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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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c h a p t e r Rotation of a Rigid Object About a Fixed Axis 10.1 Angular Displacement, Velocity, and Acceleration 10.2 Rotational Kinematics: Rotational Motion with Constant Angular Acceleration 10.3 Angular and Linear Quantities 10.4 Rotational 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 Did you know that the CD inside this player spins at different speeds, depend- ing on which song is playing? Why would such a strange characteristic be incor- porated into the design of every CD player? (George Semple) C h a p t e r O u t l i n e 292 P U Z Z L E R P U Z Z L E R
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10.1 Angular Displacement, Velocity, and Acceleration 293 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. For this reason, it is convenient to consider an extended object as a large number 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, it is an object in which the separations between all pairs of particles remain constant. All real bodies are deformable to some extent; however, our rigid-object model is use- ful in many situations in which deformation is negligible. In this chapter, we treat the rotation of a rigid object about a fixed axis, which is commonly referred to as pure rotational motion. ANGULAR DISPLACEMENT, VELOCITY, AND ACCELERATION Figure 10.1 illustrates a planar (flat), rigid object of arbitrary shape confined to the xy plane and rotating about a fixed axis through O . The axis is perpendicular to the plane of the figure, and O is the origin of an xy coordinate system. Let us look at the motion of only one of the millions of “particles” making up this object. 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 object 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 preferred direction—in this case, the positive x axis. In this representation, the only coordinate that changes in time is the angle ; r remains constant. (In cartesian coordinates, both x and y vary in time.) As the particle moves along the circle from the positive x axis ( 0) to P , it moves through an arc of length s , which is related to the angular position through the relationship (10.1a) (10.1b) It is important to note the units 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 com- monly give the artificial unit radian (rad), where s r s r 10.1
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