The process of composition of forces is used to determine whether a net unbalanced force (or forces) exists on a seg- ment, because this will determine whether the segment is at rest or in motion. Furthermore, the direction/orientation and location of the net unbalanced force or forces determine the type and direction of motion of the segment. The process of composition of forces was oversimplified in Ex- amples 1-2 and 1-3 (see Figs. 1–19 and 1–21). The process of composition depends on the relationship of the forces to each other: that is, whether the forces are in a linear, con- current, or parallel force system.
Let us return to our case example of John Alexander and the weight boot. In Figure 1–13, we identified the force of weightboot-on-legfoot (WbLf) on John’s leg-foot segment.
However, Figure 1–13 must be incomplete because this force cannot exist alone; otherwise, the leg-foot segment would accelerate downward. We also have not yet accounted for the force of gravity. Figure 1–25 is the same figure but with the
Figure 1–25 The forces of gravity-on-legfoot (GLf) and weightboot-on-legfoot (WbLf) are in the same linear force system when the leg-foot segment is at 90° of knee flexion.
mass or moment of inertia. From the law of acceleration, it can be seen that inertia (a body’s or object’s resistance to change in velocity) is resistance to acceleration and is pro- portional to the mass of the body or object. The greater the mass or moment of inertia of an object, the greater the magnitude of net unbalanced force or torque needed either to get the object moving or to change its motion. A very large woman in a wheelchair has more inertia than does a small woman in a wheelchair; an aide must exert a greater push on a wheelchair with a large woman in it to get the chair in motion than on the wheelchair with a small woman in it.
GLf (48 N)
WbLf (40 N)
force system and exert an upward force on the leg-foot segment.
Figure 1–25 indicates that the femur is potentially touching the leg-foot segment. However, the contact of the femur would be a push on the leg-foot segment and, in the position shown in Figure 1–25, would be directed away from the femur in the same direction as the two force vectors that are already shown. Also, the downward forces already on the leg-foot segment would tend to move the leg-foot segment away from the femur, minimizing or eliminating the contact of the femur with the leg-foot segment. A net force that moves a bony segment away from its adjacent bony segment is known as a distraction force. A distraction force tends to cause a separa- tion between the bones that make up a joint. Consequently, we still need to account for a force of 88 N acting upward on the leg-foot segment to have equilibrium.
In the human body, the two bones of a synovial joint (e.g., the knee joint) are connected by a joint capsule and ligaments made of connective tissue. Until we explore con- nective tissue behavior in detail in Chapter 2, capsuloliga- mentous structures are best visualized as strings or cords with some elasticity that can “pull” (not “push”) on the bones to which they attach. Figure 1–26A shows a schematic representation of the capsuloligamentous structures that join the femur and the tibia.
Side-bar: In reality, the capsule surrounds the adjacent bones, and the ligamentous connections are more complex.
We will nickname the structures “Acapsule” (anterior capsule) and “Pcapsule” (posterior capsule), understanding that these two forces are representing the pull of both the capsule and the capsular ligaments at the knee. Because cap- sules and ligaments can only pull, the forces that are created
by the contact of Acapsule and Pcapsule in Figure 1–26A through C are directed upward toward the capsuloligamen- tous structures (positive). Under the assumption that the pulls of the capsule anteriorly and posteriorly in this exam- ple are likely to be symmetrical, the vectors are given the same length in Figure 1–26A.
The vectors for Acapsule-on-legfoot (AcLf) and Pcapsule- on-legfoot (PcLf) are drawn in Figure 1-26A so that the points of application are at the points on the leg-foot segment where the fibers of the capsular segments converge (or in the center of the area where the fibers converge).
Side-bar: Although the anterior and posterior segments of the capsule also touch the femur, we are considering only the leg-foot segment at this time.
The vector arrows for the pulls of Acapsule-on-legfoot and Pcapsule-on-legfoot must follow the fibers of the capsule at the point of application and continue in a straight line. A vector, for any given snapshot of time, is always a straight line. The vector for the pull of the capsule does not change direction even if the fibers of the capsule change direction after the fibers emerge from their at- tachment to the bone.
In a linear force system, vectors must be co-linear and coplanar. These vectors (AcLf and PcLf) are not co-linear or coplanar with the vectors weightboot-on-legfoot and gravity-on-legfoot vectors. Therefore, these vectors can- not all be part of the same linear force system. If vectors Acapsule-on-legfoot and Pcapsule-on-legfoot are extended slightly at their bases, the two vectors will converge (see Fig. 1–26B). When two or more vectors applied to the same object are not co-linear but converge (intersect), the vectors are part of a concurrent force system.
GLf (48 N)
WbLf (40 N)
PcLf AcLf
A
GLf (48 N) CLf (88 N)
WbLf (40 N)
C
PcLf AcLf
B
CLf
Figure 1–26 A. Schematic representation of the pull of the anterior capsule (AcLf) and posterior capsule (PcLf) on the leg-foot segment. B. Determination of the direction and relative magnitude of the resultant (capsule-on-legfoot [CLf]) of concurrent forces AcLf and PcLf, through the process of composition by parallelogram. C. The resultant force CLf has been added to the leg-foot segment, with a magnitude equivalent to that of GLf +WbLf.
Trigonometric Solution
Let us assume that PcLf and AcLf each have a magnitude of 51 N and that the vectors are at a 60° angle (α) to each other (Fig. 1–27). As was done for the graphic solution, the parallelogram is completed by drawing AcLf′ and PcLf′
parallel to and the same lengths as AcLf and PcLf, respec- tively. The law of cosines can be used to find the length of the side opposite a known angle (when the triangle is not a right triangle).
The reference triangle (shaded) is that formed by PcLf, AcLf′, and CLf (see Fig. 1–27). To apply the law of cosines, angle βmust be known because vector CLf (whose length we are solving for) is the “side opposite” that angle. The known angle (α) in Figure 1-27 is 60°. If PcLf is extended (as shown by the dotted line in Fig. 1-27), angle αis repli- cated because it is the angle between PcLf and AcLf′(given AcLf′is parallel to AcLf). Angle β, then, is the complement of angle α, or:
β =180° – 60°=120°
By substituting the variables given in the example, the magnitude of the resultant, CLf, can be solved for using the following equation:
If the value of 51 N is entered into the equation for both PcLf and AcLf and an angle of 120° is used, vector CLf =88 N. As we shall see, the trigonometric solution is simpler when the triangle has one 90° angle (right triangle).
CLf = PcLf2+AcLf2−2(PcLf AcLf) ( ) (cosβ)
capsule-on-legfoot must be equal in magnitude and oppo- site in direction to the sum of gravity-on-legfoot and weightboot-on-legfoot because these three vectors are co- linear, coplanar, and applied to the same object. The arith- metic sum of the three forces must be zero because (1) these vectors are part of the same linear force system, (2) nothing else is touching the leg-foot segment, and (3) the leg-foot segment is not moving.
The magnitude of the resultant of two concurrent forces has a fixed proportional relationship to the original two vectors. The relationship between the two composing vectors and the resultant is dependent on both the magnitudes of the composing vectors and the angle between (orientation of) the composing vectors. In composition of forces by parallelogram, the relative lengths (the scale) of the concurrent forces being composed must be appropriately represented to obtain the correct relative magnitude of the resultant force. It is always true that the magnitude of the resultant will be less than the sum of the magnitudes of the composing forces.
Trigonometric functions can also be used to determine the magnitude of the resultant of two concurrent forces.
The trigonometric solution is presented below. A trigono- metric solution requires knowledge of both the absolute magnitudes of the two composing vectors and the angle between them. However, these values are rarely known in a clinical situation.
Concurrent Force Systems
It is quite common (and perhaps most common in the human body) for forces applied to an object to have action lines that lie at angles to each other. A common point of application may mean that the forces are literally applied to the same point on the object or that forces applied to the same object have vectors that intersect when extended in length (even if the intersection is outside the actual segment or object, as we saw with the center of mass). The net effect, or resultant, of concurrent forces appears to occur at the common point of application (or point of intersection). Any two forces in a concurrent force system can be composed into a single resultant force through a graphic process known as composition by parallelogram.
Determining Resultant Forces in a Concurrent Force System
In composition by parallelogram, two vectors are taken at a time. The two vectors and their common point of application or point of intersection form two sides of a parallelogram.
The parallelogram is completed by drawing two additional lines at the arrowheads of the original two vectors (with each new line parallel to one of the original two). The resultant has the same point of application as the original vectors and is the diagonal of the parallelogram. If there are more than two vectors in a concurrent force system, a third vector is added to the resultant of the original two through the same process. The sequential use of the resultant and one of the original vectors continues until all the vectors in the original concurrent force system are accounted for.
In Figure 1–26B, vectors AcLf and PcLf are composed into a single resultant vector (CLf). Vectors AcLf and PcLf are extended to identify the point of application of the new resultant vector that represents the combined action of AcLf and PcLf. A parallelogram is constructed by starting at the arrowhead of one vector (AcLf) and drawing a line of relatively arbitrary length that is parallel to the adjacent vector (PcLf). The process is repeated by starting at the arrowhead of PcLf and draw- ing a line of relatively arbitrary length parallel to AcLf.
Both the lengths of the two new lines should be long enough that the two new lines intersect. Because the two new lines are drawn parallel to the original two and intersect (thus closing the figure), a parallelogram is created (see Fig. 1–26B). The resultant of AcLf and PcLf is a new vector (“capsule-on-legfoot” [CLf]) that has a shared point of application with the original two vectors and has a magnitude that is equal to the length of the diagonal of the parallelogram. If the vectors were drawn to scale, the length of CLf would represent +88 N.
Example 1-5
Continuing Exploration 1-4:
Capsule-on-legfoot in Figure 1–26C is the resultant of Pcapsule-on-legfoot and Acapsule-on-legfoot in Figure 1–26B.
Assuming nothing else is touching the leg-foot segment,
When there are more than two forces in the concurrent force system, the process is the same whether a graphic or trigonometric solution is used. The first two vectors are composed into a resultant vector, the resultant and a third vector are then composed to create a second resultant vec- tor, and so on until all vectors are accounted for. Regardless of the order in which the vectors are taken, the solution will be the same.
Returning to John Alexander’s weight boot, we have es- tablished that vectors gravity-on-legfoot and weightboot- on-legfoot have a net force of –88 N and that capsule-on- legfoot has a magnitude of +88 N. In John’s case, we must also consider not only the pull of the capsule on his leg-foot segment but also the pull of his leg-foot segment on his cap- sule, because John has injured his medial collateral liga- ment (part of that capsule). We can segue to consideration of this new “object” (the capsule) by examining the prin- ciple in Newton’s law of reaction.
Newton’s Law of Reaction
Newton’s third law (the law of reaction) is commonly stated as follows: For every action, there is an equal and opposite reaction. In other words, when an object applies a force to a second object, the second object must simultane- ously apply a force equal in magnitude and opposite in direction to the first object. These two forces that are applied to the two contacting objects are an interaction pair and can also be called action-reaction (or simply reaction) forces.
Reaction forces are always in the same line and applied to the different but contacting objects. The directions of reac- tion forces are always opposite to each other because the two touching objects either pull on each other or push on each other. Because the points of application of reaction forces are never on the same object, reaction forces are never part of the same force system and typically are not part of the same space diagram. However, we will see that reaction forces can be an important consideration in human func- tion, because no segment ever exists in isolation (as in a space diagram).
Gravitational and Contact Forces
A different scenario can be used to demonstrate the some- times subtle but potentially important distinction between a force applied to an object and its reaction. We generally assume when we get on a scale that the scale shows our weight (Fig. 1–29). A person’s weight (gravity-on-person [GP]), however, is not applied to the scale and thus cannot act on the scale. What is actually being recorded on the scale is the contact (push) of the “person-on-scale”
(PS) and not “gravity-on-person.” The distinction be- tween these forces and the relation between these
PcLf
PcLf´
AcLf AcLf´
CLf α
α β
Figure 1–27 The cosine law for triangles is used to compute the magnitude of CLf, given the magnitudes of AcLf and PcLf, as well as the angle of application (α) between them. The relevant angle (β) is the complement of angle α(180° – α).
Figure 1–28 Weightboot-on-legfoot (WbLf) and legfoot-on- weightboot (LfWb) are reaction forces or an interaction pair. Both forces exist by virtue of the contact between the two objects. Al- though separated for clarity, these two vectors will be in line with each other.
WbLf (40 N)
WbLf (40 N) LfWb (40 N)
segment. If the weight boot contacts the leg-foot seg- ment, then the leg-foot segment must also contact the weight boot. Legfoot-on-weightboot (LfWb) is a reaction force that is equal in magnitude and opposite in direction to WbLf (Fig. 1–28). We did not examine LfWb initially because it is not part of the space diagram under consid- eration. It is presented here simply as an example of a reaction force.
Side-bar: In Figure 1–28, the points of application and action lines of the reaction forces are shifted slightly so that the two vectors can be seen as distinctly different and as applied to different but touching objects.
Continuing Exploration 1-5:
Reactions to Leg-Foot Segment Forces
Among the vectors in Figure 1–26C we see the force vector of weightboot-on-legfoot (WbLf). WbLf arises from the contact of the weight boot with the leg-foot