# Biomechanical Principles

## NEWTON’S LAWS: UNDERLYING PRINCIPLES OF BIOMECHANICS

### Newton’s Laws of Motion

This chapter uses Newton’s laws of motion to introduce techniques for analysis of the relationship between the forces applied to the body and the consequences of those forces on human motion and posture. (Throughout the chapter, the term *body* is used when elaborating on the concepts related to the laws of motion and the methods of quantitative analysis. The reader should be aware that this term could also be used interchangeably with the entire human body; a segment or part of the body, such as the forearm segment; an object, such as a weight that is being lifted; or the system under consideration, such as the foot-floor interface. In most cases the simpler term, *body,* is used when describing the main concepts.) Newton’s laws are described for both linear and rotational (angular) motion (Table 4-1).

#### NEWTON’S FIRST LAW: LAW OF INERTIA

In addition to the human body as a whole, each segment, such as the arm or trunk, also has a defined center of mass. In the lower extremity, for example, the major segments include the thigh, shank (lower leg), and foot. Figure 4-1 shows the center of mass of these segments for the lower extremities of a sprinter, indicated by black circles. The location of the center of mass within each segment remains fixed, approximately at its midpoint. In contrast, however, the location of the center of mass of the *entire* lower extremity changes with a change in spatial configuration of the segments (compare red circles in Figure 4-1). As shown for the left (flexed) lower extremity, the specific configuration of the segments can displace the center of mass of the lower limb *outside* the body. Additional information regarding the center of mass of body segments is discussed later in this chapter under the topic of anthropometry.

The *mass moment of inertia* of a body is a quantity that indicates its resistance to a change in *angular velocity.* Unlike inertia, its linear counterpart, the mass moment of inertia depends not only on the mass of the body, but, perhaps more important, on the distribution of its mass with respect to an axis of rotation.^{7} (Mass moment of inertia is often indicated by I and is expressed in units of kilograms-meters squared [kg-m^{2}]). Because most human motion is angular rather than linear, the concept of mass moment of inertia is very relevant and important. Consider again the two positions of the lower extremities of the sprinter in Figure 4-1. Within each segment, the individual centers of mass of the thigh, shank, and foot are in the same location in both lower extremities; however, because of the different degrees of knee flexion, the *distances* of the centers of mass of the shank and foot segments have changed relative to the hip joint. As a consequence the mass moment of inertia of each entire limb changes; the right extended (and “longer”) lower extremity has a greater mass moment of inertia than the left. (Another way of conceptualizing the increase is to note that as the knee extends, the center of mass of the entire right lower extremity, depicted by the red circle, moves farther from the hip, thereby increasing its mass moment of inertia.) The ability to actively change an entire limb’s mass moment of inertia can profoundly affect the muscle forces and joint torques necessary for movement. For example, during the swing phase of running, the entire lower limb functionally shortens by the combined movements of knee flexion and ankle dorsiflexion (as in the left lower extremity in Figure 4-1). The lower limb’s reduced mass moment of inertia reduces the torque required by the hip muscles to accelerate and decelerate the limb during swing phase. This concept can be readily appreciated during the swing phase while running with the knees held nearly extended (increased I), or almost fully flexed (decreased I).

Athletes often attempt to control the mass moment of inertia of their entire body by altering the position of their individual body segments relative to the axis of rotation. This concept is well illustrated by divers who reduce their moment of inertia in order to successfully complete multiple somersaults while in the air (Figure 4-2, *A*). The athlete can assume an extreme “tuck” position by placing the head near the knees, holding the arms and legs tightly together, thereby bringing their segments’ centers of gravity closer to the axis of rotation. Based on the principle of “conservation of angular momentum,” reducing the body’s mass moment of inertia results in an increased angular velocity. Conversely, the athlete could slow the rotation by assuming a “pike” (see Figure 4-2, *B*) position and increasing the body’s moment of inertia, or assuming a “layout” position (see Figure 4-2, *C*), which maximizes the body’s mass moment of inertia and greatly slows the body’s angular velocity.

#### NEWTON’S SECOND LAW: LAW OF ACCELERATION

Force (Torque)-Acceleration Relationship: Newton’s second law states that the linear acceleration of a body is directly proportional to the force causing it, takes place in the same direction in which the force acts, and is inversely proportional to the mass of the body. Newton’s second law generates an equation that relates force (F), mass (m), and acceleration (a) (Equation 4.1). Conceptually, Equation 4.1 defines a *force-acceleration relationship.* Considered a cause-and-effect relationship, the left side of the equation, force (F), can be regarded as a cause because it represents a pull or push exerted on a body; the right side, m × a, represents the effect of the pull or push. In this equation, ΣF designates the sum of, or net, forces acting on a body. If the sum of the forces acting on a body is zero, acceleration is also zero and the body is in linear equilibrium. As previously discussed, this case is described by Newton’s first law. If, however, the net force produces acceleration, the body will accelerate in the direction of the resultant force. In this case, the body is no longer in equilibrium.

Force is measured in newtons, where 1 newton (N) = 1 kg-m/sec^{2}.

The rotary or angular counterpart to Newton’s second law states that a *torque* will cause an *angular acceleration* of a body around an axis of rotation. Furthermore, the *angular* acceleration of a body is directly proportional to the *torque* causing it, takes place in the same *rotary* direction in which the *torque* acts, and is inversely proportional to the *mass moment of inertia* of the body. (The italicized words denote the essential differences between the linear and angular counterparts of this law.) For the rotary condition, Newton’s second law generates an equation that relates the torque (T), mass moment of inertia (I), and angular acceleration (α) (Equation 4.3). (This chapter uses the term *torque.* The reader should be aware that this term is interchangeable with terms *moment* and *moment of force.*) In this equation, ΣT designates the sum of, or net, torques acting to rotate a body. Conceptually, Equation 4.3 defines a *torque–angular acceleration relationship.* Within the musculoskeletal system, the primary torque producers are muscles. A contracting biceps muscle, for example, produces a net internal flexion torque at the elbow. Neglecting external influences such as gravity, the angular acceleration of the rotating forearm is proportional to the internal torque (i.e., the product of the muscle force and its internal moment arm) but is inversely proportional to the mass moment of inertia of the forearm-and-hand segment. Given a constant internal torque, the forearm-and-hand segment with the smaller mass moment of inertia will achieve a greater angular acceleration than one with a larger mass moment of inertia. (A smaller mass moment of inertia can be achieved by moving a cuff weight from the wrist to the mid-forearm, for example.) Understand that this inertial resistance to the angular acceleration of the limb applies even in the absence of gravity. For example, consider the positions of the lower limb in Figure 4-1 but with the person on his or her side in a “gravity eliminated” position. Because of changes in the mass moment of inertia, less muscular effort will be required to flex the hip with the knee also flexed than with the knee extended.

Torque is expressed in newton-meters, where 1 Nm = 1 kg-m^{2} × radians/sec^{2}.

Impulse-Momentum Relationship: Additional relationships can be derived from Newton’s second law through the broadening and rearranging of Equations 4.1 and 4.3. One such relationship is the *impulse-momentum relationship.*

Acceleration is the rate of change of velocity (Δv/t). Substituting this expression for linear acceleration in Equation 4.1 results in Equation 4.4. Equation 4.4 can be further rearranged to Equation 4.5.

The product of mass and velocity on the right side of Equation 4.5 defines the momentum of a moving body. *Momentum* describes the quantity of motion possessed by a body. Momentum is generally represented by the letter *p* and has units of kilogram-meters per second (kg-m/sec). An *impulse* is a force applied over a period of time (the product of force and time on the left side of Equation 4.5). The linear momentum of an object such as a moving car is changed by the application of a force over a given time. When a quick change in momentum is required (during an emergency stop, for instance), a very large brake force is applied for a short time. Less brake force for the same time, or the same brake force for even less time, results in a smaller change in momentum. Impulse and momentum are vector quantities. Equation 4.5 defines the linear *impulse-momentum relationship.*

Newton’s second law involving torque can also apply to the rotary case of the impulse-momentum relationship. Similar to the substitutions and rearrangements for the linear relationship, the angular relationship can be expressed by substitution and rearrangement of Equation 4.3. Substituting Δω/t (rate of change in angular velocity) for α (angular acceleration) results in Equation 4.6. Equation 4.6 can be rearranged to Equation 4.7—the angular equivalent of the impulse-momentum relationship. Torque and angular momentum are also vector quantities.

Work-Energy Relationship: To this point, Newton’s second law has been described using (1) the force (torque)-acceleration relationship (Equations 4.1 and 4.3) and (2) the impulse-momentum relationship (Equations 4.4 through 4.7). Newton’s second law can also be restated to provide a *work-energy relationship.* This third approach can be used to study human movement by analyzing the extent to which work causes a change in an object’s energy. Work occurs when a force or torque operates over some linear or angular displacement. *Work (W)* in a linear sense is equal to the product of the magnitude of the *force (F)* applied against an object and the linear *displacement* of the object in the direction of the applied force (Equation 4.8). If no movement occurs in the direction of the applied force, no mechanical work is done. Similar to the linear case, angular work can be defined as the product of the magnitude of the torque (T) applied against the object, and the angular displacement of the object in the direction of the applied torque (Equation 4.9). Work is expressed in joules (J).

*Average power (P)* is work (W) divided by time (Equation 4.10). Because work is the product of force (F) and displacement (d), the rate of work at any instant can be restated in Equation 4.11 as the product of force and velocity. Angular power may also be defined as in the linear case, using the angular analogs of force and linear velocity: torque (T) and angular velocity (ω), respectively (Equation 4.12). Angular power is often used as a clinical measure of muscle performance. The mechanical power produced by the quadriceps, for example, is equal to the net internal torque produced by the muscle times the average angular velocity of knee extension. Power is often used to designate the net transfer of energy between active muscles and external loads. *Positive power* reflects the rate of work done by *concentrically active muscles* against an external load. *Negative power,* in contrast, reflects the rate of work done by the external load against *eccentrically active muscles.*

Table 4-2 summarizes the definitions and units needed to describe many of the physical measurements related to Newton’s second law.

TABLE 4-2.

Physical Measurements Associated with Newton’s Second Law

^{*}To convert from English units to SI units, multiply the English value by the number in the table cell. To convert from SI units to English, divide by that number. If two equations are in the cell, the upper equation is used to convert the linear measure, and the lower equation is used to convert the angular measure.

^{†}Radians, which are unitless, may be used instead of degrees (1 radian = about 57.3 degrees).

^{‡}The English unit of a mass is a pound-mass (lbm) or slug.

#### NEWTON’S THIRD LAW: LAW OF ACTION-REACTION

Another example of Newton’s law of action-reaction is the reaction force provided by the surface on which one is walking or standing. The foot produces a force against the ground, and in accordance with Newton’s third law, the ground generates a *ground reaction force* in the opposite direction but of equal magnitude (Figure 4-5). The ground reaction force changes in magnitude, direction, and point of application on the foot throughout the stance period of gait. Newton’s third law also has an angular equivalent. For example, during an isometric exercise, the internal and external torques are equal and in opposite rotary directions.

## INTRODUCTION TO MOVEMENT ANALYSIS: SETTING THE STAGE FOR ANALYSIS

### Anthropometry

Much of the information pertaining to the body segments’ center of gravity and mass moment of inertia has been derived from cadaver studies.^{3,}4 Other methods for deriving anthropometric data include mathematic modeling and imaging techniques, such as computed tomography and magnetic resonance imaging. Table 4-3 lists data on the weights of body segments and the location of the center of gravity. (The specific details contained in this table will be needed to solve selected parts of the biomechanical problems posed in Appendix I, Part B).

TABLE 4-3.

^{*}Even though some definitions listed in this table do not represent the endpoints of the segment, they are easily identified locations on the human. The values for segment weight and center of gravity location in this table take into consideration the discrepancy between the definition of the segment and the true endpoints. For example, the segment definition for the forearm is the same for the forearm-and-hand, but the percentages listed for the segment weight and center of gravity location of the forearm-and-hand are higher, taking into consideration the mass of the hand.

### Free Body Diagram

How a free body diagram is configured depends on the intended purpose of the analysis. Consider the example presented in Figure 4-6. In this example, the free body diagram represents the shank-and-foot at the instant of initial heel contact during walking. The free body diagram involves figuratively “cutting through” the desired joint(s) to isolate or “free” the body of interest. In the example presented in Figure 4-6, the knee joint was cut through to isolate the shank-and-foot segment. The effects of active muscle force are usually distinguished from the effects of other soft tissues, such as passive tension created in stretched joint capsule and ligaments. Although the contribution of individual muscles acting across a joint may be determined, a single resultant muscle force (M) vector is often used to represent the sum total of all individual muscle forces. Other forces external to the system are added to the diagram, which may include the ground reaction force (G) and weight of the shank-and-foot segments (S and F). As specified by Newton’s third law, the ground reaction force is the force developed in response to the foot striking the earth.

An additional force is identified in Figure 4-6: the *joint reaction force (J).* This force includes joint contact forces as well as the net or cumulative effect of all other forces transmitted from one segment to another. Joint reaction forces are caused “in reaction” to other forces, such as those produced by activation of muscle, by passive tension in stretched periarticular connective tissues, and by gravity (body weight). As will be discussed, the free body diagram is completed by defining an X-Y coordinate reference frame and writing the governing equations of motion.

Clinically, reducing joint reaction force is often a major focus in treatment programs designed to lessen pain and prevent further joint degeneration in persons with arthritis. Frequently, treatments are directed toward reducing joint forces through changes in the magnitude of muscle activity and their activation patterns or through a reduction in the weight transmitted through a joint. Consider the patient with osteoarthritis of the hip joint as an example. The magnitude of the hip joint’s reaction force may be decreased by having the person reduce walking velocity, thereby lessening the magnitude of muscle activation. Highly cushioned shoes may be recommended to reduce impact forces. In addition, a cane may be used to reduce forces through the hip joint.^{1,}10,13 If obesity is a factor, a weight-reduction program may be recommended.

#### STEPS FOR CONSTRUCTING THE FREE BODY DIAGRAM

Consider the situation in which an individual is holding a weight out to the side, as shown in Figure 4-7. This free body is assumed to be in static equilibrium, and the sum of all forces and the sum of all torques acting on the body are equal to zero. One purpose of the analysis might be to determine how much muscle force is required by the glenohumeral joint abductor muscles (M) to keep the arm abducted to 90 degrees; another purpose might be to determine the magnitude of the glenohumeral joint reaction force (J) during this same activity.

Step II involves defining a coordinate reference frame that allows the position and movement of a body to be defined with respect to a known point, location, or axis (see Figure 4-7, X-Y coordinate reference frame). More detail on establishing a reference frame is discussed ahead.

Step III involves identification and inclusion of all forces that act on the free body. Internal forces are those produced by muscle (M). External forces include the force of gravity on the mass of the exercise ball (B), as well as the force of gravity on the arm segment (S). Although not relevant to Figure 4-7, other examples of external forces could include forces applied by therapists, cables, resistance bands, the ground or other surface, air resistance, and orthotics. The forces are drawn on the figure while specifying their approximate point of application and spatial orientation. For example, vector S acts at the center of gravity of the upper extremity, a location determined by using anthropometric data, such as those presented in Table 4-3.

#### SPATIAL REFERENCE FRAMES

Whether motion is measured via a relative or global reference frame, the location of a point or segment in space can be specified using a coordinate reference frame. In laboratory-based human movement analysis, the *Cartesian coordinate system* is most frequently employed. The Cartesian system uses coordinates for locating a point on a plane in 2D space by identifying the distance of the point from each of two intersecting lines, or in three-dimensional (3D) space by the distance from each of three planes intersecting at a point. A 2D coordinate reference frame is defined by two imaginary axes arranged perpendicular to each other *with the arrowheads pointed in positive directions*. The two axes (labeled, for example, X and Y) may be oriented in any manner that facilitates quantitative solutions (compare Figures 4-6 and 4-7, for example). A 2D reference frame is frequently used when the motion being described is predominantly planar (i.e., in one plane), such as knee flexion and extension during gait.

In most cases, human motion occurs in more than one plane. In order to fully describe this type of motion, a 3D coordinate reference frame is necessary. A 3D reference frame typically has three axes (X, Y, and Z), each perpendicular (or orthogonal) to another. As with the 2D system, *the arrowheads point in positive directions*. A universal convention for orienting this triplanar coordinate system in space is based on the *right-hand rule*. This rule is used throughout most quantitative biomechanical studies (see Special Focus 4-4).

Throughout most of this textbook, the terminology used to describe linear direction within planes (such as the direction of a muscle force or an axis of rotation around a joint) is less formal than that dictated by the right-hand rule. As described in Chapter 1, linear direction in space is loosely described relative to the human body standing in the anatomic position, using terms such as *anterior-posterior, medial-lateral,* and *vertical.* Although useful for most qualitative or anatomic-based descriptions, this convention is not well suited for quantitative analyses, such as those introduced later in this chapter. In these cases, the Cartesian coordinate system is used, and the orientation of its 3D axes is designated by the right-hand rule.