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equilibrium diagrams

  1. 1. Equilibrium Chapter 4
  2. 2. Chapter Topics • Discuss equilibrium, including: – The concept of static equilibrium (SE) – Force and moment conditions for SE – Center of mass – Segmental method for human center of mass – Hydrostatics and flotation
  3. 3. Bodies at Rest • A body at rest or in uniform motion is in equilibrium – External forces must give zero resultant.
  4. 4. Behold, a new definition for Force! • A vector quantity that tends to cause movement in material objects
  5. 5. Types of Equilibrium • Stable – A slight disturbance generates a restoring force to return the equilibrium • Unstable – A slight disturbance leads to an increasing departure from equilibrium • Neutral – A disturbance simply moves the object to a new position
  6. 6. Stable/Unstable/Neutral?
  7. 7. Friction • Many common objects are in equilibrium because of the effects of friction forces • Example?
  8. 8. Friction and Equilibrium • Objects at rest on a surface are held by static friction. • Surfaces moving with respect to each other are met with sliding friction or kinetic friction. • Which is greater?
  9. 9. Friction and Equilibrium cont’d • Examples of friction: – Slowing of a golf ball across a green – Stationary object on an inclined plane – Walking and not falling over
  10. 10. Friction and Equilibrium cont’d • Examples of low friction: – Ice skating – Skiing
  11. 11. Coefficient of Friction • Depends on the surfaces of both of the bodies in contact with one another • Simple experiment: at what angle does an object start to slide down a slope – Mechanical principle:
  12. 12. Question • Is friction essential for a body to be in equilibrium?
  13. 13. Moments • Moment of a force: a measure of the turning effect of a force
  14. 14. Moments of Force • Moment of force about a point is the tendency of the force to turn the body to which it is applied about that point – Also known as torque – F d – d = length of moment arm • Perpendicular distance between point and line of action of the force – Given in Nm
  15. 15. Example (page 81) • A man supports a weight of 250 N with his arms at right angles to his body. He holds the weight with both hands level with his shoulders. If is arms are 75 cm long, what is the moment of the force?
  16. 16. Example cont’d • Moment = F x d = 250 N x 0.75 m = 187.5 Nm
  17. 17. BE CAREFUL! • The book’s answer is incomplete. A moment is a vector, so the answer should have a direction!
  18. 18. Practice The following problems would be useful to review as you prepare for the first exam. Page 49: Problems 1, 2, 4, 6 Page 50: Problems 7, 8, 11b, 13 Page 73: Problem 1 Page 74: Problem 5 Page 75: Problems 11, 13, 17a, 17b Page 104: Problems 1, 2 Page 106: Problem 8 Answers are in the back of the book
  19. 19. Point of Application of Resultant Force
  20. 20. Where does the resultant force act? • Solution comes from sum of moments for equilibrium
  21. 21. Where does the resultant force act? • What one force at what one location would produce equilibrium by balancing all the distributed forces?
  22. 22. Sum of moments • Choose a point, and sum the clockwise and ccw moments about that point. For example, sum moments about point A, taking the opposite sense of P+Q to assume equilibrium (P+Q) x AC = Q x AB
  23. 23. Example and Case Study, pg. 82- 83
  24. 24. Center of Gravity (CoG) • A point within or near the object through which the resultant weight of the object passes
  25. 25. Center of Mass (CoM) • A point at which the object's mass can be assumed, for many purposes, to be concentrated
  26. 26. Whole Body CoM • How would you calculate it? • Giovani Alfonso Borelli (1608-1679) • Case study 4.3
  27. 27. Consistency of CoM as a % of height • Page 85: • Height range: 158-188 cm • CoM range 54.4-55.89 %ht
  28. 28. Force Couples • What’s the sum of the horizontal forces? • Do they therefore have no mechanical effect?
  29. 29. Couples • A pair of equal parallel forces acting in opposite senses, and not in the same line, is called a couple • Object will tend to rotate and is therefore not in equilibrium
  30. 30. Falling Over
  31. 31. CoM of a Stationary Body • Objects will balance on pivots if CoG is directly over the pivot. • Works mainly for straight objects. • Example: a cricket bat
  32. 32. CoM of a Stationary Body cont’d • Position of CoG for a bat: %100.1 L L
  33. 33. Similar expression for body segments • total arm COM = 0.53 from proximal – GH to ulnar styloid • total leg COM = ____ from proximal – Greater trochanter to medial malleolus
  34. 34. Equilibrium Under Three Forces • Parallel forces: – The sum of the forces must be zero – Equilibrium requires that BOTH ΣF and ΣM each equal zero
  35. 35. Equilibrium Under Three Forces cont’d • A uniform beam AB, 6 m long, weighing 400 N, is supported at the end A at point C. Point C is 2 m from B. Find the reaction forces at supports A and C.
  36. 36. Equilibrium Under Three Forces cont’d • R1 + R2 = 400 N • ΣM about A Moment due to 400N force at G: 400 x AG = 400 x 3 = 1200 Nm CW Moment due to R2 at C: R2 x AC = R2 x 4m CCW
  37. 37. Equilibrium Under Three Forces cont’d • Equate the moments (for equilibrium) 1200 = 4R2 R2 = 1200/4 = 300 N R1 = 400 - R2 = 400 - 300 = 100 N
  38. 38. Now, solve the problem by summing moments to determine R1
  39. 39. Equilibrium Under Three Forces cont’d • Nonparallel forces: – Consider three forces D, E, and F in equilibrium.
  40. 40. Equilibrium Under Three Forces cont’d • Triangle forces rule: – Three nonparallel forces in equilibrium – Represented in size and direction by three sides of the triangle taken in order
  41. 41. Alternative • Measure the horizontal and vertical components of each vector
  42. 42. Hydrostatics and Flotation
  43. 43. Pressure as a Function of Depth • Increasing depth gives increasing pressure. • Liquid density is: • Pressure at depth h in N/m2 given by: ρ kg/m3 P=ρgh
  44. 44. Upthrust on Immersed Body • Pressure increases with depth. • The underside of the object experiences greater force than the top side.
  45. 45. Upthrust on Immersed Body cont’d • Principle of Archimedes: – When a solid body is wholly or partially immersed in fluid, it experiences an upthrust equal to the weight of the mass of displaced fluid.
  46. 46. Upthrust on Immersed Body cont’d • For floating body in water: – Upthrust force = Weight of body – U = W • For sinking body in water: – Upthrust force < Weight of body – U < W
  47. 47. Factors: Density and Shape • Less dense than water: always float • More dense than water: only float if shaped such that the object can displace at least its own weight in water
  48. 48. Specific Gravity • SG = Mass of certain volume of substance Mass of equal volume of water • SG = Weight of certain volume of substance Weight of equal volume of water
  49. 49. Specific Gravity cont’d • Water at temperature 4°C Density is 999.97 kg/m3 • Human body: close, but not homogenous!
  50. 50. Center of Mass in Humans
  51. 51. Estimating Center of Mass • Use the segmental method if you know: – Position of the end points of all of the body’s segments – Mass of each segment – Location of CoM within each segment
  52. 52. Estimating Center of Mass cont’d • 14 body segments: – Trunk – Head and neck – Right and left thighs – Right and left lower legs
  53. 53. Estimating Center of Mass cont’d • 14 body segments cont’d – Right and left feet – Right and left upper arms – Right and left lower arms – Right and left hands • Each segment has its own CoM
  54. 54. • Calcuation is based on the moment attributable to the weight of each segment about the x and y axes.
  55. 55. Free-body Diagrams • Include forces acting on the body only • Exclude forces that the body exerts on its surroundings and internal forces
  56. 56. • What is the system? • How can you determine R1 & R2 ?
  57. 57. • Draw the FBD of: – The bucket – The left arm – The head
  58. 58. Calculation of Joint Moments • Just like with summing forces with a free- body diagram, calculate and sum moments • Consider Figure 4-20.
  59. 59. FBD: foot segment ΣM about a
  60. 60. Summary • Certain physical conditions are necessary to maintain static equilibrium – Force balance: ΣF=0 – Moment balance: ΣM=0 • Additional principles associated with equilibrium – Center of mass – Hydrostatics

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