## Qualitative And Quantitative Analysis Of Vectors

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Before moving on to the next kinetic approach to studying the causes of movement, it is a good time to review the special mathematics required to handle vector quantities like force and acceleration. The linear kinetics of a biomechanical issue called muscle angle of pull will be explored in this section. While a qualitative understanding of adding force vectors is enough for most kinesiology professionals, quantifying forces provides a deeper level of explanation and understanding of the causes of human movement. We will see that the linear kinetics of the pull of a muscle often changes dramatically because of changes in its geometry when joints are rotated.

### Qualitative Vector Analysis of Muscle Angle of Pull

While the attachments of a muscle do not change, the angle of the muscle's pull on bones changes with changes in joint angle. The angle of pull is critical to the linear and angular effects of that force. Recall that a force can be broken into parts or components. These pulls of a muscle's force in two dimensions are conveniently resolved into longitudinal and rotational components. This local or relative frame of reference helps us study how muscle forces affect the body, but do not tell us about the orientation of the body to the world like absolute frames of reference do. Figure 6.7 illustrates typical angles of pull and these components for the biceps muscle at two points in the range of motion. The linear kinetic effects of the biceps on the forearm can be illustrated with arrows that represent force vectors.

The component acting along the longitudinal axis of the forearm (FL) does not create joint rotation, but provides a load that stabilizes or destabilizes the elbow joint. The component acting at right angles to the forearm is often called the rotary component (FR) because it creates a torque that contributes to potential rotation. Remember that vectors are drawn to scale to show their magnitude with an arrowhead to represent their direction. Note that

Figure 6.7. Typical angles of pull of the biceps brachii muscle in an arm curl. The angular positions of the shoulder and elbow affect the angle of pull of the muscle, which determines the size of the components of the muscle force. Muscle forces (F) are usually resolved along the longitudinal axis of the distal segment (FL) and at right angles to the distal segment to show the component that causes joint rotation (FR).

Figure 6.7. Typical angles of pull of the biceps brachii muscle in an arm curl. The angular positions of the shoulder and elbow affect the angle of pull of the muscle, which determines the size of the components of the muscle force. Muscle forces (F) are usually resolved along the longitudinal axis of the distal segment (FL) and at right angles to the distal segment to show the component that causes joint rotation (FR).

in the extended position, the rotary component is similar to the stabilizing component. In the more flexed position illustrated, the rotary component is larger than the smaller stabilizing component. In both positions illustrated, the biceps muscle tends to flex the elbow, but the ability to do so (the rotary component) varies widely.

This visual or qualitative understanding of vectors is quite useful in studying human movement. When a muscle pulls at a 45° angle, the two right-angle components are equal. A smaller angle of pull favors the longitudinal component, while the rotary component benefits from larger angles of pull. Somewhere in the midrange of the arm curl exercise the biceps has an angle of pull of 90°, so all the bicep's force can be used to rotate the elbow and there is no longitudinal component.

Vectors can also be qualitatively added together. The rules to remember are that the forces must be drawn accurately (size and direction), and they then can be added together in tip-to-tail fashion. This graphical method is often called drawing a parallelogram of force (Figure 6.8). If the vastus later-alis and vastus medialis muscle forces on the right patella are added together, we get the resultant of these two muscle forces. The resultant force from these two muscles can be determined by drawing the two muscle forces from the tip of one to the tail of the other, being sure to maintain correct length and direction. Since these diagrams can look like parallelograms, they are called a parallelogram of force. Remember that there are many other muscles, ligaments, and joint forces not shown that affect knee function. It has been hypothesized that an imbalance of greater lateral forces in the quadriceps may contribute to patello-femoral pain syndrome (Callaghan & Oldham, 1996). Does the resultant force (FR)

Figure 6.8. Any vectors acting on the same object, like the vastus medialis (FVM) and vastus lateralis (FVL) of the right knee, can be added together to find a resultant (Fr). This graphical method of adding vectors is called a parallelogram of forces.

in Figure 6.8 appear to be directed lateral to the longitudinal axis of the femur?

Quantitative Vector Analysis of Muscle Angle of Pull

Quantitative or mathematical analysis provides precise answers to vector resolution (in essence subtraction to find components) or vector composition. Right-angle trigonometry provides the perfect tool for this process. A review of the major trigonometric relationships (sine, cosine, tangent) is provided in Appendix D. Suppose an athlete is training the isometric stabilization ability of their abdominals with leg raises in a Roman chair exercise station. Figure 6.9a illustrates a typical orientation and magnitude of the major hip flexors (the iliopsoas group) that hold their legs elevated. The magnitude of the weight of the legs and the hip flexor forces provide a large resistance for the abdominal muscles to stabilize. This exercise is not usually appropriate for untrained persons.

If an iliopsoas resultant muscle force of 400 N acts at a 55° angle to the femur, what are the rotary (FR) and longitudinal (FL) components of this force? To solve this problem, the rotating component is moved tip to tail to form a right triangle (Figure 6.9b). In this triangle, right-angle trigonometry says that the length of the adjacent side to the 55° angle (FL) is equal to the resultant force times cos 55°. So the stabilizing component of the iliopsoas force is: FL = 400(cos 55°) = 229 N, which would tenLd to compress the hip joint along the longitudinal axis of the femur. Likewise, the rotary component of this force is the side opposite the 55° angle, so this side is equal to the resultant force times sin 55°. The component of the 400-N iliopsoas force that would tend to rotate the hip joint, or in this example isometrically hold the legs horizontally, is: Fr = 400 sin 55° = 327 N upward. It is often a good idea to check calculations with a qualitative assessment of the free body diagram. Does the rotary component look larger than the longitudinal component? If these components are the same at 45°, does it make sense that a higher angle would increase the vertical component and decrease the component of force in the horizontal direction?

When the angle of pull (9) or push of a force can be expressed relative to a horizontal axis (2D analysis like above), the horizontal component is equal to the resultant times the cosine of the angle 9. The vertical component is equal to the resultant times the sine of the angle 9. Consequently, how the angle of force application affects the size of the components is equal to the shape of a sine or cosine wave. A qualitative understanding of these functions helps one

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Figure 6.9. Right-angle trigonometry is used to find the components of a vector like the iliopsoas muscle force illustrated. Notice the muscle force is resolved into components along the longitudinal axis of the femur and at right angles to the femur. The right angle component is the force that creates rotation (FR).

Figure 6.9. Right-angle trigonometry is used to find the components of a vector like the iliopsoas muscle force illustrated. Notice the muscle force is resolved into components along the longitudinal axis of the femur and at right angles to the femur. The right angle component is the force that creates rotation (FR).

understand where the largest changes occur and what angles of force application are best. Let's look at a horizontal component of a force in two dimensions. This is analogous to our iliopsoas example, or how any force applied to an object will favor the horizontal over the vertical component. A cosine function is not a linear function like our spring example in chapter 2. Figure 6.10 plots the size of the cosine function as a percentage of the resultant for angles of pull from 0 to 90°.

A 0° (horizontal) angle of pull has no vertical component, so all the force is in the horizontal direction. Note that, as the angle of pull begins to rise (0 to 30°), the cosine or horizontal component drops very slowly, so most of the resultant force is directed horizontally. Now the cosine function begins to change more rapidly, and from 30 to 60° the horizontal component has dropped from 87 to 50% the size of the resultant force. For angles of pull greater than 60°, the cosine drops off very fast, so there is a dra

Figure 6.10. Graph of the cosine of angle 0 between 0 and 90° (measured from the right horizontal) shows the percentage effectiveness of a force (F) in the horizontal direction (FH). This horizontal component is equal to F cos(0), and angle 0 determines the tradeoff between the size of the horizontal and vertical components. Note that the horizontal component stays large (high percentage of the resultant) for the first 30° but then rapidly decreases. The sine and cosine curves are the important nonlinear mathematical functions that map linear biomechanical variables to angular.

Figure 6.10. Graph of the cosine of angle 0 between 0 and 90° (measured from the right horizontal) shows the percentage effectiveness of a force (F) in the horizontal direction (FH). This horizontal component is equal to F cos(0), and angle 0 determines the tradeoff between the size of the horizontal and vertical components. Note that the horizontal component stays large (high percentage of the resultant) for the first 30° but then rapidly decreases. The sine and cosine curves are the important nonlinear mathematical functions that map linear biomechanical variables to angular.

matic decrease in the horizontal component of the force, with the horizontal component becoming 0 when the force is acting at 90° (vertical). We will see that the sine and cosine relationships are useful in angular kinetics as well. These curves allow for calculation of several variables related to angular kinetics from linear measurements. Right-angle trigonometry is also quite useful in studying the forces between two objects in contact, or precise kinematic calculations.

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### Responses

• Niina
Are vector diagrams qualitative or quantitative?
11 months ago