Equilibrium requires that the sum of all forces and sum of all moments acting on an object are equal to zero. Static equilibrium is maintained when the external forces give a zero resultant force and zero total moment. Additional concepts important to equilibrium include the center of mass, hydrostatics, friction, and moments of force.

1. Equilibrium
Chapter 4

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. Bodies at Rest
• A body at rest or in uniform motion is in
equilibrium
– External forces must give zero resultant.

4. Behold, a new definition for Force!
• A vector quantity that tends to cause movement in
material objects

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. Stable/Unstable/Neutral?

7. Friction
• Many common objects are in equilibrium because of the
effects of friction forces
• Example?

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. 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. Friction and Equilibrium cont’d
• Examples of low friction:
– Ice skating
– Skiing

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. Question
• Is friction essential for a body to be in equilibrium?

13. Moments
• Moment of a force: a measure of the turning
effect of a force

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

16. 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?

17. Example cont’d
• Moment = F x d
= 250 N x 0.75 m
= 187.5 Nm

18. BE CAREFUL!
• The book’s answer is incomplete. A moment is a vector,
so the answer should have a direction!

19. 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

20. Point of Application of Resultant Force

21. Where does the resultant force act?
• Solution comes from sum of moments for equilibrium

22. Where does the resultant force act?
• What one force at what one location would produce
equilibrium by balancing all the distributed forces?

23. 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

24. Example and Case Study, pg. 82-
83

25. Center of Gravity (CoG)
• A point within or near the object through
which the resultant weight of the object
passes

26. Center of Mass (CoM)
• A point at which the object's mass can be
assumed, for many purposes, to be
concentrated

27. Whole Body CoM
• How would you calculate it?
• Giovani Alfonso Borelli (1608-1679)
• Case study 4.3

29. Consistency of CoM as a % of height
• Page 85:
• Height range: 158-188 cm
• CoM range 54.4-55.89 %ht

30. Force Couples
• What’s the sum of the horizontal forces?
• Do they therefore have no mechanical effect?

31. 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

32. Falling Over

33. 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

35. CoM of a Stationary Body cont’d
• Position of CoG for a bat:
%100.1
L
L

36. 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

37. Equilibrium Under Three Forces
• Parallel forces:
– The sum of the forces must be zero
– Equilibrium requires that BOTH ΣF and ΣM each
equal zero

38. 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.

40. 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

41. 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

42. Now, solve the problem by summing
moments to determine R1

43. Equilibrium Under Three Forces
cont’d
• Nonparallel forces:
– Consider three forces D, E, and F in equilibrium.

45. 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

46. Alternative
• Measure the horizontal and vertical components of
each vector

47. Hydrostatics and Flotation

48. 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

49. Upthrust on Immersed Body
• Pressure increases with depth.
• The underside of the object experiences
greater force than the top side.

51. 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.

52. 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

53. 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

54. 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

55. Specific Gravity cont’d
• Water at temperature 4°C
Density is 999.97 kg/m3
• Human body: close, but not homogenous!

56. Center of Mass in Humans

57. 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

58. Estimating Center of Mass cont’d
• 14 body segments:
– Trunk
– Head and neck
– Right and left thighs
– Right and left lower legs

59. 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

60. • Calcuation is based on the moment attributable to the
weight of each segment about the x and y axes.

65. Free-body Diagrams
• Include forces acting on the body only
• Exclude forces that the body exerts on its
surroundings and internal forces

66. • What is the system?
• How can you determine
R1 & R2 ?

67. • Draw the FBD of:
– The bucket
– The left arm
– The head

68. Calculation of Joint Moments
• Just like with summing forces with a free-
body diagram, calculate and sum moments
• Consider Figure 4-20.

69. FBD: foot segment ΣM about a

70. 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