1. Chapter 12: Equilibrium and Elasticity  Conditions Under Which a Rigid Object is in Equilibrium  Problem-Solving Strategy  Elasticity

2. Equilibrium: An object at equilibrium is either... at rest and staying at rest (i.e., static equilibrium), or in motion and continuing in motion with the constant velocity and constant angular momentum. For the object in equilibrium, the linear momentum ( ) of its center of mass is constant. the angular momentum ( ) about its center of mass, or any other point, is constant.

3. Conditions of Equilibrium: Net force: Net torque: Conditions of equilibrium: Another requirements for static equilibrium:

4. The center or gravity: The gravitational force on a body effectively acts at a single point, called the center of gravity (cog) of the body. the center of mass of an object depends on its shape and its density the center of gravity of an object depends on its shape, density, and the external gravitational field. Does the center of gravity of the body always coincide with the center of mass (com)? Yes, if the body is in a uniform gravitational field.

5. How is the center of gravity of an object determined? The center of gravity (cog) of a regularly shaped body of uniform composition lies at its geometric center. The (cog) of the body can be located by suspending it from several different points. The cog is always on the line-of- action of the force supporting the object. cog

6. Problem-Solving Strategy: Define the system to be analyzed Identify the forces acting on the system Draw a free-body diagram of the system and show all the forces acting on the system, labeling them and making sure that their points of application and lines of action are correctly shown. Write down two equilibrium requirements in components and solve these for the unknowns

7. Sample Problem 12-1: Define the system to be analyzed: beam & block Identify the forces acting on the system: the gravitational forces: mg & Mg, the forces from the left and the right scales: F l & F r Draw a force diagram Write down the equilibrium requirements in components and solve these for the unknowns O

8. Elasticity Some concepts : Rigid Body: Deformable Body: elastic body: rubber, steel, rock… plastic body: lead, moist clay, putty… Stress: Deforming force per unit area (N/m 2 ) Strain: unit deformation

9. Young’s Modulus: Elasticity in Length The Young’s modulus, E, can be calculated by dividing the stress by the strain, i.e. where (in SI units) E is measured in newtons per square metre (N/m²). F is the force, measured in newtons (N) A is the cross-sectional area through which the force is applied, measured in square metres (m 2 )  L is the extension, measured in metres (m) L is the natural length, measured in metres (m)

10. Material Density  (kg/m 3 ) Young’s Modulus E (10 9 N/m 2 ) Ultimate Strength S u (10 6 N/m 2 ) Yield Strength S y (10 6 N/m 2 ) Steel7860200400250 Aluminum27107011090 Glass21906550  Concrete23203040  Wood5251350  Bone19009170  Polystyrene1050348  Table 12-1: Some elastic properties of selected material of engineering interest

11. Shear Modulus: Elasticity in Shape The shear modulus, G, can be calculated by dividing the shear stress by the strain, i.e. where (in SI units) G is measured in newtons per square metre (N/m²) F is the force, measured in newtons (N) A is the cross-sectional area through which the force is applied, measured in square metres (m 2 )  x is the horizontal distance the sheared face moves, measured in metres (m) L is the height of the object, measured in metres (m)

12. Bulk Modulus: Elasticity in Volume The bulk modulus, B, can be calculated by dividing the hydraulic stress by the strain, i.e. where (in SI units) B is measured in newtons per square metre (N/m²) P is measured in in newtons per square metre (N/m²)  V is the change in volume, measured in metres (m 3 ) V is the original volume, measured in metres (m 3 )

13. Young’s modulusShear modulusBulk modulus Under tension and compression Under shearing Under hydraulic stress Strain is

14. Summary: Requirements for Equilibrium: The cog of an object coincides with the com if the object is in a uniform gravitational field. Solutions of Problems: Elastic Moduli: tension and compression shearing hydraulic stress Define the system to be analyzed Identify the forces acting on the system Draw a force diagram Write down the equilibrium requirements in components and solve these for the unknowns

15. Sample Problem 12-2: Define the object to be analyzed: firefighter & ladder Identify the forces acting on the system: the gravitational forces: mg & Mg, the force from the wall: F w the force from the pavement: F px & F py Draw a force diagram Write down the equilibrium requirements in components and solve these for the unknowns

16. Sample Problem 12-3: Define the object to be analyzed: Beam Identify the forces acting on the system: the gravitational force (mg), the force from the rope (T r ) the force from the cable (T c ), and the force from the hinge (F v and F h ) Draw a force diagram

17. Write down the equilibrium requirements in components and solve these for the unknowns Balance of forces: Balance of torques:

18. Sample Problem 12-6: Define the system to be analyzed: table plus steel cylinder. Identify the forces acting on the object: the gravitational force (Mg), the forces on legs from the floor (F 1 = F 2 = F 3 and F 4 ). Draw a force diagram

19. Write down the equilibrium requirements in components and solve these for the unknowns Balance of forces: If table remains level: