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 = 2x 3 – 2x 2 – 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

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