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# lecture-ch12 - Contents 12 Equilibrium and Elasticity 12.1...

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Contents 12 Equilibrium and Elasticity 1 12.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 12.2 The Conditions for Equilibrium . . . . . . . . . . . . . . . . . 2 12.3 The Center of Gravity (cog) . . . . . . . . . . . . . . . . . . . 3 12.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 12.4.1 Sample Problem 12 - 1 . . . . . . . . . . . . . . . . . . 4 12.4.2 Sample Problem 12 - 2 . . . . . . . . . . . . . . . . . . 5 12.4.3 Sample Problem 12 - 3 . . . . . . . . . . . . . . . . . . 6 12.4.4 Sample Problem 12 - 4 . . . . . . . . . . . . . . . . . . 7 12.5 Indeterminate Structures . . . . . . . . . . . . . . . . . . . . . 8 12.6 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 12 Equilibrium and Elasticity In this chapter we will define equilibrium and find the conditions needed so that an object is at equilibrium. We will then apply these conditions to a variety of practical engineering problems of static equilibrium. We will also examine how a “rigid” body can be deformed by an external force. In this section we will introduce the following concepts: Stress and strain Young’s modulus (in connection with tension and com- pression) Shear modulus (in connection with shearing) Bulk modulus (in connection to hydraulic stress) 12.1 Equilibrium For an object, 1. The linear momentum vector P of its center of mass is constant. 2. Its angular momentum vector L about its center of mass, or about any other point is also constant. We say such an object is in equilibrium. The two requirements for equi- librium are then vector P = a constant and vector L = a constant. 1

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Our concern in this chapter is with situations in which vector P = 0 and vector L = 0. That is we are interested in objects that are not moving in any way (this includes translational as well as rotational motion) in the reference frame from which we observe them. Such objects are said to be in static equilibrium. In chapter 8 we differentiated between stable and unstable static equilibrium. If a body that is in static equilibrium is displaced slightly from this position, the forces on it may return it to its old position. In this case we say that the equilibrium is stable. If the body does not return to its old position, then the equilibrium is unstable. An example of unstable equilibrium is shown in the figures above. In fig. a we balance a domino with the domino’s center of mass vertically above the supporting edge. The torque of the gravitational force vector F g about the supporting edge is zero because the line of action of vector F g passes through the edge. Thus the domino is in equilibrium. Even a slight force on the domino ends the equilibrium. As the line of action of vector F g moves to one side of the supporting edge (see fig. b) the torque due to vector F g is non-zero and the domino rotates in the clockwise direction away from its equilibrium position of fig. a.
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