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Tuesday, December 14, 2010

Angular Kinematics

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Describing Objects in Angular Motion
A hammer thrower steps into the throwing circle. After a few windups, the thrower begins rotating as he swings the hammer around and moves across the circle. The turning rate increases as he approaches the front of the circle. Suddenly he lets go of the hammer, and it becomes a fast moving projectile. The 16 lb steel ball seems to fly forever before it buries itself into the ground with a thud 250 ft away. Wow! How did the thrower’s rotary motion cause the hammer to move so fast and go so far linearly when he let go of it? To answer this question, you need to know something about angular kinematics and its relationship to linear kinematics. Both of these topics are discussed in this chapter.
OBJECTIVES
When you finish this chapter, you should be able to do the following:
• Define relative and absolute angular position, and distinguish between the two
• Define angular displacement
• Define average angular velocity
• Define instantaneous angular velocity
• Define average angular acceleration
• Define instantaneous angular acceleration
• Name the units of measurement for angular position, displacement, velocity, and acceleration
• Explain the relationship between average linear speed and average angular velocity
• Explain the relationship between instantaneous linear velocity and instantaneous angular velocity
• Define tangential acceleration and explain its relationship to angular acceleration
• Define centripetal acceleration and explain its relationship to angular velocity and tangential velocity
• Describe anatomical position
• Define the three principal anatomical planes of motion and their corresponding axes
Describe the joint actions that can occur at each of the major joints of the appendages

This chapter is concerned with angular kinematics, the description of angular motion. Recall that in chapter 2, we described kinematics as part of dynamics, which is a branch of mechanics. In chapter 2, we also learned about linear kinematics. Angular kinematics is the other branch of kinematics. Angular motion occurs when all points on an object move in circular paths about the same fixed axis. Angular motion is important because most human movements are the result of angular motions of limbs about joints. An understanding of how angular motion is measured and described is important.


• Figure 6.1 An angle-(Q) formed by the intersection of two lines.

Angular Position and Displacement
Before further discussion, a definition of angle is needed. What is an angle? An angle is formed by the intersection of two lines, two planes, or a line and a plane. The term angler refers to the orientation of these lines or planes to each other (figure 6.1).
• An angle is formed by the intersection of two lines, two planes, or a line and a plane
In figure 6.1, the Greek letter Q (theta) is used to represent the angle formed by the intersections of the lines and planes. Greek letters are used to represent many of the terms used in angular kinematics.

Angular Position
Angular position refers to the orientation of a line with another line or plane. If the other line or plane is fixed and immovable relative to the earth, the angular position is an absolute angular position. The angle your forearm makes with a horizontal plane describes the absolute angular position of your forearm, because a horizontal plane is a fixed reference. These absolute angular positions will be our primary concern in the first part of this chapter.
If the other line or plane is capable of moving, the angular position is a relative angular position. The angle your forearm makes with your upper arm describes a relative angular position of your forearm, or your elbow joint angle. Angles formed by limbs at joints describe relative angular positions of limbs. Anatomists have developed special terms for describing the relative positions and movements of limbs at joints. These anatomical terms are presented in the last part of this chapter.
What units of measurement are used for angles? You probably are most familiar with measuring angles in degrees, but Acre are other units of measurement for angles besides degrees, just as there are other units of measurement for linear distance besides feet. If we wanted to measure the absolute angle of line AB, as shown in figure 6.2a, with a horizontal plane, we might imagine a horizontal line BC with the same length as AB. We could then draw a circle using the length of AB as the radius and the point B as the center (figure 6.2b). The angle of ABC could then be described as the fraction of the circle created by the pie ABC. An angle of 1° represents 1/360 of a circle because there are 360° in a circle.
Another way to describe an angle is to measure how many radii are in the arc length AC if one radius is equal to the length of line segment BC or AB. In other words, if the length BC represents 1 radius, how long is the arc from A to C measured using this radius as the unit of measure? Mathematically, an angle measured in radius units (we’ll call these radians) is
RUMUS 6.1
Where:
Q = angular measurement in radians
l = arc length
r = radius
The radian (abbreviated rad) unit of measure for an angle is really a ratio of arc length divided by the radius.
• Figure 6.2 A circle is useful in describing the angle of a line (a) if the center of the circle coincides with the intersection of the lines defining the angle (b).
Figure 6.3 graphically shows the definition of an angle of 1 rad,  (pi) rad, and 2  rad. If you recall from geometry, the circumference of a circle is 2  , so there are 2  rad in a circle or 2  rad in 360°. These conversions are shown below:
RUMUS
Angular Displacement
Angular displacement is the angular analog of linear displacement. It is the change in absolute angular position experienced by a rotating line. Angular displacement is thus the angle formed between the final position and the initial position of a rotating line. (We often speak of the angular displacement of an object when the object is not a line. To measure such a displacement, choose any two points on the object. Imagine a line connecting these two points. If the object is rigid, the angular displacement of this line segment is identical to the angular displacement of the object.)

• Figure 6.3. The relationships among-the radius of a circle, the arc length along a circle, and an angle measured in radians.
• Angular displacement is the change in absolute angular position experienced by a rotating line.
As with linear displacement, angular displacement has direction associated with it. How is angular direction described? Clockwise and counterclockwise are common terms used to describe the direction of rotation. The hands of a clock rotate in a clockwise direction when you view a clock face from the front. If you could view a clock face from the back, would you still see the hands rotating in a clockwise direction? Because your viewpoint changed, the clock hands would now appear to be rotating counterclockwise! They didn’t change the direction of rotation; you just changed your viewing position. If you describe an angular displacement as clockwise, a person must know your viewing position to be certain of the direction of the angular displacement.
One way of overcoming this possible source of confusion is to first identify the axis of rotation and the plane in which the part rotates. The axis of rotation is always perpendicular to the plane in which the motion occurs. This axis of rotation is like the axle of a bicycle wheel, and the spokes of the wheel lie in the plane of motion. Along the axis of rotation, establish a positive direction. If you situate the thumb of your right hand so that it points in the positive direction along the axis of rotation, the direction in which your fingers curl is the positive direction of rotation. This is called the right hand thumb rule.
• Figure 6.4. The right-hand thumb rule. The direction the fingers curl indicates the positive angular direction if the right thumb points in the positive linear direction along the axis of rotation.
Now consider the direction of rotation of the hands of a clock. The plane of their motion is the clock face, and the axis of rotation is a line through the plane of the clock face. If die positive direction along this axis is established as pointing out of the clock face toward you, the positive direction of rotation is counterclockwise as you face the clock. Check this by pointing your right thumb out away from the clock in the positive direction of the axis of rotation. Then observe the direction in which your fingers curl—counterclockwise (see figure 6.4).
Most screws, nuts, and bolts have right-handed threads. They follow the right-hand thumb rule. If you point your right thumb in the direction you want the screw or nut to move, your right fingers curl in the direction the screw or nut must be turned. This sign convention also applies to torques and measures of angular position.
Now let’s consider how angular displacement is measured. A pitcher is being evaluated for shoulder joint range of motion. The measurement begins with the pitcher’s arm at his side, as shown in figure 6.5. The pitcher then raises it away from his side as far as he can (he abducts his shoulder). What is the angular displacement of his arm?
The axis of rotation is the anteroposterior axis, a line through the shoulder joint with the positive direction pointing out of the page toward us. The plane of motion is the frontal plane—the plane formed by his arms, legs, and trunk. It the initial position of the arm is 5° from the vertical and its final position is 170° from the vertical, the angular displacement is
RUMUS (6.2)
Dimana:
• Figure 6.S Angular displacement of a pitcher’s arm at the shoulder joint around the anteroposterior axis.
The displacement is positive because the rotation was in the direction in which the fingers of your right hand curl if your thumb is pointed in the positive direction of the axis of rotation (anteriorly and out of the shoulder).
As a coach or teacher, you will rarely measure angular displacements precisely, but in some sports, angular displacement is an important part of the skill. The number of twists or somersaults done in diving, gymnastics, or figure skating is a measure of angular displacement and plays an important role in how many points a judge awards. The angular displacement of a swing (range of motion) in golf or tennis affects the manner in which the ball is hit in these sports.
Angular and Linear Displacement
In chapter 5, we discovered that our muscles must produce very large forces to lift modest loads. The reason for this is that most muscles attach to bones close to the joint; thus they have small moment arms about the joint. Because of the small moment arm, large forces must be produced by the muscles to produce modest torques about the joints. The muscles are at a mechanical disadvantage for producing torque. Is there any advantage to this arrangement? Try self-experiment 6.1 for some insight into the advantages of this arrangement.
How far any point on the arm moves when you flex your arm, as in self-experiment 6.1, is dependent on how far that point is from the elbow. This relationship is actually apparent in the definition of an angle measured in radians, as shown in equation 6.1:
Rumus: (6.1)
For our example, then, let’s use an angular displacement measured in radians, and
Rumus (6.3)
If the angular displacement of the arm was 1 rad, r, was 10 in., and r,, was 1 in., then
Rumus:
Where:
In this example, the hand moves 10 rimes as far as the insertion of the biceps tendon. The linear distance (arc length) traveled by a point on a rotating object is directly proportional to the angular displacement of the object and the radius, the distance that point is from the axis of rotation of the object. If the angular displacement is measured in radians, the linear distance traveled (the arc length) is equal to the product of the angular displacement and the radius. This is only true if the angular displacement is measured in radians. This relationship is expressed mathematically in equation 6.4:
RUMUS (6.4)
where
€ = arc length
AQ = angular displacement measured in radians
r = radius

SELF-EXPERIMENT 6.1
Place your forearm on the desk or table in front of you. Now flex at the elbow and bring your hand off the table. Move it toward your shoulder as far as it can go while keeping your elbow on the desktop as shown in figure 6.6.
All the parts of the arm underwent the same angular displacement, but which moved farther, your hand (point A in figure 6.6) or the point of attachment of your biceps muscle (point B in figure 6.6)? Obviously, your hand moved farther. The linear distance it traveled (arc length AA (l2 in figure 6.6) and its linear displacement (chord AA (la in figure 6.6) are larger than the linear distance traveled (arc length BB (lb) and linear displacement (chord BB (da) of the biceps insertion.
• Figure 6.6 The distance the hand or wrist (A) moves (la or da) when your elbow (B) flexes is greater than the distance the insertion point of the biceps moves (lb or db). The ratio of these distances to each other is the same as the ratio of the radius ra to the radius rb.

The linear displacement of a point on a rotating object is also directly proportional to the distance that point is from the axis of rotation (the radius). This linear displacement is also related to the angular displacement, but it is not directly proportional to the angular displacement. The relationship is more complex.
The relationship between linear displacement and radius is easily seen in figure 6.7. The linear displacement and radii form sides of similar triangles, as shown in figure 6.7. From this, the following relationship can be established:
RUMUS (6.5)
where
r = radius
d = linear displacement (or chord length)
• Figure 6.7 The ratio of the linear displacements of two points on a rotating object (d, - d^) is equal to the ratio of the radii of these two points from the axis of rotation (r^ - r^).

One advantage of muscle insertions close to joints should now be clear. The muscle only has to contract and shorten a short distance to produce a large movement (linear displacement) at the end of the limb. Because the distance a muscle can shorten is limited to approximately 50% of its resting length, the range of motion at a joint would also be further limited if muscles inserted farther away from the joints.
This concept is useful when we use sports implements. Small movements of the hands produce large linear displacements at the end of a putter, a badminton racket, a fishing pole, a hockey stick, and so on.

SAMPLE PROBLEM 6.1
The golfer’s hands move through an arc length of 10 cm during a putt. What arc length does the head of the putter move through if the hands are 50 cm from the axis of rotation and the putter head is 150 cm from the axis of rotation?
Solution:
Step 1: Identify the known quantities and inferred relationships.
rumus
The hands, the arms, and the putter move together as one pendulum, so
rumus
Step 2: Identify the unknown variable to solve for.
rumus
Step 3: Search for equations with the unknown and known variables.
rumus
Step 4: Substitute known quantities and solve for the unknown quantity.
Rumus
Step 5: Common sense check.
The putter head moves farther than the hands, and 30 cm is about a foot. This is a reasonable movement for the putter head.

Angular Velocity
Angular velocity is denned as the rate of change of angular displacement. Its units of measurement are radians per second (rad/s), degrees per second (°/s), revolutions per minute (rpm), and so on. Angular velocity is abbreviated with the Greek letter omega (w). Angular velocity is a vector quantity, just like linear velocity, so it has direction associated with it. The direction of an angular velocity is determined using die right-hand thumb rule, as with angular displacement. Because angular velocity is a vector, a change in the size of the angular velocity or in the direction of its axis of rotation results in a change in angular velocity.
• Angular velocity is defined as the rate of change of angular displacement.
Average angular velocity is computed as the change in angular position (angular displacement) divided by time. Mathematically,
Rumus (6.6)
Dimana:
W = average angular velocity
A9 = angular displacement
At = time
Q = final angular position
Qi = initial angular position

If we are concerned with how long it takes for something to rotate through a certain angular displacement, average angular velocity is the important measure. If we are concerned with how fast something is rotating at a specific instant in time, instantaneous angular velocity is the important measure. Instantaneous angular velocity is an indicator of how fast something is spinning at a specific instant in time. The tachometer on a car’s instrument panel gives a measure of the engine’s instantaneous angular velocity in revolutions per minute.
The average angular velocity of a batter’s swing may determine whether or not he contacts the ball, but it is the bat’s instantaneous velocity at ball contact that determines how fast and how far the ball will go. Similar situations exist in all racket sports and striking activities. To gymnasts, divers, and figure skaters, average angular velocity is the more important measure. It determines whether or not they will complete a certain number of twists or somersaults before landing or entering the water.
Angular and Linear Velocity
In several sports, especially ball sports, implements are used sis extensions to the athletes’ limbs. Golf, tennis, squash, lacrosse, racquetball, badminton, field hockey, and ice hockey are examples of these sports. One advantage of using these implements was described earlier—they amplify the motion (displacement) of our limbs. Now compare the velocities of the ball (or shuttlecock or puck) in each of these sports if it were thrown by hand versus if it were struck (or thrown) with the respective stick (or racket or club). Which is faster? The implements enable us to impart faster linier velocities to the ball (or shuttlecock or puck) in each of these sports. This is another advantage of using a stick, racket, or club. How is this accomplished?



Deriving the Relationship Between Linear and Angular Velocity
The relationship between angular displacement and linear distance traveled provides the answer. Consider a swinging golf club. All points on the club undergo the same angular displacement, and thus the same average angular velocity, because they all take the same time to undergo that displacement. But a point on the club closer to the club head (and farther from the axis of rotation) moves through a longer arc length than a point farther from the club head (and closer to the axis of rotation). Both points travel their respective arc lengths in the same time. The point farther from the axis of rotation must have a faster linear speed because it moves a longer distance but in the same time. Mathematically, this relationship can be derived from the relationship between angular displacement and linear distance traveled (arc length).
Rumus (6.4)
Dividing both sides by the time it takes to rotate through the displacement gives us
Rumus (6.7)
where
€ = as arc length
r = radius
Eo = angular displacement (measured in radians)
t = lime
w = average linear speed
w = average angular velocity (measured in radians per second)

The average linear speed of a point on a rotating object is equal to the average angular velocity of the object times the radius (the distance from the point on the object to the axis of rotation of the object).
At an instant in time, this relationship becomes
Rumus (6.8)
Where:
v- = instantaneous linear velocity tangent to the circular path of the point
< ar2 if v1 = v2). The centripetal acceleration of the runner in the inside lane is greater due to the smaller radius (r1 > r2).

On the other hand, a hammer thrower swinging a hammer with a 1.0 m chain (see figure 6.12a) would have to pull on the chain with greater force than if he swung a hammer with a 0.75 m chain at die same angular velocity (see figure 6.12b). The centripetal acceleration is greater (and thus the centripetal force provided by the thrower must be greater) for the hammer with the 1.0 m chain than for the one with the shorter chain because the radius of rotation is larger for the hammer with the 1.0 m chain. Equation 6.12 (a1 = w2r) would be the appropriate equation to use to evaluate this situation because the angular velocities are the same.
You can try this for yourself by turning in a circle while holding this book at arm’s length and then pulling it closer to you while still turning at the same angular velocity. You have to exert a greater force on the book when it is farther from the axis of rotation.

Anatomical System for Describing Limb Movements
The first part of this chapter dealt with strictly mechanical terminology for describing angular motion. Anatomists use their own terminology for describing the relative angular positions and movements of the limbs of the body. You probably already have some knowledge of anatomy and anatomical terminology. This part of the chapter will only present the system used by anatomists and other human movement professionals to describe relative positions and movements of the body and its parts.
Anatomical Position

Figure 6.12. A hammer thrower using a hammer with a l .0 m chain (a) must exert a greater centripetal force than a thrower using a 0.75. m chain (b) if both hammers rotate with the same angular velocity (ar1, ar2, since w1 = w2, and r1 > r2). The centripetal acceleration of the hammer with the 1.0 m chain is greater due to the larger radius.

Try to describe where a specific freckle or mole or hair is on your body. It’s a difficult task. You probably identified the location of the freckle or mole or hair by describing where it was in relation to some other body part. A similar situation occurs if you try to describe the movement of a limb. To describe the location or movement of a body part, die other parts of the body are used as a reference. But the human body can adopt many different positions, and the orientation of the limbs may change as well, so a common reference position of die body must be used. The most commonly used reference position of the human body is called anatomical position. Early anatomists suspended cadavers in this position to study them more easily. The body is in anatomical position when it is standing erect, facing forward, both feet aligned parallel to each other, toes forward, arms and hands hanging straight below the shoulders at the sides, fingers extended, and palms facing forward. Anatomical position is the standard reference position for the body when describing locations, positions, or movements of limbs or other anatomical structures. The body is shown in anatomical position in figure 6.13.
• Anatomical position is the standard reference position for the body when describing locations, positions, or movements of limbs or other anatomical structures.
Planes and Axes of Motion
Anatomists have developed names to identify specific planes that pass through the body. Each plane has a corresponding axis that passes perpendicularly through the plane. These planes are useful to anatomists in describing planes of dissections or imaginary dissections. The planes are also useful in describing relative movements of body parts, with the axes used to describe the lines around which these motions occur.
Anatomical Planes
A plane is a flat two-dimensional surface. A sagittal plane, also called an ameroposterior plane, is an imaginary plane running anterior (front) to posterior (back) and superior (top) to inferior (bottom), dividing the body into right and left parts. A frontal plane, also called a coronal or lateral plane, runs side to side and superior to inferior, dividing the body into anterior and posterior parts. A transverse plane, or horizontal plane, runs from side to side and anterior to posterior, dividing the body into superior and inferior pans. All sagittal planes arc perpendicular to all frontal planes, which arc perpendicular to all transverse planes.
Many sagittal planes can he imagined to pass through the body, but they arc all parallel to each other. Likewise, many frontal or transverse planes can be imagined to pass through the body. A cardinal plane is a plane that passes through the midpoint or center of gravity of the body. The center of gravity is the point on which the body would balance if it were supported at only one point. The cardinal sagittal plane (midsagitial or median plane) is the plane that divides the body into equal right and left halves. The cardinal sagittal, frontal, and transverse planes of the body are shown in figure 6.13.
• Figure 6.13 Cardinal anatomical planes and axes of the body.
Biomechanical, the anatomical planes may be useful for locating anatomical structures, but their greatest worth is for describing limb movements. How can a plane be useful for describing movement? Movements of most limbs occur as rotations of the limbs. Rotations occur around specific axes of rotation and within specific planes of movement. Descriptions of the limb movements relative to each other arc thus facilitated by identification of the axis of rotation around which the limb moves and the plane within which the limb moves.
• Biomechanically, the anatomical planes may be useful for locating anatomical structures, but their greatest worth is for describing limb movements.
Anatomical Axes
We’ve defined several specific anatomical planes. What are the specific anatomical axes? The anatomical axes correspond to lines that are perpendicular to the previously defined anatomical planes. An anteroposterior axis (sagittal, sagittal-transverse, or cartwheel axis) is an imaginary line running from anterior to posterior and perpendicular to the frontal planes. The anteroposterior axis is often abbreviated as AP axis. An AP axis is defined by the intersection of a transverse plane with a sagittal plane, so it may also be called a sagittal-transverse axis. A transverse axis (lateral, frontal, media lateral, frontal-transverse, or some result axis) is an imaginary line running from left to right and perpendicular to the sagittal planes. A transverse axis is defined by the intersection of a transverse plane with a frontal plane, so it may also be called a frontal-transverse axis. A longitudinal axis (vertical, frontal-sagittal, or twist axis) is an imaginary line running from top to bottom and perpendicu ar to the transverse planes. A longitudinal axis is denned by the intersection of a frontal plane with a sagittal plane, so it may also be called a frontal-sagittal axis. All AP axes are perpendicular to all transverse axes, which are perpendicular to all longitudinal axes. An infinite number of these axes pass through the body. Examples of AP, transverse, and longitudinal axes are shown in figure 6.13.

Identifying Planes and Axes of Motion
Now let’s see how axes and planes are used to describe human motion. Imagine a bicycle wheel. The wheel ti ms about an axle. The line along and through the axle of the wheel defines the axis of rotation of the wheel. This is the axis around which the wheel rotates. The spokes of the wheel are perpendicular to the axle or axis of rotation. Thus the spokes must lie in the plane of motion of the wheel. Now let’s look at an example from the human body. Stand in anatomical position and flex (bend) your right arm at the elbow without moving your upper arm, as shown in figure 6.14. Think of your forearm as a spoke of the bicycle wheel and your elbow as the axle. What plane does your forearm lie in throughout the movement? It is moving within a sagittal plane. What axis is perpendicular to a sagittal plane’ A transverse axis is perpendicular to a sagittal plane. The movement of your forearm and hand is in a sagittal plane and abound a transverse axis.
Now let’s describe the method we use to identify the plane or axis of motion for any limb movement. First, one principle should be noted. If you can identify either the plane or the axis of motion, the other is easily identified. If the plane within which the motion occurs is known, there is only one axis around which the motion can occur, and that is the axis perpendicular to the plane of motion. Likewise, if the axis around which the motion occurs is known, there is only one plane within which the motion can occur, and that is the plane perpendicular to the axis of motion. Table 6.1 lists each of the three planes of motion along with the corresponding axis that is perpendicular to that plane of motion.

• Figure 6.14. Imagining a bicycle wheel can help you identify the plane and axis of motion.
Tabel 6.1. Anatomical Planes of Motion. and Their Corresponding Axes of Motion
Plane:
- Sagittal
- Frontal
- Transverse
Axis:
- Transverse
- Anteroposterior (AP)
- Longitudinal

• If you can identify either the plane or the axis of motion, the other is easily identified.
Prior to describing techniques for determining the plane of motion, a more precise and accurate definition of a plane may be helpful. Just as two points geometrically define a line in space, three noncolinear points or two intersecting lines define a plane in space. If a table or chair has only three legs, each of those legs will always touch a plane, such as the floor, because it only likes three points to define a plane. A table with four legs will always have three legs on one plane, but the fourth leg will only touch that plane if its end lies on the plane defined by the other three legs.
Most of our limbs are longer in one dimension than in the others, so they can be thought of as long cylinders or even line segments. These line segments move by swinging or rotating around joints. If you can imagine the line segment define by the limb at the start of motion and the line segment defined by the limb at any other instant during its motion, these line segments will intersect at the joint (if it’s a single-joint motion). The plane of motion is defined by these two line segments. For example, stand in anatomical position. Imagine the line segment defined by a line drawn from your right shoulder to your right wrist. Now abduct at the shoulder (lift your right arm up and to the side, laterally away from your body, until it is shoulder height) as shown in figure 6.15. Imagine a line segment drawn through your shoulder and wrist now. What planes do both line segments fall within? When you were in anatomical position, your right arm was in a sagittal plane and a frontal plane. When you completed the movement, your arm was in a transverse plane and a frontal plane. Your arm was in the frontal plane at the start and end of the movement and throughout the movement), so the movement occurred within a frontal plane. Because frontal planes are perpendicular to AP axes, the movement occurred around an anteroposterior axis.
Figure 6.15 Frontal view of shoulder abduction.
Here’s another procedure for determining the plane of motion. If you could view the movement from any vantage point, what vantage point would give you the best view so that you could always see the entire length of the moving limb? What vantage point would you view from so that the moving limb was not moving toward or away from you, but across your field of view? II the best viewpoint is from in front or behind, the view of the body is a frontal view and the movement is in a frontal plane. If the best viewpoint is from the left or right side, the view of the body is a sagittal view and the movement is in a sagittal plane. If the best viewpoint is from above or below (admittedly difficult vantage points to view from, but imagine the view if you could get into a position above or below a person), the view of the body is a transverse view and the movement is in a transverse plane.
For example, if you viewed the movement described in the previous example (abduction of the right arm at the shoulder) from a position in front of (anterior to, as shown in figure 6.15) or behind (posterior to) the person moving, you would see the full length of the arm throughout the movement, and the arm wouldn’t move away from or toward you. If you viewed the motion from the side (sagittally, as shown in figure 6.16a), the full length of the arm would be in view at the start, but as the arm moved upward, it would appear to shorten until you would only see the hand when the arm reached shoulder height. The mil would move toward you. If you viewed the motion from above (transversely, as shown in figure 6.16b), only the shoulder would be in view at the start, but the full length of the arm would be in view when it reached shoulder height. The arm would move toward you. The best view was from the front, so the view was a frontal view and the movement was in a frontal plane. The axis perpendicular to a frontal plane is an anteroposterior axis.
In some limb movements, the length of the limb doesn’t swing around an axis, but the limb rotates or twists about its length. In this case, it is easier to determine the axis of rotation of the limb first and then determine its plane of motion. The axis of rotation of a limb twisting about its length is denned by the direction of the line running the length of the limb from the proximal to the distal end. If this line is parallel to a longitudinal axis, it is a longitudinal axis. If it is parallel to an anteroposterior axis, it is an anteroposterior axis. If it is parallel to a transverse axis, it is a transverse axis.
• Figure 6.16. Sagittal (a) and transverse (b) views, of shoulder abduction.
For example, stand in anatomical position and turn the palm of your right hand toward your side and then to the rear as shown in figure 6.17. Your arm twisted around an axis through its length. A line drawn from the proximal to the distal end of your arm is a vertical line and parallel to a longitudinal axis. The axis of rotation for this movement was a longitudinal axis. A longitudinal axis is perpendicular to a transverse plane, so the twisting motion of your right arm occurred in the transverse plane.
So far, the examples we’ve used have all been movements that occurred within one of the three planes of motion we defined. Sagittal, transverse, and frontal planes are primary planes of motion. Other planes exist that are not primary planes of motion, but movements can occur within these planes. For example, what plane do your arms move in when you swing a golf club as shown in figure 6.18? The best viewpoint wouldn’t be directly in front or directly above but in front and above. The plane of motion is a diagonal plane between the transverse plane and the frontal plane, and the axis of motion is a diagonal axis between a longitudinal and an AP axis. An infinite number of diagonal planes and axes exist within or around which our limbs can move. The principal planes and axes give us standard planes and axes from which other, diagonal planes and axes can be described.
• Figure 6.17. Movement around a longitudinal axis.
• Figure 6.18 Movement in a diagonal plane.

Joint Actions
We can identify the plane and action of limb motions, but what are the terms that describe the limb motions? Human movement description uses terminology that describes the relative movements of two limbs on either side of a joint (relative angular motion) rather than terminology that describes individual limb movements (absolute angular motion). The terms thus describe joint actions, the relative angular movements of the limbs on the distal and proximal sides of a joint. From anatomical position, the joint actions that occur when limbs move around transverse joint axes and within sagittal planes are flexion, extension, hyperextension, plantar flexion, and dorsiflexion. The joint actions that occur when limbs move around anteroposterior joint action and within frontal planes are abduction, adduction, radial deviation (radial flexion), ulnar deviation (ulnar flexion), inversion, eversion, elevation, depression, lateral flexion to the right, and lateral flexion to the left. The joint actions that occur when limbs move around longitudinal axes and within transverse planes are internal (inward or medial) rotation, external (outward or lateral) rotation, pronation, supination, horizontal abduction (horizontal extension), horizontal adduction (horizontal flexion), rotation left, and rotation right.
• Human movement description uses terminology that describes the relative movements of two limbs on cither side of a joint (relative angular motion) rather than terminology that describes individual limb movements (absolute angular motion).
Movements Around Transverse Axes
Flexion, extension, and hyperextension are joint actions occurring at the wrist, elbow, shoulder, hip, knee, and intervertebral joints. Starring from anatomical position, flexion is the joint action that occurs around the transverse axes through these joints and causes limb movements in sagittal planes through the largest range of motion. Extension is the joint action that occurs around the transverse axes through these joints and causes the opposite limb movements in sagittal planes that return the limbs to anatomical position. Hyper extension is the joint action that occurs around the transverse axes and is a continuation of extension past anatomical position. Elbow flexion thus occurs when the forearm is moved forward and upward and the angle between the forearm and the upper arm at die anterior side of the elbow joint gets smaller. Elbow extension occurs when the forearm is returned to anatomical position. Elbow hyperextension would occur if the forearm could continue ex-tending past anatomical position. Figure 6.19, a through g, shows the flexion, extension, and hyperextension joint actions that occur at the wrist, elbow, shoulder, hip, knee, trunk, and neck joints, respectively.
Dorsiflexion and plantar flexion are joint actions that occur at the ankle. Starting from anatomical position, dorsiflexion is the joint action that occurs around the transverse axis through the ankle joint and causes die foot to move in a sagittal plane such that it moves forward and upward toward the leg. When you lift your toes off the ground and put your weight on your heels, you are dorsiflexing at your ankles. Plantar flexion is the joint action that occurs around the transverse axis through the ankle joint and causes the opposite movement of the foot in a sagittal plane so that the foot moves downward away from the leg. When you stand on your toes, you are plantar flexing at your ankles. Dorsiflexion and plantar flexion are also shown in figure 6.19g.
Figure 6.19. Sagittal plane joint actions at the wrist, elbow, shoulder, hip, knee, trunk, and neck and ankle.
Figure 6.19. (Continued)

Movements Around Anteroposterior Axes
Abduction and adduction arc joint actions occurring at the shoulder and hip joints. Starting from anatomical position, abduction is the joint action that occurs around the AP axes through these joints and causes limb movements in frontal planes through the largest range of motion. Adduction is the joint action that occurs around the AP axes through these joints and causes limb movement in a frontal plane back toward anatomical position. Shoulder abduction thus occurs when the arm is moved upward and laterally away from the body. Shoulder adduction occurs when the arm is returned to anatomical position. Figure 6.20, a and b, shows the abduction and adduction joint actions that occur at the shoulder and hip joints.
Ulnar deviation (adduction or ulnar flexion) and radial deviation (abduction or radial flexion) are joint actions occurring at the wrist joint (see figure 6.20 c).
Starting from anatomical position, ulnar deviation is the joint action that occurs around the AP axis through the wrist and causes hand movement in a frontal plane toward the little finger. Radial deviation is the joint action that occurs around the AP axis through the wrist joint and causes the opposite movement of the hand in a frontal plane, moving it laterally toward the thumb.
Inversion and aversion arc frontal plane movements that occur at the ankle joint (see figure 6.20c). These joint actions occur around an AP axis through the foot. Starting from anatomical position, inversion occurs when the medial side of the sole of the foot is lifted. The return to anatomical position and the movement of the foot beyond anatomical position where the lateral side of the sole of the foot is lifted is eversion.
The movements of the scapula (the shoulder blade) and shoulder girdle occur primarily in the frontal plane as well. These movements include abduction (movement of the scapula away from the midline) and adduction (movement or the scapula toward the midline), elevation (superior movement of the scapula) and depression (inferior movement of the scapula), and upward rotation (such that the medial border moves superiorly and the shoulder joint moves superiorly) and downward rotation (such that the medial border moves superiorly and the shoulder joint moves inferiorly).
Lateral flexion to the left or right also occurs in the frontal plane. Lateral flexion of the trunk to the left occurs when you lean to the left, and lateral flexion to the right occurs when you lean to the right. Likewise, lateral flexion of the neck to the left occurs when you lean your head toward your left shoulder, and lateral flexion of the neck to the right occurs when you lean your head toward your right shoulder. Lateral flexion is shown in figure 6.20, d and e.


Movements Around Longitudinal Axes
Internal rotation (inward or medial rotation) and external rotation (outward or lateral rotation) are joint actions occurring at the shoulder and hip joints. Starting from anatomical position, internal rotation is the joint action that occurs around the longitudinal axes through these joints and causes limb movements in the transverse plane such that the knees turn inward toward each other or the palms of the hands turn toward the body. External rotation is the joint action that occurs around the longitudinal axes through these joints and causes the opposite limb movements in transverse planes and returns the limbs to anatomical position or moves them beyond anatomical position. Figure 6.21, a and b, shows the internal and external rotation joint actions that occur at the hip and shoulder joints.
Pronation and supination are joint actions occurring at the radioulnar joint in the forearm. Starting from anatomical position, pronation is the joint action that occurs around the longitudinal axis of the forearm and through the radioulnar joint and causes the palm to turn toward the body. This motion is similar to internal rotation at the shoulder joint except that it occurs at the radioulnar joint. Supination is the joint action that occurs around the longitudinal axis through the radioulnar joint and causes the opposite limb movement in a transverse plane and returns the forearm and hand to anatomical position or moves them beyond anatomical position. Figure 6.21 d also shows the supination and pronation actions that occur at the radioulnar joint.
Horizontal abduction (horizontal extension or transverse abduction) and horizontal adduction (horizontal flexion or transverse adduction) are joint actions occurring at the hip and shoulder joints. These joint actions do not commence from anatomical position. First, hip or shoulder flexion must occur and continue until the arm or thigh is in the transverse plane. Horizontal abduction is then the movement of the arm or leg in the transverse plane around a longitudinal axis such that the arm or leg moves away from the midline of the body. The return movement is horizontal adduction. These joint actions are also shown in figure 6.21c.
Rotations of the head, neck, and trunk also occur around a longitudinal axis. Turning your trunk so that you face left is rotation to the left, and turning your trunk so that you face right is rotation to the right. These actions are shown in figure 6.21, e and f.
Circumduction is a multiple-axis joint action that occurs around the transverse and AP axes. Circumduction is (a) flexion combined with abduction and then adduction or (b) extension or hyperextension combined with abduction and then adduction. The trajectory of a limb being circumducted forms a cone-shaped surface. If you abduct your arm at the shoulder joint and then move your arm and forearm so that your hand traces the shape of a circle, the joint action occurring at the shoulder joint is circumducdon.
Just as there are diagonal planes and axes, joint actions may also occur in diagonal planes around diagonal axes. Such diagonal joint actions may be combinations of joint actions if they occur at multiple-axis joints, or they may be one of the joint actions described above if the joint or limb has been moved into a diagonal plane by the action at a more proximal joint. Each of the terms describing joint actions really specifics the direction of relative angular motion at a joint. The joint actions and the corresponding planes and axes of motion for these actions are summarized in table 6.2.
• Each of the terms describing joint actions really specifies the direction of relative angular motion at a joint.
Figure 6.21. transverse plane joint actions at the hip, shoulder, shoulder and hip, radioulnar joint, neck, and trunk.
Table 6.2. Joint actions and their corresponding plane and axes of motion
Plane of motion:
- Sagittal
- Frontal
- Transverse
Axis:
- Transverse
- Anteroposterior
- Longitudinal
Joint axtions:
- Flexion
- Extension
- Hyperextension
- Plantar flexion
- Dorsiflexion

- Abduction
- Abduction
- Ulnar deviation
- Radial deviation
- Inversion
- Eversion
- Elevation
- Depression
- Upward rotation
- Downward rotation
- Lateral flexion to the left
- Lateral flexion to the right

- Internal rotation
- External rotation
- Pronation
- Supination
- Horizontal abduction
- Horizontal abduction
- Rotation to the right
- Rotation to the left


Summary
Angular kinematics is concerned with the description of angular motion. Angles describe the orientation of two lines. Absolute angular position refers to the orientation of an object relative to a fixed reference line or plane, such as horizontal or vertical. Relative angular position refers to the orientation of an object relative to a no fixed reference line or plane. Joint angles are relative, whereas limb positions may be relative or absolute. The angular movements of limbs around joints are described using terminology developed by anatomists using the anatomical position of the body as a reference. The three principal anatomical planes (sagittal, frontal, and transverse) along with their corresponding axes (transverse, ameroposterior, and longitudinal) are also useful for describing movements of the limbs.
When an object rotates, it undergoes an angular displacement. To define the angular displacement, the axis and plane of rotation must be known. The direction of the angular displacement (and all other angular motion and torque vectors) is then established using the right-hand thumb rule. The definitions of angular displacement, angular velocity, and angular acceleration are similar to their linear counterparts.
The linear displacement and distance traveled by a point on a rotating object are directly proportional to the radius of rotation. The linear distance traveled equals the product of the angular displacement measured in radians times the radius of rotation.
Tangential linear velocity and acceleration of a point on a rotating object are directly proportional to the radius as well. The tangential linear velocity is equal to the product of the angular velocity times the radius of rotation. Lengthening the radius while maintaining the angular velocity is an important principle in a variety of striking skills. Tangential linear acceleration is equal to the product of angular acceleration times the radius of rotation.
Centripetal acceleration (also called radial acceleration) of an object rotating in a circular path is the component of linear acceleration directed toward the axis of rotation. It is directly proportional to the square of the tangential linear velocity or the square of the angular velocity. Centripetal force is the force exerted on the rotating object to cause the centripetal acceleration.

KEY TERMS
abduction (p. 165)
absolute angular position (p. 149)
adduction (p. 165)
anatomical position (p. 159)
angular acceleration (p. 1 56)
angular displacement (p. 149)
angular position (p. 149)
angular velocity (p. 153)
anteroposterior axis (p. 160)
average angular velocity (p. 153)
cardinal plane (p. 1 59)
centripetal acceleration (p. 157)
centripetal force (p. 157)
circumduction (p. 167)
depression (p. 167)
dorsiflexion (p. 164)
downward rotation (p. 167)
elevation (p. 167)
eversion (p. 167)
extension (p. 164)
external rotation (p. 167)
flexion (p. 164)
frontal plane (p. 159)
horizontal abduction (p. 167)
horizontal adduction (p. 167)
hypercxtension (p. 164)
instantaneous angular velocity (p. 15 3)
internal rotation (p. 167)
inversion (p. 167)
lateral flexion (p. 167)
longitudinal axis (p. 160)
plantar flexion (p. 164)
pronation (p. 167)
radial deviation (p. 166)
radian (p. 149)
relative angular position (p. 149)
sagittal plane (p. 159)
Stipulation (p. 167)
Tangential acceleration (p. 156)
transverse axis (p. 160)
transvese plane (p. 159)
ulnar deviation (p. 166)
upward rotation (p. 167)

REVIEW QUESTIONS
1. In golf, the longest club is the driver and the shortest club is the pitching wedge. Why is it easier to hit the bail farther with a driver than with a pitching wedge?
2. What advantages does a longer-limbed individual have in throwing and striking activities?
3. Explain how step length (a linear kinematic variable) might increase if stretching exercises increase die range of motion at die hip joint, thus increasing the angular displacement (an angular kinematic variable) of the hip joint during a step.
4. What is the plane of motion for most of the joint actions that occur during sprint running? What is the corresponding axis of motion for these joint actions?
5. When you swing a baseball bat, what is the plane of motion for the action occurring at your leading shoulder? What is the axis of motion? What joint action occurs at die leading shoulder during the swing?
6. During the deliver phase of a baseball pitch, what joint actions occur at the shoulder and elbow joints of the throwing arm?
7. What joint action occurs at the shoulder joint and what are the plane and axis of motion during the pulling-up phase of a wide-grip pull-up?
8. How do the actions at the shoulder joint differ between a wide-grip pull-up and a narrow-grip pull-up?
9. When you spike a volley ball, what joint action occurs at the elbow joint of your hitting arm?

PROBLEMS
1. A therapist examines the range of motion of an athlete’s knee joint during her rehabilitation from a knee injury. At hill extension, the angle between the leg and thigh is 178°. At full flexion, the angle between the leg and thigh is 82°. During the lest, the thigh was held in a fixed position and only the leg moved. What was the angular displacement of the leg from full extension in lull flexion? Express your answer in (a) degrees and (b) radians.
2. Figure skater Michelle Kwan performs a triple twisting jump. She rotates around her longitudinal axis three times while she is in the air. The time it takes to complete the jump from takeoff to landing is 0.8 s. What was Michelle’s average angular speed in twisting for this jump?
3. When Josh begins his discus throwing motion, he spins with an angular velocity of 5 rad/s. Just before lie releases the discus, Josh’s angular velocity is 25 rad/s. If the lime from the beginning of the throw to just before release is 1 s, what is Josh’s average angular acceleration?
4. The tendon from the knee extensor muscles attaches to the tibia bone 1,5 in. away from the center of the knee joint, and the foot is 15 in. away from the knee joint. What are length does the foot move through when the knee extensor muscles contract and their point of insertion on the tibia moves through an arc length of 2 in.?
5. A hammer thrower spins with an angular velocity of 1700 °/s. The distance from his axis of rotation to the hammer head is 1.2 in.
a. What is the linear velocity of the hammer head?
b. What is the centripetal acceleration of the hammer head?
6. A hook in boxing primarily involves horizontal flexion of the shoulder, while maintaining a constant angle at the elbow. During this punch, the horizontal flexor muscles of the shoulder contract and shorten at an average speed of 75 cm/s. They move through an arc length of 5 cm during the hook, while the fist moves through an arc length of 100 cm. What is the average speed of the fist during the hook?
7. A baseball pitcher pitches a fastball with a horizontal velocity of 40 m/s. The horizontal distance from the point of release to home plate is 17.50 m. The batter decides to swing die bat 0.30 s after the ball was released by the pitcher. The average angular velocity of the bat is 12 rad/s. The angular displacement of die bat from the batter’s shoulder to hitting positions above the plate is between 1.5 and 1.8 rad.
a. Will the bat be in a hitting position above the plate when the ball is above the plate? Assume the pitch is in the strike zone.
b. Assume that the batter does hit the ball. If the bat’s instantaneous angular velocity is 30 rad/ s at the instant of contact, and the distance from the sweet spot on the bat to the axis of rotation is 1.25 m, what is the instantaneous linear velocity of the sweet spot at the instant of ball contact?