forces acting upon the system must be identified and drawn. Figure 1019 is a free body diagram of a total body sagittal view of a runner. The external forces acting on the runner
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are the GRF, friction, fluid or air resistance, and gravity as reflected in the runner's BW. Vector representations of the external forces are drawn on the stick figure at the approximate point of application. If the runner is carrying an implement, such as a wrist weight, another force vector representing the weight of this implement must be added to the free body diagram (Fig. 1020). For the most part, however, the four external forces noted earlier are the only ones identified on a total body diagram.
FIGURE 1019 A free body diagram of a runner with the whole body defined as the system. 
FIGURE 1020 A free body diagram of a runner's wrist weight system. 
When a specific segment, not the total body, is defined as the system, the interpretation of what constitutes an external force must be clarified. In drawing a free body diagram of a particular segment, the segment must be isolated from the rest of the body. The segment is drawn disconnected from the rest of the body and all external forces acting on that segment are drawn. The muscle forces that cross the proximal or distal joints of that segment are external to the system and must be classified as external forces. As noted earlier, it is not possible to identify all of the muscles and their forces acting across a joint. An idealized net muscle force, that is, a single force vector, is used to represent the sum total of all muscle forces.
Figure 1021 is a free body diagram of the forearm of an individual doing a biceps curl. The four external forces acting on this system that must be identified are the net biceps muscle force, the joint reaction force, the force of gravity on the arm represented by the weight of the forearm, and the force of gravity acting on the barbell or the weight of the barbell. These are drawn as they would act during this movement. In many instances, the joint reaction force and the net muscle force are not known but must be calculated. All other forces, such as the friction forces in the joints and the forces of the ligaments and tendons, are assumed to be negligible. Air resistance is also ignored.
FIGURE 1021 A free body diagram of the forearm during a biceps curl. 
Free body diagrams are extremely useful tools in biomechanics. Drawing the system and identifying the forces acting upon the system define the problem and determine how to undertake the analysis.
Analysis Using Newton's Laws of Motion
Multiple conceptual forms and variations of Newton's laws can describe the relationship between the kinematics and the kinetics of a movement. From Newton's law of acceleration (F = ma) arise three general approaches to exploring kinematic and kinetic interactions. These approaches can be categorized as the effect of a force at an instant in time, the effect of a force applied over a period of time, and the effect of a force applied over a distance (54). None of these methods can be considered better or worse than any other. The choice of which relationship to use simply depends upon which method will best answer the question that you are asking. Using the appropriate analytical technique, however, makes it possible to investigate the forces causing motion more effectively.
Effects of a Force at an Instant in Time
When considering the effects of a force and the resulting acceleration at an instant in time, Newton's second law of motion is considered:
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Two situations based on the magnitude of the resulting acceleration can be defined. In the first situation, the resulting acceleration has a zero value. This is the branch of mechanics known as statics. In the second case, the resulting acceleration is a nonzero value. This area of study is known as dynamics.
Static Analysis
The static case is devoted to systems at rest or moving at a constant velocity. In both of these situations, the acceleration of the system is zero. When the acceleration of a system is zero, the system is said to be in equilibrium. A system is in equilibrium when, as stated in Newton's first law, it remains at rest or it is in motion at a constant velocity.
In translational motion, when a system is in equilibrium, all forces that are acting on the system cancel each other out, and the effect is zero. That is, the sum of all forces acting on the system must total zero. This is expressed algebraically as:
The forces in this equation can be further expressed in terms of the twodimensional x and ycomponents as:
and
Here the sum of the forces in the horizontal (x) direction must equal zero and the sum of the forces in the vertical (y) direction must equal zero.
The static case is simply a particular example of Newton's second law and can be described in terms of a causeandeffect relationship. The lefthand side of these equations describes the cause of the motion, and the righthand side describes the product or the result of the motion. Because all forces in the system are in balance, there is no acceleration. If the forces were not in balance, some acceleration would occur.
Figure 1022 presents a free body diagram of a linear force system in which a 100N box rests on a table. Gravity acts to pull the box downward on the table with a force of 100 N. Because the box does not move vertically, an equal and opposite force must act to support the box. In this coordinate system, up is positive and down is negative. The weight of the box, acting downward, thus has a negative sign. There are no horizontal forces acting in this example. Thus, the reaction force R is:
FIGURE 1022 A free body diagram of a box on a table. The box is in equilibrium because there are no horizontal forces and the sum of the vertical forces is zero. 
R_{y} is the reaction force equal to the weight of the box. Because the weight of the box acts negatively, the reaction force must act positively or in the opposite direction to the weight of the box.
Consider a system with multiple forces acting. In Figure 1023, a tugofwar is presented as a linear force system. In this example, the two contestants on the right balance the three contestants on the left. The contestants on the left exert forces of 50 N, 150 N, and 300 N, respectively. These forces can be considered to act in a negative horizontal direction. Assuming a static situation, the reaction force (R) to produce equilibrium can be calculated. Thus:
FIGURE 1023 Tugofwar. The system is in equilibrium because the sum of the forces in the horizontal direction is zero. No movement to the left or the right can occur. 
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FIGURE 1024 A. A force system in which the sum of the forces is zero. B. A free body diagram of the force system shows the horizontal and vertical components of all forces. 
The two contestants on the right must exert a reaction force of 500 N in the positive direction to produce a state of equilibrium.
The linear force system previously presented is a relatively simple example of the static case, but in many instances in human motion, the forces are nonparallel. In Figure 1024A, two nonparallel forces F_{1} and F_{2}, act n a rigid body in addition to the weight of the rigid body. For this system to be in equilibrium, a third force (F_{3}) must act through the intersection of the two nonparallel forces. The free body diagram in Figure 1024B illustrates that the horizontal component of F_{3}, acting in a positive direction, must counterbalance the sum of the horizontal components of the nonparallel forces F_{1} and F_{2}. Also, the vertical components F_{1} and F_{2} must be counterbalanced by the weight of the rigid body and by the vertical component of F_{2}. If F_{1} = 100 N, the components of F_{1} are:
and if F_{2} = 212.13 N, the components of F_{2} are:
The weight of the rigid body, 50 N, also acts in a negative vertical direction. Thus:
F_{3}y must have a magnitude of 150 N to maintain the system in equilibrium in the vertical direction. In the horizontal direction:
To balance the two nonparallel forces in the horizontal direction, a force of 236.6 N is required. The resultant force, F_{3}, can be determined using the Pythagorean relationship:
The F_{3} force orientation can be determined using trigonometric functions:
The forces F_{1} and F_{2} and the weight of the rigid body are counteracted by the force F_{3}, thus keeping the system in equilibrium.
A second condition that determines whether a system is in equilibrium occurs when the forces in the system are not concurrent. Concurrent forces do not coincide at the same point, so they cause rotation about some axis. These rotations all sum to zero, however. Because this is a static case, no rotation occurs. This will be discussed in more detail in Chapter 11.
Static models have been developed to evaluate such tasks such as material handling and lifting. A free body diagram of the joint reaction forces and forces acting at the center of mass of the segment is created. Figure 1025 is a static lifting model (18) that shows the linear forces acting on the body at the shoulder, elbow, wrist, hip, knee, ankle joints and the ground contact. This model is not complete until the angular components are also included (see Chapter 11).
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FIGURE 1025 A free body diagram of a sagittal view static lifting model showing the linear forces at the joints and segments. (Adapted from Chaffin, D. B., Andersson, G. B. J. [1991]. Occupational Biomechanics [2nd ed.]. New York: Wiley ). 
Dynamic Analysis
A static analysis may be used to evaluate the forces on the human body when acceleration is insignificant (4). When accelerations are significant, however, dynamic analysis must be undertaken. A dynamic analysis should be used, therefore, when accelerations are not zero. The equations for a dynamic analysis were derived from Newton's second law of motion and expanded by the famous Swiss mathematician Leonhard Euler (17071783). The equations of motion for a twodimensional case are based on:
Linear acceleration may be broken down into horizontal (x) and vertical (y) components. As in the static twodimensional analysis, independent equations are used in a dynamic twodimensional linear kinetic analysis:
where x and y represent the horizontal and vertical coordinate directions, respectively, a is the acceleration of the center of mass, and m is the mass. The forces acting on a body may be any one of the previously discussed forces such as muscular, gravitational, contact, or inertial. The gravitational forces are the weights of each of the segments. The contact forces can be reactionsforces with another segment, the ground, or an external object–and the inertial forces are ma_{x} and ma_{y}. Using the equations of dynamic motion, the forces acting on a segment can be calculated.
In moving from static to dynamic analysis, the problem becomes more complicated. In the static case, no accelerations were present. In the dynamic case, linear accelerations and the inertial properties of the body segments resisting these accelerations must be considered. In addition, there is a substantial increase in the work done to collect the data necessary to conduct a dynamic analysis. Because the forces that cause the motion are determined by evaluating the resulting motion itself, a technique called an inverse dynamics approach will be used. This method is often referred to as a NewtonEuler inverse dynamics approach. This approach calculates the forces based on the accelerations of the object instead of measuring the forces directly.
In using the inverse dynamics approach, the system under consideration must be determined. The system is usually defined as a series of segments. The analysis on a series of segments is generally conducted beginning with the most distal segment, proceeding proximally up to the next segment, and so on. Several assumptions must be made when using this approach. The body is considered to be a rigid linked system with frictionless pin joints. Each link, or segment, has a fixed mass and a center of mass at a fixed point. Finally, the moment of inertia about any axis of each segment remains constant. Moment of inertia is discussed in Chapter 11.
As stated previously, the dynamic case is more complicated than the static case. As a result, only a limited example of a single segment will be presented. In Figure 1026 a free body diagram of the foot of an individual during the swing phase of the gait cycle is presented for the linear forces acting on the segment.
During the swing phase of gait, no external forces other than gravity act on the foot. It can be seen that the only linear forces acting on the foot are the horizontal and vertical components of the joint reaction force and the weight of the foot acting through the center of mass. The joint reaction force components can be computed with the twodimensional linear kinetic equations defining the dynamic analysis. First, the horizontal joint reaction force may be defined:
Because there are no horizontal forces other than the horizontal joint reaction force, this equation becomes:
If the mass of the foot is 1.16 kg and the horizontal acceleration of the center of mass of the foot is –1.35 m/s^{2}, the horizontal reaction force is:
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FIGURE 1026 Free body diagram of the foot segment during the swing phase of a walking stride showing linear forces and accelerations. 
Next, the vertical joint reaction force component may be defined from:
There is, however, a vertical force other than the vertical joint reaction force. This force is the weight of the foot itself, so the vertical forces are described as:
and solving for R_{y}, the equation becomes:
If the vertical acceleration of the center of mass of the foot is 7.65 m/s^{2}, then:
Refer to the walking data in Appendix C. Identify the maximum vertical peak force. Using the body mass of the participant (50 kg), compute the vertical acceleration at that point. Repeat for the maximum anteroposterior and mediolateral forces.
Effects of Force Applied Over a Period of Time
For motion to occur, forces must be applied over time. Manipulating the equation describing Newton's second law of motion allows generation of an important physical relationship in human movement that describes the concept of forces acting over time. This relationship relates the momentum of an object to the force and the time over which the force acts. This relationship is derived from Newton's second law:
Because a = dv/dt, this equation can be rewritten as:
and further:
If each side is multiplied by dt to remove the fraction on the righthand side of the equation, the resulting equation is:
or
The quantity mv (mass ^{*} velocity) refers to the momentum of the object. The right hand side of this equation then refers to the change in momentum. The lefthand side of this equation, the product of F ^{*} dt, a quantity known as an impulse and has units of newtonseconds (N ^{*} s). Impulse is the measure that is required to change the momentum of an object. The derived equation describes the impulsemomentum relationship.
Figure 1027 illustrates the vertical component of the GRF of a single footfall of a runner. This figure represents a force applied over time as the foot is in contact with the surface, generating a downward force and receiving an equal and opposite reaction force. Impulse may be expressed graphically as the area under a forcetime curve.
Consider an individual with a mass of 65 kg jumping from a squat position into the air. The velocity of the person at the beginning of the jump is zero. Video analysis reveals
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that the velocity of the center of mass at takeoff was 3.4 m/s. The impulse can then be calculated as:
FIGURE 1027 Vertical GRF of a subject running at 5 m/s. The vertical impulse is the shaded area under the forcetime curve. 
If we assume that the force application took place over 0.2 s, the average force applied would be:
The nature of the force applied and the time over which it is applied determine how the momentum of the object is changed. To change the momentum of an object, a force of a large magnitude can be applied over a short period of time, or a force of a smaller magnitude over a long period of time. Of course, the tactic used depends on the situation. For example, in landing from a jump, the performer must change momentum from some initial value to zero. The initial momentum value at impact is a function of body mass times the velocity created by the force of gravity that causes the jumper to accelerate toward the ground at a rate of 9.81 m/s^{2}. At floor impact, an impulse is generated to change the momentum, and reduce it to zero as the jumper stops. If an individual lands with little knee flexion (locked knees), the impact force occurs over a very short period. If the individual lands and flexes the joints of the lower extremity, however, the impact force is smaller and occurs over an extended time. If in both cases, the jumper is landing with the same velocity and consequently the same momentum, the resulting impulses will also be the same, even though there are different force (large vs. smaller force) and time (short vs. extended) components.
An example of soft and rigid landings is presented in Figure 1028. This example, taken from a study by DeVita and Skelly (23), shows a large peak force in the rigid landing compared with the smaller force in the soft landing. Because both appear to take place over about the same amount of time, there is a greater impulse in the rigid landings.
Using the data in Appendix C, graph the vertical, anteroposterior, and mediolateral ground reaction curves for the support phase of walking and shade in the impulse for each curve.
One interesting research application for the use of the impulsemomentum relationship has been in vertical jumping. This area of research has used the force platform to determine the GRF during the vertical jump. Remember that the GRF reflects the force acting on the center of mass of the individual. Thus, researchers have used the impulsemomentum relationship on data collected from the force platform to determine the parameters necessary to investigate the height of the center of mass above its starting point during vertical jumping (44). Figure 1029A shows the vertical GRF profile of an individual starting at rest on the platform and jumping into the air. This is called a countermovement jump because the subject flexes at the knees and then swings the arms up as the knees are extended during the jump. Figure 1029B illustrates a squat jump, in which the subject begins in a squat (knees flexed) and simply forcefully extends the knees to jump into the air.
FIGURE 1028 Vertical ground reaction force for soft and stiff landings. Stiff landings have a 23% greater linear impulse than the soft landing. (Adapted from DeVita, P., Skelly, W. A. [1992]. Effect of landing stiffness on joint kinetics and energetics in the lower extremity. Medicine and Science in Sports and Exercise, 24:108115 ). 
In both cases represented in Figure 1029, the impulsemomentum relationship can be used to calculate the peak height of the center of mass above the initial height during the jump. Consider the vertical GRF curve of a counterjump in Figure 1030. The constant vertical force in the initial portion of the curve is the individual's BW. If the BW line is extended to the instant of takeoff, the area beneath the curve describes the BW impulse (BW_{imp}). It may be calculated as an integral (i.e., determining the area under the curve:
where t_{i} to t_{f} represents the time interval when the subject is standing still on the force platform until the instant of takeoff. The total area under the forcetime curve until the subject leaves the force platform may be designated as total, and can be calculated as an integral:
where t_{i} to t_{jump} represents the time when the subject is on the platform before the jump. The impulse that propelled the subject into the air can then be determined by:
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FIGURE 1029 The vertical GRF of two types of vertical jumps: the countermovement jump (A) and the squat jump (B). 
The impulsemomentum relationship can then be formulated using the impulse that the subject generated to perform the jump. Thus:
where v_{i} is the initial velocity of the center of mass and v_{f} is the takeoff velocity of the center of mass. Because v = 0:
By substituting the impulse of the jump (jump_{imp}) and the subject's body mass (m) in this equation, the velocity of the center of mass at takeoff for the vertical jump can be calculated. The height of the center of mass during the jump is then calculated based on the projectile equations elaborated upon in Chapter 8. Thus:
This calculation has proved to be in good agreement with values calculated from highspeed film data. Komi and Bosco (44) reported an error of 2% from the computation on the force platform.
FIGURE 1030 The vertical ground reaction force of a countermovement jump illustrating the body weight impulse and the jump impulse. 
Consider the forcetime profile of a countermovement jump in Figure 1030. The BW impulse, the rectangle formed by the BW and the time of force application, was calculated to be:
The total impulse, that is, the total force application from time 0 to time = 1.5 s, including the BW, is:
The jump impulse, therefore, is:
Substituting this value in the impulsemomentum relationship, it is possible to solve for the velocity of the center of mass at takeoff. Thus, with the body mass of the jumper of 56.2 kg, the velocity of takeoff is:
The height of the center of mass during the jump can be calculated by:
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Therefore, in this particular jump, the center of mass was elevated 37.3 cm above the initial height of the center of mass.
This calculation technique was used by Dowling and Vamos (25) to identify the kinetic and temporal factors related to vertical jump performance. They found a large variation in the patterns of force application between the subjects that made it difficult to identify the characteristics of a good performance. Interestingly, they reported that a high maximum force was necessary but not sufficient for a good performance. They concluded that the pattern of force application was the most important factor in vertical jump performance.
Refer to the walking data in Appendix C. Calculate the vertical, anteroposterior, and mediolateral impulse from contact (frame 0) to the first vertical peak (frame 18). Using the impulse values, calculate the velocities generated up to that point.
Effect of a Force Applied Over a Distance
The term work is used to mean a variety of things. Generally, we consider work to be anything that demands mental or physical effort. In mechanics, however, work has a more specific and narrow meaning. Mechanical work is equal to the product of the magnitude of a force applied against an object and the distance the object moves in the direction of the force while the force is applied to the object. For example, in moving an object along the ground, an individual pushes the object with a force parallel to the ground. If the force necessary to move the object is 100 N and the object is moved 1 m, the work done is 100 Nm. The case cited, however, is a very specific one. More generally, work is:
where F is the force applied, s is the displacement, and 9 is the angle between the force vector and the line of displacement. The unit of mechanical work is derived from the product of force in Newtons and displacement in meters. The most commonly used units are the Newtonmeter and the joule (J). These are equivalent units:
In Figure 1031 A, the force is applied to a block parallel to the line of displacement, that is, at an angle of 0° to the displacement. Because cos 0° = 1, the work done is simply the product of the force and the distance the block is displaced. Thus, if the force applied is 50 N and the block is displaced 0.1 m, the mechanical work done is:
FIGURE 1031 The mechanical work done on a block. A. A force is applied parallel to the surface (0 = 0°; thus cos 0 = 1). B. A force is applied at an angle to the direction of motion (0 = 30°; thus, cos 0 = 0.866). 
If the same force is applied at an angle of 30° over the same distance, d, (Fig. 1031B), the work done, is:
Therefore, more work is done if the force is applied parallel to the direction of motion than if the force is applied at an angle.
As this discussion of work implies, work is done only when the object is moving and its motion is influenced by the applied force. If a force acts on an object and does not cause the object to move, no mechanical work is done because the distance moved is zero. During an isometric contraction, for example, no mechanical work is done because there is no movement. A weightlifter holding an 892N (200lb) barbell overhead does no mechanical work. In lifting the barbell overhead, however, mechanical work is done. If the barbell is lifted 1.85 m, the work done is:
This value assumes that the bar was lifted completely vertically.
Power
In evaluating the amount of work done by a force, the time over which the force is applied is not taken into account. For example, when the work done by the weight lifter to raise the barbell overhead was calculated, the time it took to raise the barbell was not taken into consideration.
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Regardless of how long it took to raise the barbell, the amount of work done was 1650.2 J. The concept of power takes into consideration the work done per unit of time. Power is defined as the rate at which a force does work:
where W is the work done and dt is the time period in which the work was done. Power has units of watts (W). The change in work is expressed in joules and the change in time in seconds. Thus:
If power is plotted on a graph as a function of time, the area under the curve equals the work done. If the weight lifter in the previous example raised the bar in 0.5 s, the power developed is:
Decreasing the time over which the bar is lifted to 0.35 s increases the power developed by the weight lifter to 4714.86 W. Although the work done remains constant, greater power must be developed to do the mechanical work more quickly.
Another definition for power can be developed by rearranging the formula. If the product of the force (F) and the distance over which it was applied (s) is substituted for the mechanical work done, the equation becomes:
and rearranging this equation:
Because ds/dt was defined in a previous chapter as the velocity in the sdirection, it can be readily seen that:
where F is the applied force and v is the velocity of the force application.
Power is often confused with force, work, energy, or strength. Power, however, is a combination of force and velocity. In many athletic endeavors, power, or the ability to use the combination of force and velocity, is paramount. One such activity, weight lifting, has already been discussed, but there are many others, such as shot putting, batting in baseball, and boxing. Jumping also requires power. To generate a takeoff velocity of 2.61 m/s in a vertical jump, Harman et al. (40) reported a peak power generation of 3896 W. In comparisons of jump techniques, researchers have noted differences in peak power output between countermovement jumps (men = 4708 W; women, 3069 W) and squat jumps (men, 4620 W; women, 2993 W) (68).
Energy
As with work, the mechanical term energy is often misused. Simply stated, energy is the capacity to do work. The many types of energy include light, heat, nuclear, electrical, and mechanical; the main concern of biomechanics is mechanical energy. The unit of mechanical energy in the metric system is the joule. The two forms of mechanical energy that will be discussed here are kinetic and potential energy.
Kinetic energy
(KE) refers to the energy resulting from motion. An object possesses kinetic energy when it is in motion, that is, when it has some velocity. Linear kinetic energy is expressed algebraically as:
where m is the mass of the object and v is the velocity. Because this expression includes the square of the velocity, any change in velocity greatly increases the amount of energy in the object. If the velocity is zero, the object has no kinetic energy. An approximate value for the kinetic energy of a 625 N runner would be 3600 J; a swimmer of comparable BW would have a value of 125 J.
A moving body must have some energy because a force must be exerted to stop it. To start an object moving, a force must be applied over a distance. Kinetic energy, therefore, is the ability of a moving object to do work resulting from its motion. The generation of a sufficient level of kinetic energy is especially important when projecting an object or body, such as in long jumping, throwing, and batting. For example, kinetic energy is developed in a baseball over the collision phase with the bat and projects the ball at velocities more than 100 mph. The kinetic energy before the collision has been demonstrated to be in the range of 320 J and 115 J for the bat and ball, respectively (30). After the contact, the kinetic energy of the bat is reported to be reduced to 156 J, and the ball's kinetic energy increased to 157 J (28). With bat speeds in the range of 55 to 80 mph and incoming ball speeds of 85 to 100 mph, there is considerable interchange of kinetic energy.
Potential energy
(PE) is the capacity to do work because of position or form. An object may contain stored energy, for example, simply because of its height or its deformation. In the first case, if a 30kg barbell is lifted overhead to a height of 2.2 m, 647.5 J of work is done to lift the barbell. That is:
As the barbell is held overhead, it has the potential energy of 647.5 J. The work done to lift it overhead is also the potential energy. Potential energy gradually increases as the bar is lifted. If the bar is lowered, the potential energy decreases. Potential energy is defined algebraically as:
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where m is the mass of the object, g is the acceleration due to gravity, and h is the height. The more work done to overcome gravity, therefore, the greater the potential energy
An object that is deformed may also store potential energy This type of potential energy has to do with elastic forces. When an object is deformed, the resistance to the deformation increases as the object is stretched. Thus, the force that deforms the object is stored and may be released as elastic energy. This type of energy, strain energy (SE), is defined as:
where k is a proportionality constant and Ax is the distance over which the object is deformed. The proportionality constant, k, depends on the material deformed. It is often called the stiffness constant because it represents the object's ability to store energy.
It has already been discussed how certain tissues, such as muscles and tendons, and certain devices, such as springboards for diving, may store this strain energy and release it to aid in human movement. In athletics, numerous pieces of equipment achieve such an end. Examples are trampolines, bows in archery, and poles in pole vaulting. Perhaps the most sophisticated use of elastic energy storage is the design of the tuned running track at Harvard University. McMahon and Greene (52) analyzed the mechanics of running and the energy interactions between the runner and the track to develop an optimum design for the track surface. In the first season on this new track, an average speed advantage of nearly 3% was observed. Furthermore, it was determined that there was a 93% probability that any given individual will run faster on this new track (51).
Many instances in human motion can be understood in terms of the interchanges between kinetic and potential energy. The mathematical relationship between the different forms of energy was formulated by the German scientist von Helmholtz (18211894). In 1847, he defined what has come to be known as the law of conservation of energy. The main point of this law is that energy cannot be created or destroyed. No machine, including the human machine, can generate more energy than it takes in. It follows, therefore, that the total energy of a closed system is constant because energy does not enter or leave a closed system. A closed system is one that is physically isolated from its surroundings. This point can be phrased mathematically by stating:
where TE is a constant representing the total energy of the system. In human movement, this occurs only when the object is a projectile, whereby the only external force acting upon it is gravity, because fluid resistance is neglected.
Consider the example of a projectile traveling up into the air. At the point of release, the projectile has zero potential energy and a large amount of kinetic energy. As the projectile ascends, the potential energy increases and the kinetic energy decreases because gravity is slowing the upward flight of the projectile. At the peak of the trajectory, the velocity of the projectile is zero and the kinetic energy is zero, but the potential energy is at its maximum. The total energy of the system does not change because increases in potential energy result in equal decreases in kinetic energy. On the downward flight, the reverse change in the forms of energy occurs. These changes in energy are illustrated in Figure 1032.
FIGURE 1032 Changes in potential energy (PE) and kinetic energy (KE) as a ball is projected straight up and as it falls back to earth. 
When an object is moved, mechanical work is said to have been done on the object. Thus, if no movement occurs, no mechanical work is done. Because energy is the capacity to do work, intuitively, there should be some relationship between work and energy. This very useful relationship is called the workenergy theorem. This theorem states that the work done is equal to the change in energy:
where W is the work done and AE is the change in energy. That is, for mechanical work to be done, a change in the energy level must occur. The change in energy refers to all types of energy in the system, including kinetic, potential, chemical, heat, light, and so on. For example, if a trampolinist weighs 780 N and is at a peak height of 2 m above a trampoline bed, the potential energy based on the height above the trampoline bed is:
Assuming no horizontal movement, at impact the kinetic energy is 1560 J, while the potential energy is zero. The kinetic energy has this initial value, and as the trampoline bed deforms, the potential strain energy increases while the kinetic energy goes to zero. The work done on the trampoline bed is:
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This value, 1560 J, is also the value of the potential strain energy of the trampoline bed. As the bed reforms, the potential strain energy changes from this value to zero and constitutes the work done by the trampoline bed on the trampolinist.
To evaluate the work done, the energy level of a system must be evaluated at different instants in time. This change in energy represents the work done on the system. For example, a system has an energy level of 26.3 J at position 2 and 13.1 J at position 1. The work done is:
Because this work has a positive value, it is considered positive work. On the other hand, if the energy level is 22.4 J at position 1 and 14.5 J at position 2, then:
Now the work done is negative and work is said to have been done on the system.
The workenergy relationship is useful in biomechanics to analyze human motion. Researchers have used this analytical method to determine the work done during a number of movements; however, the greatest use of this technique has been in the area of locomotion. The calculation of the total work done resulting from the motion of all of the body's segments is called internal work. This calculation has been used by many researchers, particularly those studying locomotion (60,64,85,87,90). For a single segment, the linear work done on the segment is:
where W_{s} is the work done on the segment, δKE is the change in linear kinetic energy of the center of mass of the segment, and δPE is the change in potential energy of the segment center of mass. Internal work of the total body can then be calculated as the sum of the work done on all segments:
where W_{b} is the total body work and W_{is} the work done on the i^{th} segment. One major limitation of the calculation of internal work is that it does not account for all of the energy of a segment and so does not account for the energy of the total body. For example, the strain energy due to the deformation of tissue and the angular kinetic energy are not considered. Angular work is discussed in Chapter 11.
Internal work for the duration of an activity can be derived by summing the changes in the segment energies over time. That is, the change in energy at each instant in time is summed for the length of time that the movement lasts. Usually, this period in locomotion studies is the time for one stride. The analysis of the mechanical work done is extremely valuable as a global parameter of the body's behavior without a detailed knowledge of the motion. A variety of algorithms may be used to calculate internal work (15,60,64,85,87,90). These models have incorporated factors to quantify such parameters as positive and negative work, the effect of muscle elastic energy, and the amount of negative work attributable to muscular sources. The major difference among these algorithms is the way energy is transferred within a segment and between segments. Energy transfer within a segment refers to changes from one form of energy to another, as in the change from potential to kinetic. Energy transfer between segments refers to the exchange of the total energy of a segment from one segment to another. Presently, no consensus exists as to which model is most appropriate; it has been argued, in fact, that none of these methods is correct (13). Values of mechanical work for a single running stride may range from 532 W to 1775 W (86), although these values should be interpreted with caution.
If it is considered that all energy is transferred between segments, the point in the stride at which the transfer occurs can be illustrated. Figure 1033 graphically illustrates the magnitude of betweensegment energy transfer during the running stride. The magnitude of energy transfer decreases during support and increases from midstance to a maximum after toeoff (86).
When work is calculated over time, such as a locomotor stride, the result is often presented as power with units of watts. These power values have generally been scaled to body mass, resulting in units of watts per kilogram of body mass. Hintermeister and Hamill (42) investigated the relationship between mechanical power and energy expenditure. They reported that mechanical power significantly influenced energy expenditure independent of the running speed. Using several algorithms representing different methods of energy transfer, mechanical power explained at best only 56% of the variance in energy expenditure. Note that these algorithms are often questioned as mechanically incorrect (13).
Two possible explanations have been offered for this rather weak relationship between mechanical work and energy expenditure. The first is that the methods of calculation are incomplete (13). The second involves the law of conservation of energy. This law states that energy is neither created nor destroyed but may be changed from one form to another, that is, potential energy may be changed to kinetic energy or heat energy. Not all of the energy can be or is used to perform mechanical work. In fact, most of the available energy of the muscle will go to maintain the
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metabolism of the muscle; only about 25% of the energy is used for mechanical work (74). Thus, the workenergy theorem, when used to determine the mechanical work of the body, does not account for all of the energy in the system.
FIGURE 1033 The magnitude of betweensegment transfer during a single running stride. (Adapted from Williams, K. R. [1980]. A biomechanical and physiological evaluation of running efficiency. Unpublished doctoral dissertation, The Pennsylvania State University ). 
External work may be defined as the work done by a body on an object. For example, the work done by the body to elevate the total body center of mass while walking up an incline is considered to be external work. External work is often calculated during an inclined treadmill walk or run as:
where BW is the BW of the subject, percent grade is the incline of the treadmill, and duration is the length of time of the walk or run. The product of the treadmill speed, percent grade, and the duration is the total vertical distance traveled. If the percent grade is zero, that is, the walking surface is level, the vertical distance traveled is zero. Thus, walking on a level surface results in no external work being done.
Special Force Applications
Centripetal Force
In Chapter 9, linear and angular kinematics were shown to be related using the situations in which an object moves along a curved path. It was demonstrated that centripetal acceleration acts toward the center of rotation when an object moves along a curved path. This is radial acceleration toward the center of the circle. The radial force occurring along a curved path that generates the acceleration is called the centripetal force. Using Newton's second law of motion, F = ma, a formula for the centripetal force can be generated. The force is no different from other forces and is generated by a push or pull. The force is called centripetal because of the effect: The force generates a change in the direction of the velocity. The magnitude of the centripetal or centerseeking force is calculated by:
where FC is the centripetal force, m is the mass of the object, co is the angular velocity, and r is the radius of rotation. Centripetal force may also be defined as:
where v is the tangential velocity of the segment.
Newton's third law states that for every action there is an equal and opposite reaction. For example, a runner moving along the curve of the running track applies a shear force to the ground, resulting in a shear GRF equal and opposite to the applied force. The shear reaction force constitutes the centripetal force. Figure 1034 is a free body diagram of the runner moving along the curved path, showing the centripetal force, the vertical reaction force, and the resultant of these two force components. This centripetal force at the runner's foot tends to rotate the runner outward. To counteract this outward rotation, the runner leans toward the center of the curve. Hamill et al. (39) reported that this shear GRF increased as the radius of rotation decreased.
The resultant of the vertical reaction force and the centripetal force must pass through the center of mass of the runner. If the centripetal force increases, the runner leans more toward the center of rotation, and the resultant vector becomes less vertical. As mentioned in Chapter 9, banked curves on tracks reduce the shear force applied by the runner and thus reduce the centripetal force. As the centripetal force is reduced, the runner reduces the lean toward the center of the track. The resultant force thus
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acts more vertically as when the runner moves along a straight path.
FIGURE 1034 A free body diagram of a runner on the curve of a running track. F_{C} is the centripetal force, F_{V} is the vertical reaction force, and R is the resultant of F_{C} and F_{V}. 
Pressure
Up to this point in the discussion of force, the way a force causes an object to accelerate to achieve a state of motion has been considered. It is also necessary to discuss how forces, particularly impact forces, are distributed. The concept of pressure is used to describe force distribution. Pressure is defined as the force per unit area. That is:
where F is a force and A is the area over which the force is applied. Pressure has units of N/m^{2}. Another unit of pressure often used is the pascal (Pa) or the kilopascal (kPa). One pascal is equal to 1 N/m^{2}. If an individual with a BW of 650 N is supported on the soles of the feet with an approximate area of 0.018 m^{2}, the pressure on the soles of the feet would be:
Another individual with a smaller BW, 500 N, and soles of the feet with an identical area would have a pressure of:
If the heavier individual's foot soles had a larger area, 0.02 m^{2}, for example, the pressure would be 32.50 kPa. The pressure on the soles of the feet of the heavier individual is less than the pressure on the soles of the lighter individual, even though the BW is different. These pressures appear quite large, but imagine if these individuals were women wearing spikeheeled shoes, which have much less surface area than ordinary shoes. A more dramatic example would be if these individuals were wearing ice skates, which have a distinctly smaller area of contact with the surface than the sole of a normal shoe or a spikeheeled shoe. On the other hand, if these individuals were using skis or snow shoes to walk in deep snow, the pressure would be quite small because of the large area of the skis or snowshoes in contact with the snow. In this way, individuals walk on snow without sinking into it.
The concept of pressure is especially important in activities in which a collision results. Generally, when a force of impact is to be minimized, it should be received over as large an area as possible. For example, when landing from a fall, most athletes attempt a roll to spread the impact force over as large an area as possible. In the martial arts, considerable time is spent in learning how to fall correctly, specifically applying the pressure as force per unit area.
A number of sporting activities in which collisions abound have special protective equipment designed to reduce pressure. Examples are shoulder pads in football and ice hockey; shin pads in ice hockey, field hockey, soccer, and baseball (for the catcher); boxing gloves; and batting helmets in baseball. In all of these examples, the point of the design of the protective padding is to spread the impact force over as large an area as possible to reduce the pressure.
With use of a force platform, it is possible to obtain a measure of the center of pressure (COP), a displacement measure indicating the path of the resultant GRF vector on the force platform. It is equal to the weighted average of the points of application of all of the downwardacting forces on the force platform. Because the COP is a general measurement, it may be nowhere near the maximal areas of pressure. However, it does provide a general pattern and has been extensively used in gait analysis. Cavanagh and Lafortune (17) showed different COP patterns for rear foot and midfoot strikers. Representative COP patterns are illustrated in Figure 1035. Cavanagh and Lafortune suggested that COP information may be useful in shoe design, but these patterns have not been related successfully to foot function during locomotion. Miller (55) noted that COP data provide only restricted information on the overall pressure distribution on the sole of the foot.
Methods of measuring the local pressure patterns under the foot or shoe have been developed. An example
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of the type of data available on these systems is presented in Figure 1036. Cavanagh et al. (16) developed such a measuring system and reported distinct local areas of high pressure on the foot throughout the ground contact phase. The greatest pressures were measured at the heel, on the metatarsal heads, and on the hallux. Cavanagh et al. (16) also compared the pressure patterns during running barefoot and with various foam materials attached to the foot. They reported that peak pressures were reduced when wearing the foam materials, but the changes in the pressure pattern over the support period were similar. Foti et al. (32), using an inshoe pressure measurement device, reported that softer midsole shoes distributed the foottoshoe pressure at heel contact during walking better than a hard midsole shoe. The implication is that softer midsole shoes provide a more cushioned feel to the wearer.
FIGURE 1035 Center of pressure patterns for the left foot. A. A heeltoe footfall pattern runner. B. A midfoot foot strike pattern runner. 
FIGURE 1036 Pressure distribution pattern of a normal foot during walking. (Adapted from Cavanagh, P. R. [1989]. The biomechanics of running and running shoe problems. In B. Segesser, W. Pforringer [Eds.]. The Shoe in Sport. London: Wolfe, 315 ). 
Linear Kinetics of Locomotion
GRF profiles continually change with time and are generally presented as a function of time. The magnitude of the GRF components for running are much greater than for walking. The magnitude of the GRFs also varies as a function of locomotor speed (38,56), increasing with running speed. The vertical GRF component is much greater in magnitude than the other components and has received the most attention from biomechanists (Fig. 1037). In walking, the vertical component generally has a maximum value of 1 to 1.2 BW, and in running, the maximum value can be 2 to 5 BW. The vertical force component in walking has a characteristic bimodal shape, that is, it has two maximum values. The first modal peak occurs during the first half of support and characterizes the portion of support when the total body is lowered after foot contact. The force rises above BW as full weight bearing takes place and the body mass is accelerated upward. The force then
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lowers as the knee flexes, partially unloading. The second peak represents the active push against the ground to move into the next step. Figure 1038 presents a comparison of walking and running vertical GRF component profiles.
FIGURE 1037 Ground reaction force for walking (A) and running (B). Note the difference in magnitude between the vertical component and the shear components. 
FIGURE 1038 The vertical ground reaction force for both walking and running. 
In running, the shape of the vertical GRF component depends on the footfall pattern of the runner (Fig. 1039). These footfall patterns are generally referred to as the heeltoe and midfoot patterns. The heel strike runner's curve has two discernible peaks. The first peak occurs very rapidly after the initial contact and is often referred to as the passive peak. The term passive peak refers to the fact that this phase is not considered to be under muscular control (58) and is influenced by impact velocity, contact area between the surface and the foot, the joint angles at impact, surface stiffness, and the motion of the segments (24). This peak is also referred to as the impact peak. The midfoot strike runner has little or no impact peak. The second peak in the vertical GRF component occurs during midsupport and generally has greater magnitude than the impact peak. Nigg (58) referred to the second peak as the active peak, indicating the role the muscles play in the force development to accelerate the body off the ground. Runners with either type of footfall pattern exhibit this peak.
FIGURE 1039 Vertical ground reaction force profiles of a runner using a heeltoe footfall pattern and a runner who initially strikes the ground with the midfoot. 
FIGURE 1040 Anteroposterior (front to back) ground reaction force for both walking and running. 
The anteroposterior GRF component also exhibits a characteristic shape similar in both walking and running but of different magnitude (Fig. 1040). The Fy component reaches magnitudes of 0.15 BW in walking and up to 0.5 BW in running. During locomotion, this component shows a negative phase during the first half of support as a result of a backward horizontal friction force between the shoe and the surface. This moves to positive near midstance, as force is generated by the muscles pushing back against the ground.
The mediolateral GRF component is extremely variable and has no consistent pattern from individual to individual. It is very difficult to interpret this force component without a video or film record of the foot contact. Figure 1041 illustrates walking and running profiles for the same
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individual. The great variety in foot placement regarding toeing in (forefoot adduction) and toeing out (forefoot abduction) may be a reason for this lack of consistency in the mediolateral component. The range of foot placement was shown in one study to be from 12° of toeing in to 29° of toeing out, and toeing out at heel strike has been shown to generate greater medial lateral forces and impulses (72). The magnitude of the mediolateral component ranges from 0.01 BW in walking to 0.1 BW in running.
FIGURE 1041 Mediolateral (side to side) GRF for both walking and running. 
Biomechanists have investigated GRFs to attempt to relate these forces to the kinematics of the lower extremity, particularly to foot function. Efforts have been made to relate these forces to the rear foot supination and pronation profiles of runners to identify possible injuries or aid in the design of athletic footwear (35,37). Because the GRF is representative of the acceleration of the total body center of mass, the use of GRF data for these purposes is probably extrapolating beyond the information provided by the GRFs.
To illustrate this point, a method of calculating the vertical GRF component proposed by Bobbert et al. (8) will be presented. In this method, Bobbert and associates used the kinematically derived values of the accelerations of the centers of mass of each of the body's segments. The vertical GRF component reflects the accelerations of the individual body segments resulting from the motion of the segments. The sum of the vertical forces of all body segments, including the effect of gravity, is the vertical GRF component. That is:
where Fz is the vertical force component (forces directed upward are defined as positive), mi is the mass of the i^{th} segment, n is the number of segments, a i is the vertical acceleration of the i^{th} segment (upward accelerations are defined as positive), and g is the acceleration due to gravity. The anteroposterior GRF component reflects the horizontal (i.e., in the direction of motion) accelerations of the individual body segments. Using similar methods, this force component can be computed as:
where a_{yi} is the horizontal acceleration of the i^{th} segment. Similarly, the mediolateral GRF component reflects the sidetoside accelerations of the individual body segments:
where a_{xi} is the sidetoside acceleration of the i^{th} segment. If the center of mass is a single point that represents the mass center of all the body's segments, the vertical component is:
where m is the total body mass, a is the vertical acceleration of the center of mass, and g is the acceleration due to gravity. Similarly, the other components may be represented as the total body mass times the acceleration of the center of mass. That is:
Therefore, the GRF represents the force necessary to accelerate the total body center of gravity (55) and cannot be directly associated with lower extremity function. Caution, then, should be exercised in describing lower extremity function using GRF data.
Because the GRF relates to the motion of the total body center of mass, the anteroposterior force profile can be related to the acceleration profile of the center of mass during support. Chapter 8 discussed a study by Bates et al. (6), illustrating the horizontal velocity pattern of the center of mass during the support phase of the running stride (Fig. 1042A). When this curve is differentiated, an acceleration curve is generated (Fig. 1042B). This curve has the characteristic shape of the anteroposterior force component in that it has negative acceleration followed by positive acceleration. According to Newton's second law of motion, if each point along this curve was multiplied by
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the runner's body mass (m), the anteroposterior GRF component would be generated as follows:
FIGURE 1042 A. Horizontal velocity of the center of mass of a runner. B. If the velocity curve in A is differentiated, a horizontal acceleration curve of the center of mass is generated. C. Multiplying each point along this curve by the runner's body mass, the anteroposterior ground reaction force is generated. 
Conversely, the acceleration curve could be calculated by dividing the F force component by the runner's body mass.
Either generating the curves using the kinematic procedure or collecting the anteroposterior GRF component leads to the same conclusion. The negative portion of the force component is often referred to as the braking phase and indicates a force against the runner serving to decrease velocity of the runner. The positive portion of the component is called the propelling phase and indicates a force in the direction of motion serving to increase velocity of the runner. If the running speed is constant, the negative and positive phases will be symmetrical, indicating no loss in velocity. If the negative portion of the curve is greater than the positive portion, the runner will slow down more than speed up. Conversely, if the positive portion is greater than the negative, the runner is speeding up.
Applying the impulsemomentum relationship again confirms that the runner does indeed slow down during the first portion of support and speed up in the latter portion (Fig. 1043). The area under the negative portion of the force component or the negative impulse serves to slow the runner down, that is, to change incoming velocity to some lesser velocity value. The positive area or the positive impulse of the component serves to accelerate the runner, that is, to change velocity from some lesser value to some outgoing velocity. If the positive change in velocity equals the negative change in velocity, the individual is running at a constant speed. Figure 1044 illustrates the changes in the braking and propelling impulses across a range of running speeds (38). In many instances in the laboratory during collection of GRF data, the ratio of the negative impulse to the positive impulse is checked to determine if the runner is at a constant velocity, speeding up, or slowing down. Even if the individual is maintaining a constant running speed, the ratio of the positive to the negative impulse is rarely 1.0 for any given support period of a stride. The average ratio over a number of footfalls approaches the ratio of 1.0, however.
FIGURE 1043 Anteroposterior ground reaction force illustrating the braking and propulsion impulses. 
An exchange of mechanical energy occurs during both running and walking, although the energy fluctuations differ. External work in walking has two components, one caused by inertial forces as a result of speed changes in the forward direction and the other caused by the cyclic
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upward displacement of the center of gravity. The work done to accelerate in the lateral direction is only a small fraction of the total work, as is evidenced by the small forces and small displacements (75). The motion of the center of gravity in walking has been modeled as an inverted pendulum. In each step, the center of gravity is either behind or in front of the contact point between the foot and the ground (22). When the center of gravity is behind the point of contact, as during the heel strike phase of support, the GRFs cause a negative acceleration and kinetic energy decreases because of loss of forward speed. A concomitant occurrence with the loss of kinetic energy is an increase in the center of gravity as the body vaults over the support limb. This increases the gravitational potential energy, which reaches a maximum level in the middle of stance. As the center of gravity moves forward of the point of the contact, the kinetic energy increases because gravitational potential energy decreases with the reduction in the height of the center of gravity. This pendulumlike exchange between potential and kinetic energy allows savings as much as 65% of muscular work (75). This conservation of energy is not perfect, so the total energy of the center of gravity fluctuates (22). The net change in the overall mechanical energy in walking is actually small.
FIGURE 1044 Changes in the anteroposterior ground reaction force as a function of running speed. Area 1 is the braking impulse, and area 2 is the propulsion impulse. (Adapted from Hamill, J., et al. [1983]. Variations in GRF parameters at different running speeds. Human Movement Science, 2:4756 ). 
In running, the mechanical energy fluctuates more than in walking. Kinetic energy in running is similar to walking, reaching minimum levels at midstance because of deceleration caused by the horizontal GRF and increasing in the latter half of the support phase. The potential energy is different than in walking because it is at a minimum at midstance because of compliance and flexion in the support limb (31). The overall vertical excursion of the center of mass is also less as the speed of running increases (31). There is not the pendulumlike exchange between potential and kinetic energy seen in running because the energies are in phase with each other compared with walking, in which they are 180° out of phase (31). The exchange of energy conserves less than 5% of the mechanical work required to lift and accelerate the center of mass (31). However, substantial mechanical energy is conserved through the storage and return of elastic energy in the tissues.
FIGURE 1045 Ground reaction forces and center of pressure for the front (left) and back (right) foot during the golf swing. (Adapted from Williams, K. R., Sih, B. L. [1999]. Ground reaction forces in regularspike and alternativespike golf shoes. In M. R. Farrally, A. J. Cochran [Eds.]. Science and Golf III: Proceedings of the 1998 World Scientific Congress of Golf. Champaign, IL: Human Kinetics, 568575 ). 
Linear Kinetics of the Golf Swing
In the golf swing, substantial linear forces are generated on the ground in response to segmental accelerations. Other important force application sites are between the hand and the club and–most important–between the club head and the ball at contact. The GRFs vary between right and left limbs. High vertical GRFs are generated between the right foot and the ground in the backswing (righthanded golfer), and a rapid transfer of force to the left foot occurs before impact, resulting in a peak force that is more than 1 BW (88). In the mediolateral direction, a lateral GRF develops in the right foot up through the backswing and from the top of the backswing to just before impact and a medial force propels the body toward the direction of the ball. At impact, this reverses to slow the movement of the body from right to left (88). In the anteroposterior direction, a force is generated as the body rotates around a vertical axis, resulting in a backward force on the left foot and a forward force on the right foot in the backswing. This reverses in the downswing as the body rotates to impact (88). The pattern of GRF generation using the different clubs is essentially the same. However, there are changes in force magnitude with different clubs. A sample of the GRFs and the COP patterns for both feet are illustrated in Figure 1045. Maximum vertical GRF values for a representative subject wearing regular spikes were 1096 N at the front foot and 729 N at the rear foot (89). The maximum anteroposterior force in the anterior was 166 N generated in the front foot and was 143
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N in the posterior direction, generated in the back foot. The maximum lateral force was generated in the front foot and was in the range of 161 N (89).
The forces acting on the body as a result of the swing have been shown to range between 40% and 50% of BW (49). These forces must be controlled by the golfer to produce an effective swing. A good golfer starts the downswing slowly, and the forces acting on the golfer consequently produce a smooth acceleration. An inexperienced golfer initiates the downswing with greater acceleration. The centripetal force produced by this rapid acceleration rotates the club, reduces the acceleration, and can actually cause negative acceleration later in the swing. Forces acting on the body as a result of the swing reach values around 40% of BW in the final third of the downswing (Fig. 1046). These forces are directed backward and must be resisted by knee flexion and a wide base of support. The force produced by the swing is maximum at contact and is vertically directed and easily resisted by the body (49).
As a result of forces acting on the body, significant forces are generated at the knee joint. Compressive forces in the front and back knees reach values approximating 100% and 72% of BW for the front and back knee joints, respectively (33). Shear forces are also developed. For a righthanded golfer, an anterior shear force is developed in the right knee (10% BW) and posterior shear forces are present in both the left and right knee joints with approximately 39% and 20% of BW in the left and right knee joints, respectively (33). For a lefthanded golfer, the values for the right and left knee would be reversed.
FIGURE 1046 Forces generated as a result of the swing are shown for the golfer Bobby Jones. (Adapted from Mather, J. S. B. [2000]. Innovative golf clubs designed for the amateur. In A. J. Subic, S. J. Haake [Eds.]. The Engineering of Sport: Research, Development and Innovation. Malden Blackwell Science, 6168 ). 
FIGURE 1047 Forces generated at the wrist joint in all three directions (Adapted from Neal, R. J., Wilson, B. D. [1985]. 3D kinematics and kinetics of the golf swing.International Journal of Sports Biomechanics, 1:221–232 ). 
Linear forces have also been measured at the wrist and shoulder. Resultant forces acting at the wrist and shoulder are shown in Figure 1047. It is important to measure these forces because they determine the eventual acceleration of the club. Peak forces in the direction of the ball are greater for the arm segment (650 N) than the club segment (approximately 300 N), and the shoulder peak force occur 85 ms from impact compared with 60 ms from impact for the club (57). Peak forces in the vertical and anteroposterior directions also occur earlier in the arm segment, suggesting some timing interaction between the segments.
When using an iron, the golf ball travels upward as a reaction to the club head action downward with the face of the club held in place and as a result of the angle of the club face. The contact with the ball is not upward, and if the swing is up, the likely result is a topped ball that goes down. The magnitude of the impact force has been reported in one study to be up to 15 kN applied for about 500 ms (50).
Linear Kinetics of Wheelchair Propulsion
To propel a wheelchair, the hand grasps the rim of the wheel and generates a pushing force. After the push phase, the hands return to the initial position before contact is again made with the rim. In this passive recovery phase, the inertial forces from the upper body movements can continue to influence the motion of the wheelchair (81) so that the backward swing of the trunk causes a reaction force that can propel the wheelchair forward.
The hand pushes on the rim at an angle, but only the force component tangential to the rim contributes to the
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propulsion (67). The propulsion force vector tangential to the rim is directed upward at the hand position of –15° to top dead center and directed downward at +60° from top dead center (79). This force has been shown to be only 67% of the total force applied to the hand rim. A representative sample of the hand rim forces produced at a velocity of 1.39 m/s and a power output of 0.5 W/kg is illustrated in Figure 1048 (82). In this example, the downward forces applied to the hand rim are nearly twice the horizontal forward directed forces. The outward force is the lowest of the three forces and increases only in the last third of the push phase. The actual propelling force can be calculated by dividing the torque at the hand rim by the radius from the wheel axle to the hand rim (83).
One of the main factors that determines the force application direction is the cost associated with each particular force application. If a force is applied perpendicular to a line from the hand to the elbow or from the hand to the shoulder, the cost increases at each joint (67). The posture of each individual influences the cost and effect as a result of each individual sitting in the wheelchair and holding the rim at a certain point. Large joint reaction forces generated in the shoulder joint change with hand position. For example, the shoulder joint mean forces at top dead center and at 15° relative to top dead center have been computed to be 1900 and 1750 N, respectively, and are approximately 10 times the mean net forces at the joint (79).
The propulsion technique also influences the output. A circular propulsion technique has been shown to generate less mean power (37.1 watts) than the pumping technique (44.4 watts) (84). Individual differences in shoulder and elbow positions can also influence propulsion effectiveness. For example, if an individual lowers the shoulder during the push phase, a more vertical and less effective propulsive force might result.
FIGURE 1048 Vertical forces (black), anteroposterior forces (red), and mediolateral forces (dotted) applied to the rim of the wheelchair by the hand. (Adapted from Veeger, H. E., et al. [1989]. Wheelchair propulsion technique at different speeds. Scandinavian Journal of Rehabilitation Medicine, 21:197203 ). 
The construction of the wheelchair can also influence propulsion. The axle position relative to the shoulder changes the push rim biomechanics significantly. If the vertical distance between the axle and the shoulder is increased, the push angle is decreased and the force available for propulsion is diminished (10). Improvements in wheelchair propulsion have been shown to be associated with a wheelchair with a more forward axle position (10). Also, cambered rear wheels whose top distance between the wheels is smaller than the bottom orient the hand rim to more closely resemble the force application. This facilitates a more effective use of elbow extension (83).
Summary
Linear kinetics is the branch of mechanics that deals with the causes of linear motion, or forces. All forces have magnitude, direction, point of application, and line of action. The laws governing the motion of objects were developed by Sir Isaac Newton and form the basis for the mechanical analysis of human motion:
Forces may be categorized as noncontact or contact. The most important noncontact force acting during human movement is gravity. The contact forces include the GRF, joint reaction force, friction, fluid resistance, inertial force, muscle force, and elastic force.
The GRF, a direct application of Newton's third law, has three components: a vertical component and two shear components acting parallel to the surface of the ground. The joint reaction force, the net force acting across a joint, has compressive and shear components. Friction results from interaction between two surfaces and is a force that acts parallel to the interface of the two surfaces and in a direction opposite to the motion. The coefficient of friction is the quantification of the interaction of the two surfaces. Fluid resistance refers to the transfer of energy from an object to the fluid through which the object is moving. The fluid resistance vector has two components, lift and drag. Drag acts in a direction opposite to the direction of motion, and lift is perpendicular to the drag component. Inertia results from the force applied by one segment on another that is not caused by muscle actions. A muscle force is the pull of the muscle on its insertion, resulting in motion at a joint. Muscle forces are generally calculated as net forces, not individual muscle forces, although intricate mathematical procedures can evaluate individual muscle
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forces. An elastic force results from the rebound of a material to its original length after it has been deformed.
A free body diagram is a schematic illustration of a system with all external forces represented by vector arrows at their points of application. Internal forces are not presented on free body diagrams. Muscle forces are generally not represented on these diagrams unless the system involves a single segment.
Analyses using Newton's law are usually conducted using one of three calculations: the effect of a force at an instant in time (F = ma), the effect of a force applied over time (impulsemomentum relationship), and the effect of a force applied over a distance (workenergy theorem). In the first technique, the analysis may be a static case (when a = 0) or a dynamic case (when a = 0). The static twodimensional linear case is determined using the following equations:
The 2D dynamic case uses the following equations:
The impulsemomentum relationship relates the force applied over time to the change in momentum:
The lefthand side of the equation (F ^{*} dt) is the impulse, and the righthand side (mv_{final} – mv_{initial}) describes the Final initial' change in momentum. Impulse is defined as the area under the forcetime curve and is thus equal to the change in momentum. This type of analysis has been used in research to evaluate the jump height of the center of mass in vertical jumping in association with the equations of constant acceleration.
Work is the product of the force applied and the distance over which the force is applied. Energy, the capacity to do work, has two forms, kinetic and potential. The relationship between work and energy is defined in the workenergy theorem, which states that the amount of work done is equal to the change in energy. Mechanical work is calculated via the change in mechanical energy. That is:
where KE is the translational kinetic energy and PE is the potential energy. Work can be calculated for either a single segment or for the total body. When this is done segment by segment, internal work or the work done on the segments by the muscles to move the segments is calculated. When the amount of work done is related to the time over which the work is done, the power developed is being evaluated.
Special force applications include definitions for centripetal force and pressure. Centripetal force is applied toward the center of rotation. Pressure is the force per unit area.
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Review Questions
True or False
Multiple Choice
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References
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Glossary
Glossary
Active Peak
The second of the two peaks in a vertical ground reaction force curve during running.
Angle of Application
The angle at which a force vector acts.
Bernoulli's Principle
The relationship between pressure and velocity that states that pressure is inversely proportional to velocity.
BoneonBone Force
The force at a joint that includes the joint reaction force and the forces due to muscles and ligaments.
Boundary Layer
The thin layer of a fluid adjacent to the surface of an object moving through the fluid.
Center of Pressure
The point of application of a force.
Centripetal Force
The force acting toward the center of rotation resulting when an object moves along a curved path.
Coefficient of Friction
The ratio of the friction force to the normal force pressing two surfaces together.
Collinear Forces
Forces whose lines of action are the same.
Composition
The process by which the resultant of two vectors is determined.
Concurrent Forces
Forces whose lines of action operate at the same point.
Contact Force
The pushes or pulls exerted by one object in direct contact with another object.
Coplanar Forces
Forces whose lines of action all act in the same plane.
Density
Mass per unit volume.
Dynamic Analysis
A calculation of the forces and moments when there are significant linear and/or angular accelerations.
Dynamics
The branch of mechanics that studies accelerating systems.
Energy
The capacity to do work.
External Work
The work done by a body on another body.
Fluid Resistance
The transfer of energy from an object to the fluid through which the object is moving.
Force
Interaction between two objects in the form of a push or pull that may or may not cause motion.
Force Platform
A device that measures ground reaction force.
Form Drag
Net resistance force caused by a pressure differential between the leading and trailing edges of an object moving through a fluid.
Free Body Diagram
A diagram in which all force vectors are drawn.
Friction
The force that resists the motion of one surface on another.
Impact Peak
The initial peak on a vertical ground reaction force.
Impulse
The product of the magnitude of a force and its time of application; the area under a forcetime curve.
ImpulseMomentum Relationship
The relationship stating that the impulse is equal to the change in momentum.
Inertia
The resistance of a body to a change in its state of motion.
Internal Work
The total work done resulting from the motion of all of the body's segments.
Inverse Dynamics
An analytical approach calculating the forces and moments based on the accelerations of the object.
Joint Reaction Force
The force acting across a joint.
Kinetic Energy
The ability of a body to do work by virtue of its motion.
Kinetics
The branch of mechanics that deals with forces acting on a system.
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Kinetic Friction
Friction between two surfaces as they slide on each other.
Laminar Flow
At slow flow velocities, the flow of a fluid smoothly over the surface of an object.
Law of Conservation of Energy
The law that states that energy can neither be created nor destroyed but can only change form.
Linear Kinetics
The branch of kinetics that deals with translational motion.
Line of Action
The line along which a force acts.
Lift Force
A component of fluid resistance force that acts perpendicular to the direction of motion.
Loading Peak
See Active Peak.
Magnus Effect
The curve in the path of a spinning ball caused by a pressure differential on either side of the ball.
Mass
The measure of a body's inertia.
Maximum Static Friction Force
The maximum friction force measured just before the impending motion of an object.
Noncontact Force
A force that acts at a distance from an object.
Passive Peak
See Impact Peak.
Point of Application
The point at which a force acts.
Point of Separation
The point at which the boundary layer separates from an object.
Potential Energy
The ability of a body to do work by virtue of its position.
Power
The quantity of work done per unit time.
Pressure
Force per unit area.
Propulsive Drag
A drag force caused by the pitch of the hand that acts to propel a swimmer through the water.
Propulsive Lift
A lift force caused by the pitch of the hand that acts to propel a swimmer through the water.
Resolution
The breakdown of a vector into its horizontal and vertical components.
Rotational Friction
The resistance in rotation of one surface upon another.
Separated Flow
The type of fluid flow in which the boundary layer separates from the object, creating turbulence and thus resisting the motion of the object.
Shear Force
A force that acts parallel to the surface.
Statics
The branch of mechanics in which the system being studied undergoes no acceleration.
Strain Energy
The capacity to do work by virtue of the deformation of an object.
Surface Drag
The fluid drag force acting on a body resulting from the friction between the surface of the object and the fluid.
System
A defined set of forces.
Turbulent Flow
The type of fluid flow in which the boundary layer becomes so turbulent that the point of separation moves farther back on the object, thus reducing drag.
Viscous Drag
See Surface Drag.
Viscosity
The measure of a fluid's resistance to flow.
Weight
The force of the earth's gravitational attraction to a body's mass.
Work
The product of the force applied to a body and the distance through which the force is applied.
WorkEnergy Theorem
The relationship between work and energy stating that the work done is equal to the change in energy.