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ROTATIONAL MOTION
11-1 Translation and Rotation The graceful movement of figure skaters can be used to illustrate two kinds of pure, or unmixed, motion. If a skater is gliding across the ice in a straight line with constant speed, her motion is one of pure translation. If she spinning at a constant rate about a vertical axis, in a motion of pure rotation. Translation is motion along a straight line, which has been our focus up to now. Rotation is the motion of wheels, gears, motors, planets, the hands of clocks, the rotors of jet engines, and the blades of helicopters. It is our focus in this chapter. 11-2 The Rotational Variables We wish to examine the rotation of a rigid body about a fixed axis. A rigid body is a body that can rotate with all its parts locked together and without any change in its shape. A fixed axis means that the rotation occurs about an axis that does not move. Thus, we shall not examine an object like the Sun, because the parts of the Sun (a ball of gas) are not locked together. We also shall not examine an object like a bowling ball rolling along a bowling alley, because the ball rotates about an axis that moves (the ball's motion is a mixture of rotation and translation). Figure 11-2 shows a rigid body of arbitrary shape in rotation about a fixed axis, called the axis of rotation or the rotation axis. Every point of the body moves in a circle whose center lies on the axis of rotation, and every point moves through the same angle during a particular time interval. In pure translation, every point of the body moves in a straight line, and every point moves through the same linear distance during a particular time interval. (Comparisons between angular and linear motion will appear throughout this chapter.) We deal now - one at a time - with the angular equivalents of the linear quantities position, displacement, velocity, and acceleration. Angular Position
Figure 11-2 shows a reference line, fixed in the body, perpendicular to the rotation axis, and rotating with the body. The angular position of this line is the angle of the line relative to a fixed direction, which we take as the zero angular position. In Fig. 11-3, the angular position Here An angle defined in this way is measured in radians (rad) rather than in revolutions (rev) or degrees. The radian, being the ratio of two lengths, is a pure number and thus has no dimension. Because the circumference of a circle of radius
and thus We do not reset
Angular Displacement If the body of Fig. 11-3 rotates about the rotation axis as in Fig. 11-4, changing the angular position of the reference line from 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 An angular displacement in the counterclockwise direction is positive, and one in the clockwise direction is negative. Angular Velocity Suppose (see Fig. 11-4) that our rotating body is at angular position
in which The (instantaneous) angular velocity
Equations 11-5 and 11-6 hold not only for the rotating rigid body as a whole but also for every particle of that body because the particles are all locked together. The unit of angular velocity is commonly the radian per second (rad/s) or the revolution per second (rev/s). If a particle moves in translation along an Angular Acceleration If the angular velocity of a rotating body is not constant, then the body has an angular acceleration. Let
in which
Equations 11-7 and 11-8 hold not only for the rotating rigid body as a whole but also for every particle of that body. The unit of angular acceleration is commonly the radian per second-squared (rad/s2) or the revolution per second-squared (rev/s2). Date: 2015-01-12; view: 1427
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