## Angular Motion

Angular kinematics is the description of angular motion. Angular kinematics is particularly appropriate for the study of human movement because the motion of most human joints can be described using one, two, or three rotations. Angular kinematics should also be easy for biomechanics students because for every linear kinematic variable there is a corresponding angular kinematic variable. It will even be easy to distinguish angular from linear kinematics because the adjective "angular" or a Greek letter symbol is used instead of the Arabic letters used for linear kinematics.

Angular displacement (0: theta) is the vector quantity representing the change in angular position of an object. Angular displacements are measured in degrees, radians (dimensionless unit equal to 57.3°), and revolutions (360°). The usual convention to keep directions straight and be consistent with our 2D linear kinematic calculations is to consider counterclockwise rotations as positive. Angular displacement measured with a goniometer is one way to measure static flexibility. As in linear kinematics, the frames of reference for these angular measurements are different. Some tests define complete joint extension as 0° while other test refer to that position as 180°. For a review of several physical therapy static flexibility tests, see Norkin & White (1995).

In analyzing the curl-up exercise shown in Figure 5.10, the angle between the thoracic spine and the floor is often used. This exercise is usually limited to the first 30 to 40° above the horizontal to limit the involvement of the hip flexors (Knudson, 1999a). The angular displacement of the thoracic spine in the eccen

Figure 5.10. The angular kinematics of a curl-up exercise can be measured as the angle between the horizontal and a thoracic spine segment. This is an example of an absolute angle because it defines the angle of an object relative to external space. The knee joint angle (0K) is a relative angle because both the leg and thigh can move.

tric phase would be -38° (final angle minus initial angle: 0 - 38 = -38°). This trunk angle is often called an absolute angle because it is measured relative to an "unmoving" earth frame of reference. Relative angles are defined between two segments that can both move. Examples of relative angles in biomechanics are joint angles. The knee angle (0K) in Figure 5.10 is a relative angle that would tell if the person is changing the positioning of their legs in the exercise.

### Angular Velocity

Angular velocity (to: omega) is the rate of change of angular position and is usually expressed in degrees per second or radians per second. The formula for angular velocity is w = 0/t, and calculations would be the same as for a linear velocity, except the displacements are angular measurements. Angular velocities are vectors are drawn by the right-hand rule, where the flexed fingers of your right hand represent the rotation of interest, and the extended thumb would be along the axis of rotation and would indicate the direction of the angular velocity vector. This book does not give detailed examples of this technique, but will employ a curved arrow just to illustrate angular velocities and torques (Figure 5.11). 0=0°

Figure 5.11. The average angular velocity of the first half of a knee extension exercise can be calculated from the change in angular displacement divided by the change in time.

The angular velocities of joints are particularly relevant in biomechanics, because they represent the angular speed of anatomical motions. If relative angles are calculated between anatomical segments, the angular velocities calculated can represent the speed of flexion/extension and other anatomical rotations. Biomechanical research often indirectly calculates joint angles from the linear coordinates (measurements) derived from film or video images, or directly from electrogoniometers attached to subjects in motion. It is also useful for kinesiology professionals to be knowledgeable about the typical angular velocities of joint movements. This allows professionals to understand the similarity between skills and determine appropriate training exercises. Table 5.2 lists typical peak joint angular speeds for a variety of human movements.

Let's calculate the angular velocity of a typical knee extension exercise and compare it to the peak angular velocity in the table. Figure 5.11 illustrates the exercise and the data for the example. The subject extends a knee, taking their leg from a vertical orientation to the middle of the range of motion. If we measure the angle of the lower leg from the vertical, the exerciser has moved their leg 40° in a 0.5-second period of time. The average knee extension angular velocity can be calculated as follows: = 0/t = 40/0.5 = 80 deg/s. The angular velocity is positive because the rotation is counterclockwise. So the exercise averages 80° per second of knee extension velocity over the half-second time interval, but the instantaneous angular velocity at the position shown in the figure is likely larger than that. The peak knee extension angular velocity in this exercise likely occurs in the midrange of the movement, and the knee extension velocity must then slow to zero at the end of the range of motion. This illustrates some limitations of free-weight exercises. There is a range of angular velocities (which have an affect on the linear Force-Velocity Relationship of the muscles), and there must be a decrease in the angular velocity of the movement at the end of the range of motion. This negative acceleration (if the direction of motion is positive) protects the joints and ligaments, but is not specific to many events where peak speed is achieved near the release of an object and other movements can gradually slow the body in the follow-through.

### Angular Acceleration

The rate of change of angular velocity is angular acceleration (a = rn/t). Angular acceleration is symbolized by the Greek letter alpha (a). The typical units of angular acceleration are deg/s/s and radians/s/s. Like linear acceleration, it is best to think about

 Table 5.2 ## Dealing With Back Pain

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