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22 angular position let us first consider the

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Unformatted text preview: icle dynamics. 2.2 Angular Position Let us first consider the rotation of a perfectly rigid body about a fixed axis of rotation. Figure 1 Top view of a rotating system. For this situation, all particles undergo the same angular displacement θ (See Figure 1 for a top view of this motion). From geometry, we know that θ is given by (2) where s is the length of a circle arc that extends from the x axis (the zero angular position) and r is the radius of the circle. An angle defined this way is measured in radians (rad) rather than in revolutions (rev) or degrees (deg). The radian, being the ratio of two lengths, is a pure number and has no dimension. Because the circumference of a complete circle with a radius of r is 2πr radians this gives us: 1 rev 360° 2 2 rad (3) Therefore by dividing equation (3) by 2π we can say that 1 rad 57.3° 0.159 rev (4) Note that, we do not reset the θ to zero with each complete rotation of the reference line about the axis of rotation. If the reference line completes two revolutions from the zero angular position, then the angular position of the line is θ=4πr. 2 For pure translation along the x axis, we can know all there is to know about a moving body if we know x(t), its position as a function of time, Similarly, for pure rotation, we can know all there is to know about a rotating body if we know θ(t), the angular position of the body’s reference line as a function of time. 2.3 Angular Displacement When a rigid body rotates about the axis of rotation, as shown in Figure 2, changing the angular position of the reference line from θ1 to θ2, the body undergoes an angular displacement Δθ given by ∆ (5) This definition of angular displacement holds not only for the rigid body as a whole, but also for every particle within that body because the particles are all locked together. If a body is in translational motion along an x axis, its displacement Δx is either positive or negative, depending on whether the body is moving in the positive or negative direction of the axis. Similarly, the angular displacement Δθ of a rotating body is either positive or negative according to the following rule: An angular displacement in the counterclockwise direction is positive, and one in the clockwise direction is negative. 2.4 Angular Velocity Suppose that our rotating body is at angular position θ1 at a time t1 and at angular position θ2 at time t2 as shown in Figure 2. Thus we define the average angular velocity of the body in the time interval Δt from t1 to t2 to be Figure 2 displacement Diagram of the angular (6) The (instantaneous) angular velocity ω which is what we will be most concerned with is lim ∆ ∆ ∆ (7) or for those that have had calculus (8) Equations (6)-(8) are valid for not only the rotating rigid body as a whole, but also for every particle of that body b...
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