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