Chapter 12: Equilibrium and Elasticity

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Chapter 12 Equilibrium and Elasticity

• Conditions Under Which a Rigid Object is in
  Equilibrium
• Problem-Solving Strategy
• Elasticity

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

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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 Fl
  Fr
• Draw a force diagram

• Write down the equilibrium requirements in
  components and solve these for the unknowns

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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/m2)
• Strain unit deformation

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Youngs Modulus Elasticity in Length
The Youngs 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 (m2) ?L is the extension,
measured in metres (m) L is the natural length,
measured in metres (m)
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Table 12-1 Some elastic properties of selected
material of engineering interest
Material Density ? (kg/m3) Youngs Modulus E (109N/m2) Ultimate Strength Su (106N/m2) Yield Strength Sy (106N/m2)
Steel 7860 200 400 250
Aluminum 2710 70 110 90
Glass 2190 65 50 ?
Concrete 2320 30 40 ?
Wood 525 13 50 ?
Bone 1900 9 170 ?
Polystyrene 1050 3 48 ?
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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 (m2) ?x is the horizontal
distance the sheared face moves, measured in
metres (m) L is the height of the object,
measured in metres (m)
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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 (m3) V is
the original volume, measured in metres (m3)
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Youngs modulus Shear modulus Bulk modulus
Under tension and compression Under shearing Under hydraulic stress

Strain is Strain is Strain is

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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

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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 Fw
• the force from the pavement Fpx Fpy
• Draw a force diagram

• Write down the equilibrium requirements in
  components and solve these for the unknowns

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Sample Problem 12-3
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• Write down the equilibrium requirements in
  components and solve these for the unknowns

Balance of torques
Balance of forces
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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 (F1 F2 F3 and
  F4).

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• Write down the equilibrium requirements in
  components and solve these for the unknowns

Balance of forces
If table remains level
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Chapter 12 Recitation
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Problem-Solving Tips

• Try to guess the correct direction for each
  force
• The choice of the origin for the torque equation
  is arbitrary. Choose an origin that will simplify
  your calculation as much as possible.
• You must have as many independent equations as
  you have unknowns in order to obtain a complete
  solution.

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Homework 12-22

• Define the system to be analyzed the rod plus
  the uniform square sign

• Identify the forces acting on the system
• the gravitational force (mg), the force from the
  cable (Tv and Th), and the force from the hinge
  (Fv and Fh)

• Draw a force diagram

• Write down the equilibrium requirements

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Tv
Fv
Fh
O
Th
2
3
mg
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Homework 12-34

• Define the system to be analyzed the beam plus
  the package

• Identify the forces acting on the system
• the gravitational force (mbg mpg),
• the force from the cable (T),
• and the force from the hinge (Fv and Fh)

• Draw a force diagram

• Write down the equilibrium requirements

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Homework 12-35

• Define the system to be analyzed two sides of
  the ladder are considered separately

• Identify the forces acting on the system
• Left side the gravitational force of the man
  (mg), the tension force of the tie rod (T), the
  force of the floor on the left feet (FA), and the
  force exerted by the right side of the ladder (Fv
  and Fh)
• Right side the tension force of the tie rod (T),
  the force of the floor on the right feet (FE),
  and the force exerted by the right side of the
  ladder (Fv and Fh)

O
O

• Draw a force diagram

?
?
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• Write down the equilibrium requirements in
  components and solve these for the unknowns

(a) First we solve for T by eliminating the other
unknowns. The first equations of two sides give
FAmg-Fv and FVFE . Substituting them into the
remaining three equations to obtain
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The last of these gives We substitute this
expression into the second equation and solve for
T. The result is
(b) We now solve for FA.
(c) We now solve for FE. We have already obtained
an expression for FE.
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Homework 12-40

1. , (b). Since the brick rests horizontally on
   cylinders now and the cylinders had identical
   length l before the brick was placed on them,
   then both cylinders have been compressed an equal
   amount ?l. Thus,

(c) Computing torques about the center of mass,
we found,
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Homework 12-43
Choose x-axis is parallel to the incline
(positive uphill) Balance of forces in
x-direction
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Basic Pulley Physics
Fixed Pulley
Movable Pulley
2P
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Homework 12-47
Cable 3
T8F
Cable 2
4F
2F
Cable 1
F
2F
4F
mg
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Homework 12-54
(a) With Fma-?kmg, the magnitude of the
deceleration is
(b), (c)
(d), (e)