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chapter_8

# chapter_8 - Rotational motion is key in the harnessing of...

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226 8 CHAPTER Rotational Equilibrium and Rotational Dynamics O U T L I N E 8.1 Torque 8.2 Torque and the Two Conditions for Equilibrium 8.3 The Center of Gravity 8.4 Examples of Objects in Equilibrium 8.5 Relationship between Torque and Angular Acceleration 8.6 Rotational Kinetic Energy 8.7 Angular Momentum © Kevin Fleming/Corbis In the study of linear motion, objects were treated as point particles without structure. It didn’t matter where a force was applied, only whether it was applied or not. The reality is that the point of application of a force does matter. In football, for example, if the ball carrier is tackled near his midriff, he might carry the tackler several yards before falling. If tackled well below the waistline, however, his center of mass rotates toward the ground, and he can be brought down immediately. Tennis provides another good example. If a tennis ball is struck with a strong horizontal force acting through its center of mass, it may travel a long distance before hitting the ground, far out of bounds. Instead, the same force applied in an upward, glancing stroke will impart topspin to the ball, which can cause it to land in the opponent’s court. The concepts of rotational equilibrium and rotational dynamics are also important in other disciplines. For example, students of architecture benefit from understanding the forces that act on buildings and biology students should understand the forces at work in muscles and on bones and joints. These forces create torques, which tell us how the forces affect an object’s equilibrium and rate of rotation. Rotational motion is key in the har- nessing of energy for power and propulsion, as illustrated by a steam- boat. The rotating paddlewheel, driven by a steam engine, pushes water backwards, and the reaction force of the water thrusts the boat forward.

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8.1 Torque 227 We will find that an object remains in a state of uniform rotational motion unless acted on by a net torque. This principle is the equivalent of Newton’s first law. Further, the angular ac- celeration of an object is proportional to the net torque acting on it, which is the analog of Newton’s second law. A net torque acting on an object causes a change in its rotational energy. Finally, torques applied to an object through a given time interval can change the object’s angular momentum. In the absence of external torques, angular momentum is conserved, a property that explains some of the mysterious and formidable properties of pulsars— remnants of supernova explosions that rotate at equatorial speeds approaching that of light. 8.1 TORQUE Forces cause accelerations; torques cause angular accelerations. There is a definite relationship, however, between the two concepts. Figure 8.1 depicts an overhead view of a door hinged at point O. From this view- point, the door is free to rotate around an axis perpendicular to the page and passing through O. If a force is applied to the door, there are three factors that determine the effectiveness of the force in opening the door: the magnitude
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