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Unformatted text preview: Chapter 12 Equilibrium and Elasticity In this chapter we will define equilibrium and find the conditions for an object at equilibrium We will then apply these conditions to a variety of engineering problems of static equilibrium We will also examine elasticity how a “rigid” body can be deformed by an external force. We will introduce the following concepts: Stress and strain Young’s modulus (in connection with tension and compression) Shear modulus (in connection with shearing) Bulk modulus (in connection to hydraulic stress) (121) n object is in equilibrium when the following two conditions are satisfied: The of the center of mass is constant The about the center of mass Equilibrium : A 1. linear momentum 2. angular momentum P L r r or any other point is a constant We focus on situations in which 0 and (no translational as well as rotational motion). Such objects are said to be in In chapter 8 we diffe = = P L r r static equilibrium rentiated between and static equilibrium If a body in equilibrium is displaced slightly, and the forces on it returns it to its old position, then the equilibrium is . If the bo stable unstable stable dy does not return to its old position, then the equilibrium is unstable (122) An example of (fig.a) We balance a domino with its center of mass vertically above the supporting edge. The torque of the gravitational force about the supporting edge is ze g F r unstable equilibrium because the line of action of passes through the rotation axis r =0 ⊥ → ro g F r Thus the domino is in . What happens if we perturb the domino? As the line of action of moves to the right side of the supporting edge (see fig.b) the torque of is nonzero, and equilibrium g g F F r r the domino rotates in the clockwise direction away from its equilibrium position of fig.a. Therefore the domino in fig.a is in a position of . How about the domino is fig.c? If we unstable equilibrium give a slight push to the right, the line of action of is still to the left side of the supporting edge, the torque of causes the domino to rotate in the anticlockwise direction back to its g g F F r r equilibrium position of fig.a Therefore the domino in fig.c is in a position of To topple the domino the applied force has to rotate it through and beyond the position of fig.a. stable equilibrium (123) In chapter 9 we know Newton's second law for translational motion is . If an object is in translational equilibrium then constant In = = = → = → The Conditions of equilibrium ne t t ne dP F dt dP P d F t r r r r r chapter 11 we know that Newton's second law for rotational motion is: For an object in rotational equilibrium we have: constan t The two requirements for a b o = = → → = = n et et n dL dt dL L dt τ τ r r r r r dy to be in equilibrium are: 1. The vector sum of all the external forces on the body must be zero 2. The vector sum of all the external torques that act on the body measured about any point must be zero (124) 0 net F = r net τ = r ,...
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 Spring '10
 IA
 Equilibrium, Force, Fnet

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