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12 - Static Equilibrium and Elasticity

12 - Static Equilibrium and Elasticity - 362 Chapter 12 C H...

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Balanced Rock in Arches National Park, Utah, is a 3 000 000-kg boulder that has been in stable equilibrium for several millennia. It had a smaller companion nearby, called “Chip Off the Old Block,” which fell during the winter of 1975. Balanced Rock appeared in an early scene of the movie Indiana Jones and the Last Crusade. We will study the conditions under which an object is in equilibrium in this chapter. (John W. Jewett, Jr.) Static Equilibrium and Elasticity Chapter 12 362 CHAPTE R OUTLI N E 12.1 The Conditions for Equilibrium 12.2 More on the Center of Gravity 12.3 Examples of Rigid Objects in Static Equilibrium 12.4 Elastic Properties of Solids
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363 I n Chapters 10 and 11 we studied the dynamics of rigid objects. Part of this current chapter addresses the conditions under which a rigid object is in equilibrium. The term equilibrium implies either that the object is at rest or that its center of mass moves with constant velocity relative to the observer. We deal here only with the for- mer case, in which the object is in static equilibrium . Static equilibrium represents a common situation in engineering practice, and the principles it involves are of special interest to civil engineers, architects, and mechanical engineers. If you are an engineering student, you will undoubtedly take an advanced course in statics in the future. The last section of this chapter deals with how objects deform under load condi- tions. An elastic object returns to its original shape when the deforming forces are re- moved. Several elastic constants are defined, each corresponding to a different type of deformation. 12.1 The Conditions for Equilibrium In Chapter 5 we found that one necessary condition for equilibrium is that the net force acting on an object must be zero. If the object is modeled as a particle, then this is the only condition that must be satisfied for equilibrium. The situation with real (ex- tended) objects is more complex, however, because these objects often cannot be mod- eled as particles. For an extended object to be in static equilibrium, a second condition must be satisfied. This second condition involves the net torque acting on the ex- tended object. Consider a single force F acting on a rigid object, as shown in Figure 12.1. The effect of the force depends on the location of its point of application P . If r is the position vector of this point relative to O , the torque associated with the force F about O is given by Equation 11.1: r F Recall from the discussion of the vector product in Section 11.1 that the vector is per- pendicular to the plane formed by r and F . You can use the right-hand rule to deter- mine the direction of as shown in Figure 11.2. Hence, in Figure 12.1 is directed toward you out of the page. As you can see from Figure 12.1, the tendency of F to rotate the object about an axis through O depends on the moment arm d , as well as on the magnitude of F .
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