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Understanding Static Equilibrium: Concepts and Examples

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This chapter explores the principles of static equilibrium, detailing key concepts such as stable and unstable equilibrium. It presents the conditions required for equilibrium, including the balance of forces and torques. We analyze examples, such as a crane supporting loads, and the intricacies of indeterminate structures. Through mathematical models, the chapter illustrates how to determine stable and unstable equilibrium points using potential energy functions. Additionally, it provides insights into the concept of elasticity and its relevance to real-life applications.

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Understanding Static Equilibrium: Concepts and Examples

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  1. Chapter 12 Static Equilibrium

  2. Equilibrium • We already introduced the concept of equilibrium in Chapter 7: dU(x)/dx = 0 • More general definition of equilibrium: • Static equilibrium: • Stable equilibrium: the body returns to the state of static equilibrium after having been displaced from that state. Unstable equilibrium: the state of equilibrium is lost after a small force displaces the body

  3. Equilibrium • Static equilibrium: • Stable equilibrium: the body returns to the state of static equilibrium after having been displaced from that state. Unstable equilibrium: the state of equilibrium is lost after a small force displaces the body

  4. Equilibrium • Stable equilibrium: • Unstable equilibrium:

  5. Chapter 12 Problem 25 A particle’s potential energy as a function of position is given by U= 2x3– 2x2– 7x+ 10,with x in meters and U in joules. Find the positions of any stable and unstable equilibria.

  6. Center of mass: stable equilibrium • We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

  7. Center of mass: stable equilibrium • We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

  8. Center of mass: stable equilibrium • We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

  9. Center of mass: stable equilibrium • We consider the torque created by the gravity force (applied to the CM) and its direction relative to the possible point(s) of rotation

  10. The requirements of equilibrium • For an object to be in equilibrium, we should have two requirements met • Balance of forces: the vector sum of all the external forces that act on the body is zero • Balance of torques: the vector sum of all the external torques that act on the body, measured about any possible point, is zero

  11. Equilibrium: 2D case • If an object can move only in 2D (xy plane) then the equilibrium requirements are simplified: • Balance of forces: only the x- and y-components are considered • Balance of torques: only the z-component is considered (the only one perpendicular to the xy plane)

  12. Examples of static equilibrium

  13. Examples of static equilibrium

  14. Chapter 12 Problem 38 A crane in a marble quarry is mounted on the quarry’s rock walls and is supporting a 2500-kg marble slab as shown in the figure. The center of mass of the 830-kg boom is located one-third of the way from the pivot end of its 15-m length, as shown. Find the tension in the horizontal cable that supports the boom.

  15. Indeterminate structures • Indeterminate systems cannot be solved by a simple application of the equilibrium conditions • In reality, physical objects are • not absolutely rigid bodies • Concept of elasticity is employed

  16. Questions?

  17. Answers to the even-numbered problems Chapter 12 Problem 24: (a) 47 m from the origin (b) unstable

  18. Answers to the even-numbered problems Chapter 12 Problem 36: 1.2 m

  19. Answers to the even-numbered problems Chapter 12 Problem 54: L/6 of the bottom book can overhang the desk

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