1. Advanced Physics Chapter 9—Bodies in Equilibrium

2. 9-1 Statics—The Study of Forces in Equilibrium 9-2 The Conditions for Equilibrium 9-3 Solving Static Problems 9-4 Applications to Muscles and Joints 9-5 Stability and Balance 9-6 Elasticity; Stress and Strain 9-7 Fracture 9-8 Spanning a Space

3. Statics—The Study of Forces in Equilibrium Statics—study of an object at rest Deals with the forces within objects at rest Equilibrium—when the net force on an object is equal to zero This means?

4. The Conditions for Equilibrium First Condition for Equilibrium The sum of all forces acting on an object (x, y, z axis) is equal to zero

5. The Conditions for Equilibrium Second Condition for Equilibrium The sum of all torques acting on an object (x, y, z axis) is equal to zero

6. Solving Static Problems Need to know how to solve five types of Statics problems: Sign Lever Seesaw Beam (with supports) Beam and wire What five steps do you employ?

7. Solving Static Problems Need to know how to solve five types of Statics problems: Sign Lever Seesaw Beam (with supports) Beam and wire

8. 9-4 Applications to Muscles and Joints We can apply the study of mechanics and statics to human bodies and motion Parts of the body of interest: bones, tendons, and muscles

9. 9-4 Applications to Muscles and Joints Insertions—point where muscle is attached to bone by use of tendons Joint—place where two bones are flexibly attached

10. 9-4 Applications to Muscles and Joints Flexor muscles— muscles that are used to bring bones closer together (biceps) Extensor muscles— muscles that are used to extend a limb (triceps)

11. 9-4 Applications to Muscles and Joints Since muscles exert a torque equal in magnitude to the torque supplied by a lifted object and the distance from joint to muscle is usually less than joint to object, muscles usually must apply a much larger force than what is being lifted. The lowest vertebra on the spinal column (fifth lumbar vertebra) acts as the fulcrum for bending, due to its angle in the spine the force on it is equal to about 2 ½ times your body weight.

12. 9-5 Stability and Balance A body in static equilibrium, if left undisturbed, will undergo no translational or rotational acceleration WHY?

13. 9-5 Stability and Balance If a body in static equilibrium is disturbed three possible outcomes are possible: It will return to its original position Object is said to be in stable equilibrium

14. 9-5 Stability and Balance If a body in static equilibrium is disturbed three possible outcomes are possible: It will move further from its original position Object is said to be in unstable equilibrium

15. 9-5 Stability and Balance If a body in static equilibrium is disturbed three possible outcomes are possible: It will remain in its new position Object is said to be in neutral equilibrium

16. 9-5 Stability and Balance To maintain balance of an object a body’s center of gravity must be above its base. The object will be stable if a vertical line projected down from the object’s CG falls within the base of support. If not it will fall over, WHY?

17. 9-5 Stability and Balance When a human carries a heavy load (i.e. backpack full of physics books) he/she shifts his/her body to maintain balance, WHY?

18. 9-6 Elasticity; Stress and Strain Elasticity—how an object will stretch without changing its original shape once the force is removed Fracture—when an object breaks due to an applied force

19. 9-6 Elasticity; Stress and Strain Elasticity Hooke’s Law F = k  L describes the change in length of an object that has a force applied to it and it stretches elastically

20. 9-6 Elasticity; Stress and Strain Elasticity Proportionality limit—point up to which an object will behave elastically and linearly when stretched. Elastic limit—the object will return to its original length if force is removed.

21. 9-6 Elasticity; Stress and Strain Plasticity—a object is permanently deformed after it is stretched Elastic region vs. plastic region for forces applied and elongation of an object

22. 9-6 Elasticity; Stress and Strain Breaking point—the minimum force applied to break an object Ultimate strength— maximum force that can be applied to an object without breaking it

23. 9-6 Elasticity; Stress and Strain Stretching an Object There is a relationship between the force applied to an object and how much its length will change from its original length and characteristics of the object  L = (1/E)(F/A)L o Elastic (Young’s) modulus (E)—constant of proportionality (N/m 2 ) that depends on material only

24. 9-6 Elasticity; Stress and Strain Stretching an Object Stress—force applied to an surface per area Stress = Force/area = F/A Strain—ratio of change in length to original length Strain =  L/L o Since  L = (1/E)(F/A)L o  E = (F/A)/(  L/L o )  E = stress/strain

25. 9-6 Elasticity; Stress and Strain An object can be subjected to three common types of stress: Tensile Stress (tension)—pulled apart Compressional Stress—pushed together Shear Stress—an equal but opposite force is applied across opposite faces of an object

26. 9-6 Elasticity; Stress and Strain Tensile Stress (tension)—pulled apart See preceding slides (Young’s modulus)

27. 9-6 Elasticity; Stress and Strain Compressional Stress—pushed together Causes a change in the length but in the opposite direction as tensile stress (E is same) Causes a change in the volume of an object  V/V = -(1/B)  P Pressure (P) is force per area Bulk modulus (B)—constant of proportionality for Compressional stress (N/m 2 )

28. 9-6 Elasticity; Stress and Strain Shear Stress—an equal but opposite force is applied across opposite faces of an object Causes the top of an object to move relative to the bottom of an object  L = (1/G)(F/A)L o Area is parallel to force applied  L is perpendicular to L Shear modulus (G)—constant of proportionality for a shifting object (N/m 2 ) G is usually about 1/3 E

29. 9-6 Elasticity; Stress and Strain Solids can have E, B and G Liquids have only B WHY?

30. 9-7 Fracture When the stress on a solid object is too great the object breaks or fractures

31. 9-7 Fracture There are three types of Ultimate Strengths depending on type of force applied Tensile strength Compressive strength Shear strength Chart p.258 (N/m 2 )

32. 9-7 Fracture Because the chart values for ultimate strengths can differ substantially from the actual values it is necessary to maintain a safety margin Safety factor of three means that the actual stress on the object should not exceed 1/3 the chart value

33. 9-7 Fracture There are ways of increasing the ultimate strength of a material Examples: Reinforced concrete Prestressed concrete

34. 9-8 Spanning a Space There are many architectural methods available to span a space

35. 9-8 Spanning a Space Post-and-Beam Two upright post support a horizontal beam Very limited due to all three types of force are exerted on the beam and material used not strong under all three Greek Temples

36. 9-8 Spanning a Space Arch Semicircular span Many types (p.261) Made up of wedge shaped material that experiences mostly Compressional forces (and material used has good Compressional strength)

37. 9-8 Spanning a Space Arch Could be used to span a large distance Arch transfers force both vertically and horizontally so must be buttressed on the sides to support horizontal force

38. 9-8 Spanning a Space Arch The more pointed the arch the less horizontal component of force is exerted If the arch is parabolic the stress within the arch is purely compressive

39. 9-8 Spanning a Space Dome An arch that spans a three-dimensional space Has mostly compressive forces A pointed dome like a pointed arch exerts a smaller force horizontally against its base.