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n this chapter, the author describes almost the same physical properties, characteristics regarding to rotational (angular) motion.Besides distance, displacement, velocity, and acceleration, which have been described in Chapter 6, other physical properties such as inertia, and different variables including tangential, centrifugal, and centripetal accelerations related to angular motion will be described in this chapter.
The reader should know the following (information) facts related to angu-lar motion. Any object/mass that is going through a uniform circuangu-lar motion with a constant speed will be accelerated by the force driving it because a rotary motion is related to acceleration, velocity, and direction.
How should be this understood? The object/mass basically has nothing to do with speed, but has to do with velocity. Since the velocity is a vector that has both magnitude and direction, a change in either the magnitude or the direction constitutes a change in the velocity.
So, the conclusion of the above statement is that an object moving in a circle at constant speed is indeed accelerating. It is accelerating because the direction of the velocity vector is changing. If the magnitude of the velocity is changing, then acceleration occurs too. Putting this in simple words, the rotary motion constitutes a change in direction even if that direction is a perfect circle, and if the rotation is going on and on, it still constitutes a change of direction.
7.1 DISTANCE AND DISPLACEMENT
When a body rotates in a 2-D plane, the rotation is characterized by the fact that the rotating body (its external point) moves around the perimeter of a circle or a cylinder at the same distance from the center of rotation. The rotat-ing body describes a certain angle that can be measured by angles or radians.
The angle is measured from the center point of the circle using the x axis as an arbitrary reference line that moves around the path of the circle.
The end point of x should coincide with the initial point of the particle of the body, and then the measurement will end at the end point of the particle rotating on the circle. This path on the circle is named as angular displacement. What is then the distance in a circular motion, and how is it measured? The distance in angular rotation in fact represents a dis-tance separation between two objects observed from a location different from either of these two objects. In other words, angular distance is thus synonymous to the angle itself.
When we deal with movements of human beings, the rotations of the human segments and the entire body describes absolutely only rotations and no rectilinear movements at all. Linear movements of humans are acceptable theoretically. How should this be understood? Imagine if you approach or depart one of your body segments such as flexion or extension of your forearm, the rotation will occur at the center point of the rotation, which is the joint of those body segments. When you flicker with one of your finger to move a small object from the table, there is a rotary move-ment between two phalanges. Recall that linear movemove-ments of human beings are acceptable theoretically or can be considered only when the human body segment is related to an object or scope.
For instance, walking is related to the ground or to the arrival point, so the body is walking using a rectilinear movement. In this case, we can talk about linear speed, distance, or even linear acceleration and so on. The measurement of the distance/displacement in 2-D plane can be done to the positive direction, which is counterclockwise, or negative direction, which is clockwise. Both measurements usually start from the x axis line. Figures 7.1 and 7.2 show the measurement of the angular distance and displacement.
The equation for the displacement is the following: s = r · θ or θ = s/r.
The unit of the radian represents the dimensionless measure of the ratio of the circle’s arc length to its radius. The author will further describe radi-ans, angles, circumference, radius, pi (π), and other mathematical signs and expressions (see Section 7.2).
A body can change its movement from a translation movement to a rotational movement and vice versa. For instance, from martial arts of karate, a simple direct punch of the forearm where the fist moves through a straight line approximately for a distance of 45–50 cm, and then the fist will rotate from supine to prone position again for a distance of approximately 12–15 cm when the fist should reach the target.
7.2 CIRCLES, QUADRANTS, AND THEIR ANGLES
When we speak in general about martial arts, it is well known that almost all the motions are in effect within 3-D in space, also describing a circular
C B
Y
A
X A1
FIGURE 7.1 The angular distance is measured from point A or point A1 through-out point B, point C, and returning to point B as the terminal site. The addition of the angular distance is made θ d = A(A1) + B + C.
r s
θ
FIGURE 7.2 The angular displacement is measured by using radians rather than degrees. The arc length (s) equals of the length of the radius (r) of the circle and the subtended central angle (θ) equal 1 radian (rad).
motion following a circular path the rotation that describes an angle is mea-sured by degrees or by radians. Usually the circle is divided into four parts named quadrants by using only the axis of x and y. Using the x and y axes, the measurement is 2-D in space. An angle with its initial side on the x axis is said to be in standard position. Angles that are in standard position are said to be quadrantal if their terminal side coincides with a coordinate axis.
A circle has four quadrant positions. Each quadrant measures 90° angle.
Therefore, a circle has 360° angle. Figure 7.3a−c shows the quadrants of cir-cle and the negative or positive directions measured by angles.
Point A represents a reference point. From this point, the calculation of the angles are counted to be counterclockwise or clockwise (−) direction.
Figure 7.3b represents a full circle of 360° (−45°) in a positive direction measured from point A toward the (x) axis all around. Figure 7.3c shows a counterclockwise direction of a 495° route.
In human mechanics, we sometimes have to use different calculation methods using a circle as a reference for the joint’s angular motions. In this case, it is important to know the geometric terminology and symbols such as circumference, revolution, diameter, radius, tangent, radian, and degrees.
The author assumes that the reader knows these terminologies. The following table serves us for fast calculations of the revolutions, radians, and degrees:
Revolutions Radians Degrees circumference (C) = 2πr, r = radius.
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FIGURE 7.3 (a) Counterclockwise (positive) direction of a circle. (b) Clockwise (negative) direction of a circle. (c) Quadrants of a circle (I, II, III, IV).
7.3 SPEED, VELOCITY, AND ACCELERATION
The average angular speed (ω) tells us how many complete rotations or revolutions are there per unit of time. More precisely, the angular speed tells us the change in angle per unit of time, which is measured in radians/s.
Angular speed can also be measured in degrees, for example, 360o/s. Even if the term angular speed is equivalent to rotational velocity, there is a dif-ference, which is the rotation (revolution) per minute, for example, 60 rpm.
Angular speed represents the magnitude of angular velocity.
The angular velocity, whose sign is also a Greek letter omega (ω), is a vector quantity; that is why in most physics books or other books, the authors use the terms angular velocity and angular speed interchangeably.
The magnitude of angular velocity can be described in two terms. One is the average angular velocity, which represents the angular displacement of an object between two points of an angle such as θ1 and θ2 at the time intervals t1 and t2, respectively.
Then, the equation for average angular velocity is ω = (θ2 −θ1/t2 −t1) (= ∆ ∆θ/ t);the other term for the magnitude is the instan-taneous angular velocity, which is the limit of the magnitude ratio as Δt approaches 0. The formula is: ω = lim as Δt → 0 Δθ/Δt = dθ/dt, both ω and ω being measured in m/s. To find out the average angular acceleration and the instantaneous angular acceleration, we proceed in an analogous fashion to linear velocities and accelerations. The average angular acceleration ( )α of a rotating body in the interval from t1 to t2 can be defined using the fol-lowing formula, where ω represents the instantaneous angular velocities.
The formula for average angular acceleration α ω= 2 −ω1/t2 − =t1
∆ ∆ω/ t,and the instantaneous angular acceleration α = lim as Δt → 0 Δω/
Δt = dω/dt, both α and α being measured in rad/s2.
Speaking about angular motion, we should mention two different and important accelerations. One of these is the so-called radial component of the linear acceleration that is named radial acceleration, also called centripe-tal acceleration. This component acts inwardly toward the center of rotation.
The mathematical equation is ar = v2/r, where v is the linear velocity and r is the length of the radius of rotation. The formula can also be written as ω2r.
The second kind of component of the acceleration is named the tangen-tial acceleration that is directed along a tangent to the path of the body trav-eling in angular motion that indicates change in linear speed: at = v2 – v1/t, where at is the tangential acceleration, v2 is the final time of the linear veloc-ity, and t is the time interval over which the body is assessed, simply m/s2.
7.4 STUDY QUESTIONS
1. A karateka stands in front stance (Zenkutsu-dachi) and executes a roundhouse kick at the middle level of the opponent. He increases his speed from “0” radian to 3 rad/s in a time of 0.5 s. What is his angular acceleration? We use the equation for angular acceleration (α) = rad/s2. One radian (rad) = 57.3°. If we multiply 57.3 × 6 = 343.8°, which means he can turn almost 360° in 1 s.
Answer: 3 – 0/0.5 s = 3/0.5 = 6 rad/s2.
2. As in the previous example, the karateka leg from the coxo-femoral joint to the toes of the kicking leg has a radius of about 110 cm and a speed of 0.5 s. What is his centripetal acceleration? a = v2/r.
Answer: 0.52/110 cm = 0.25/1.1 m = 0.22 m/s2.
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