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10 - Rotation of a Rigid Object About a Fixed Axis

10 - Rotation of a Rigid Object About a Fixed Axis -...

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292 Chapter 10 Rotation of a Rigid Object About a Fixed Axis CHAPTE R OUTLI N E 10.1 Angular Position, Velocity, and Acceleration 10.2 Rotational Kinematics: Rotational Motion with Constant Angular Acceleration 10.3 Angular and Linear Quantities 10.4 Rotational Kinetic Energy 10.5 Calculation of Moments of Inertia 10.6 Torque 10.7 Relationship Between Torque and Angular Acceleration 10.8 Work, Power, and Energy in Rotational Motion 10.9 Rolling Motion of a Rigid Object The Malaysian pastime of gasing involves the spinning of tops that can have masses up to 20 kg. Professional spinners can spin their tops so that they might rotate for hours before stopping. We will study the rotational motion of objects such as these tops in this chapter. (Courtesy Tourism Malaysia)
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293 W hen an extended object such as a wheel rotates about its axis, the motion cannot be analyzed by treating the object as a particle because at any given time different parts of the object have different linear velocities and linear accelerations. We can, however, analyze the motion by considering an extended object to be composed of a collection of particles, each of which has its own linear velocity and linear acceleration. In dealing with a rotating object, analysis is greatly simplified by assuming that the object is rigid. A rigid object is one that is nondeformable—that is, the relative loca- tions of all particles of which the object is composed remain constant. All real objects are deformable to some extent; however, our rigid-object model is useful in many situa- tions in which deformation is negligible. 10.1 Angular Position, Velocity, and Acceleration Figure 10.1 illustrates an overhead view of a rotating compact disc. The disc is rotating about a fixed axis through O . The axis is perpendicular to the plane of the figure. Let us investigate the motion of only one of the millions of “particles” making up the disc. A particle at P is at a fixed distance r from the origin and rotates about it in a circle of radius r . (In fact, every particle on the disc undergoes circular motion about O .) It is convenient to represent the position of P with its polar coordinates ( r , ), where r is the distance from the origin to P and is measured counterclockwise from some reference line as shown in Figure 10.1a. In this representation, the only coordinate for the particle that changes in time is the angle ; r remains constant. As the particle moves along the circle from the reference line ( 0), it moves through an arc of length s , as in Figure 10.1b. The arc length s is related to the angle through the relationship (10.1a) (10.1b) Note the dimensions of in Equation 10.1b. Because is the ratio of an arc length and the radius of the circle, it is a pure number. However, we commonly give the arti- ficial unit radian (rad), where Because the circumference of a circle is 2 corresponds to an angle of (2 one radian is the angle subtended by an arc length equal to the radius of the arc.
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