Chapter 12 Equilibrium and Elasticity

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Chapter 12 (Static) Equilibrium and Elasticity

• Objects in static equilibrium dont move.
• Of special interest to civil and mechanical
  engineers and architects.
• Well also learn about elastic (reversible)
  deformations (rubber).
• Plastic deformations are irreversible (like
  play dough)

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12-2 Equilibrium

• Equilibrium

• Static equilibrium two requirements
  (the constants are equal
  to zero)

unstable
stable
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12-3 Requirements of static equilibrium

• The net force acting on the particle must be
  zero.
• The net torque about any axis acting on the
  particle must be zero.
• The angular and linear speeds must be zero.

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12-3 Requirements of static equilibrium
Is this object in static equilibrium?
A force couple is acting on an object. A force
couple is a pair of forces of equal magnitude and
opposite direction along parallel lines of action
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It matters at which point the force is applied!!
If equal and opposite forces are applied at
different points ? object is not in equilibrium,
since there is a net torque.
If equal and opposite forces are applied at the
same point or along the same axis ? object is in
equilibrium
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Requirements of static equilibrium (in x-y plane
2D)

• The net force acting on the particle must be
  zero.
• The net torque about any axis acting on the
  particle must be zero.
• The angular and linear speeds must be zero.

We restrict ourselves to forces in the x-y plane.
Thus
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The figure gives six overhead views of a uinform
rod on which two or more forces act perpendicular
to the rod. If the magnitudes of the forces are
adjusted properly (but kept nonzero), in which
situations can the rod be in static equilibrium?
Checkpoint 12-1
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12-4 The Center of Gravity (cog)

• Consider an extended object.
• The gravitational force Fg always acts on the
  center of gravity!
• The center of gravity (cog) is equal to the
  center of mass (com).

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Balanced rock
For this system to be in static equilibrium, the
center of gravity must be directly over the
support point. Why??
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• Problem-solving hints
• Objects in static equilibrium
• Draw a sketch of the problem
• Select the object/system to which you will apply
  the laws of equilibrium.
• Show and label all the external forces acting on
  the system/object.
• Indicate where the forces are applied.
• Establish a convenient coordinate system for
  forces. Then apply condition 1 Net force must
  equals zero.
• Establish a convenient coordinate system for
  torque. Then apply condition 2 Net torque must
  equals zero.

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A uniform beam, of length L and mass m 1.8 kg,
is at rest with its ends on two scales. A uniform
block, with mass M 2.7 kg, is at rest on the
beam, with its center a distance L/4 from the
beam's left end. What do the scales read?
12-5 Some Examples of Static Equilibrium
Sample Problem 12-1
Choose the rotation axis at the left end of the
beam
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A ladder of length L 12 m and mass m 45 kg
leans against a slick (frictionless) wall. Its
upper end is at height h 9.3 m above the
pavement on which the lower end rests (the
pavement is not frictionless). The ladder's
center of mass is L/3 from the lower end. A
firefighter of mass M 72 kg climbs the ladder
until her center of mass is L/2 from the lower
end. What then are the magnitudes of the forces
on the ladder from the wall and the pavement?
Sample Problem 12-2
Choose the rotation axis at O
Note that Fpx is the static friction from the
pavement. It may not be equal to the maximum
value of µs Fpy.
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A safe, of mass M 430 kg, is hanging by a rope
from a boom with dimensions a 1.9 m and b 2.5
m. The boom consists of a hinged beam and a
horizontal cable that connects the beam to a
wall. The uniform beam has a mass m of 85 kg the
mass of the cable and rope are negligible.
Sample Problem 12-3
Note that Tc is not equal to Tr . Take the
rotation axis at O.
Since Tr Mg, we have
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Sample Problem 12-3
(b)  Find the magnitude F of the net force on the
beam from the hinge.
Sample Problem 13-3
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12-7 Elastic properties of solids
Definitions of Stress and Strain. Stress Force
per unit cross sectional area. Strain Measure of
the degree of deformation. These two quantities
are related by the following equation that
defines the modulus of elasticity
stress modulus x strain
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12-7 Elastic properties of solids

• We will consider three types of deformations and
  define an elastic modulus for each.
• Change in length. YOUNGS MODULUS, E measures the
  resistance of a solid to a change in its length.
• Shearing. SHEAR MODULUS, G measures the
  resistance to shearing.
• Change in volume. BULK MODULUS, B measures the
  resistance to changes in volume.

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Youngs modulus, E
Tension or compression
Note the force F is perpendicular to area A
Youngs modulus, E
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Stress-strain curve
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Shear modulus, G
Shear modulus, G
Note that the force lies in the plane of the area
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Bulk modulus, B
Hydraulic compression or stress
Bulk modulus, B
F/AP is the fluid pressure!
Note that the force acts all around the body
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A steel rod has a radius R of 9.5 mm and a length
L of 81 cm. A 62 kN force stretches it along its
length. What are the stress on the rod and the
elongation and strain of the rod?
Sample Problem 12-5
(Use E from table)