08/06/04

1. Angular position, angular distance, angular displacement

         It is very important to understand the angular motion to deal with human motion because most of the human movement involves rotation about the joints.

        Angle in angular kinematics is compatible to the position in linear kinematics. An angle is composed of two sides (lines) and a vertex at which the side intersect each other.

        Angles that are measured relative to the positive direction of the x axis are positive if they are measured in the counterclockwise direction, and negative if in the clockwise direction. For example, 120° and -210° are the same angles as shown in Figure 1-2.

The rotating body can be described by specifying the angular position q is the angle of the line relative to the fixed direction. The angular position q is measured relative to the positive direction of the x axis, the q is described by 

                Figure 1-3. Measure of angular position in radian

           The radian unit is widely used in science fields. There are two different units of angle, which are degrees (°) and radians (rad). You can associate two measurement remembering that 360° is equal to 2π rad. For example, 45° can be converted to radian unit as follows;

        Figure 1-4. Calculation of radians from degrees

           If the body rotates from one angular position t1 to another t2, the angular displacement Dq is given by Figure 1-5 and described in Figure 1-6. The sign of angular displacement can be positive or negative depending on the direction of rotation. If the body is rotating counterclockwise or clockwise, the sign will be positive or negative compatibly.

                                                        Figure 1-5. Measure of angular displacement

If the body rotates  from t1, and moves t3 and then comes back to t2 as shown in Figure 1-7, and q1, q2 and q3 are corresponding to t1, t2 and t3, the angular distance are given by

                    Figure 1-7. Measure of angular distance

while the angular displacement is still .

2. Angular speed, angular velocity and angular acceleration

        If a body is at angular position q1 at time t1 to q2 at time t2, the average angular velocity during Dt from t1 to t2 can be defined by

                 Figure 2-1. Definition of average angular velocity

The instantaneous angular velocity can be defined as

                 Figure 2-2. Definition of instantaneous angular velocity

When you know q(t), it is easy to find by differentiating q(t). The sign of angular velocity can be either positive or negative depending on whether q is increasing (counterclockwise) or decreasing (clockwise).  Angular speed is just the magnitude of angular velocity. The unit of angular velocity is rad/s.

        Average angular acceleration of a rotating body in the interval from t1 to t2 can be defined as

              Figure 2-3. Definition of average angular acceleration

The instantaneous angular acceleration , with which we shall be most concerned, is limit of this quantity as is made to approach zero as follows;

                Figure 2-4. Definition of instantaneous angular acceleration

When the velocity of an object increases, the angular acceleration is positive and vice versa. The unit of angular acceleration is rad/s².

3. Motion in constant linear and angular acceleration

Table 2-1. Formula for motion in constant linear and angular acceleration

Motion	Formula	Missing Variable
Linear
Angular

4. Linear and angular variables

        It is sometimes needed to relate linear variables s, v and a for a particular point in a rotating body to the angular variables  , and . The two sets of variables are related by r which the perpendicular distance of the point from the rotation axis.

When a rigid body is rotate through an angle and a point in the body is moved a distance s along a circular arc, s is given from Figure 1-3 as

                Figure 4-2. Linear displacement expressed by angular components

As you already know, the differentiation of displacement produces velocity. The differentiation of Figure 4-2 gives you

                Figure 4-3.

Since and , we can rewrite Figure 4-3 as

                    Figure 4-4. Linear velocity expressed by angular components

When you derive Figure 4-3, you can get

            Figure 4-5.

Because and , We can again rewrite Figure 4-5 as

                    Figure 4-6. Tangential component of acceleration

        When the angular speed is constant, All points within the body experiences the constant linear speed. The period of revolution T for the motion of the rigid body can be given by

                Figure 4-6. Period of revolution expressed with linear velocity

The period or period of revolution T is defined as "the time for a particle to go around a closed path exactly once" . Figure 4-6 can also be expressed from Figure 4-4 as

                Figure 4-7. Period of revolution expressed with angular velocity

        There is another acceleration which is composing the actual acceleration a other than the tangential acceleration a_t. That is a_r, radial component of acceleration which passes through the rotational axis in its direction. It is given by

         Figure 4-8. Radial component of acceleration