Static Equilibrium; Elasticity and Fracture

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Chapter 9 Static Equilibrium Elasticity and
Fracture
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Equilibrium

• First Condition of Equilibrium
• The net external force must be zero
• Second Condition of Equilibrium
• The net external torque must be zero

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Torque and Equilibrium, cont

• To ensure mechanical equilibrium, you need to
  ensure rotational equilibrium as well as
  translational
• The Second Condition of Equilibrium states
• The net external torque must be zero

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9-1 The Conditions for Equilibrium
The first condition for equilibrium is that the
forces along each coordinate axis add to zero.
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9-2 Solving Statics Problems
The previous technique may not fully solve all
statics problems, but it is a good starting point.
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9.2 Rigid Objects in Equilibrium
Example 3 A Diving Board A woman whose weight
is 530 N is poised at the right end of a diving
board with length 3.90 m. The board
has negligible weight and is supported by a
fulcrum 1.40 m away from the left end. Find the
forces that the bolt and the fulcrum exert on
the board.
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9.2 Rigid Objects in Equilibrium
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9.2 Rigid Objects in Equilibrium
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9-2 Solving Statics Problems
If there is a cable or cord in the problem, it
can support forces only along its length. Forces
perpendicular to that would cause it to bend.
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9-1 The Conditions for Equilibrium
The second condition of equilibrium is that there
be no torque around any axis the choice of axis
is arbitrary.
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9-2 Solving Statics Problems

1. Choose one object at a time, and make a
   free-body diagram showing all the forces on it
   and where they act.
2. Choose a coordinate system and resolve forces
   into components.
3. Write equilibrium equations for the forces.
4. Choose any axis perpendicular to the plane of
   the forces and write the torque equilibrium
   equation. A clever choice here can simplify the
   problem enormously.
5. Solve.

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9-2 Solving Statics Problems
If a force in your solution comes out negative
(as FA will here), it just means that its in the
opposite direction from the one you chose. This
is trivial to fix, so dont worry about getting
all the signs of the forces right before you
start solving.
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9-3 Applications to Muscles and Joints
These same principles can be used to understand
forces within the body.
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9-3 Applications to Muscles and Joints
The angle at which this mans back is bent places
an enormous force on the disks at the base of his
spine, as the lever arm for FM is so small.
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9-4 Stability and Balance
If the forces on an object are such that they
tend to return it to its equilibrium position, it
is said to be in stable equilibrium.
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9-4 Stability and Balance
If, however, the forces tend to move it away from
its equilibrium point, it is said to be in
unstable equilibrium.
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9-4 Stability and Balance
An object in stable equilibrium may become
unstable if it is tipped so that its center of
gravity is outside the pivot point. Of course, it
will be stable again once it lands!
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9-4 Stability and Balance
People carrying heavy loads automatically adjust
their posture so their center of mass is over
their feet. This can lead to injury if the
contortion is too great.
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9-5 Elasticity Stress and Strain
Hookes law the change in length is proportional
to the applied force.
(9-3)
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9-5 Elasticity Stress and Strain
This proportionality holds until the force
reaches the proportional limit. Beyond that, the
object will still return to its original shape up
to the elastic limit. Beyond the elastic limit,
the material is permanently deformed, and it
breaks
at the breaking point.
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9-5 Elasticity Stress and Strain
The change in length of a stretched object
depends not only on the applied force, but also
on its length and cross-sectional area, and the
material from which it is made. The material
factor is called Youngs modulus, and it has been
measured for many materials.
The Youngs modulus is then the stress divided by
the strain.
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9-5 Elasticity Stress and Strain
In tensile stress, forces tend to stretch the
object.
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9-5 Elasticity Stress and Strain
Compressional stress is exactly the opposite of
tensional stress. These columns are under
compression.
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9-5 Elasticity Stress and Strain
Shear stress tends to deform an object
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9-6 Fracture
If the stress is too great, the object will
fracture. The ultimate strengths of materials
under tensile stress, compressional stress, and
shear stress have been measured.
When designing a structure, it is a good idea to
keep anticipated stresses less than 1/3 to 1/10
of the ultimate strength.
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9-6 Fracture
A horizontal beam will be under both tensile and
compressive stress due to its own weight.
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9-6 Fracture
What went wrong here? These are the remains of an
elevated walkway in a Kansas City hotel that
collapsed on a crowded evening, killing more than
100 people.
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9-6 Fracture
Here is the original design of the walkway. The
central supports were to be 14 meters long.
During installation, it was decided that the long
supports were too difficult to install the
walkways were installed this way instead.
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9-1 The Conditions for Equilibrium
An object with forces acting on it, but that is
not moving, is said to be in equilibrium.
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9-6 Fracture
The change does not appear major until you look
at the forces on the bolts
The net force on the pin in the original design
is mg, upwards.
When modified, the net force on both pins
together is still mg, but the top pin has a force
of 2mg on it enough to cause it to fail.
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9-7 Spanning a Space Arches and Domes
The Romans developed the semicircular arch about
2000 years ago. This allowed wider spans to be
built than could be done with stone or brick
slabs.
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9-7 Spanning a Space Arches and Domes
The stones or bricks in a round arch are mainly
under compression, which tends to strengthen the
structure.
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9-7 Spanning a Space Arches and Domes
Unfortunately, the horizontal forces required for
a semicircular arch can become quite large this
is why many Gothic cathedrals have flying
buttresses to keep them from collapsing.
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9-7 Spanning a Space Arches and Domes
Pointed arches can be built that require
considerably less horizontal force.
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9-7 Spanning a Space Arches and Domes
A dome is similar to an arch, but spans a
two-dimensional space.
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Summary of Chapter 9

• An object at rest is in equilibrium the study
  of such objects is called statics.
• In order for an object to be in equilibrium,
  there must be no net force on it along any
  coordinate, and there must be no net torque
  around any axis.
• An object in static equilibrium can be either in
  stable, unstable, or neutral equilibrium.

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Summary of Chapter 9

• Materials can be under compression, tension, or
  shear stress.
• If the force is too great, the material will
  exceed its elastic limit if the force continues
  to increase the material will fracture.