Download 1 Chapter 12 Static Equilibrium Equilibrium Summary Static vs

Static Equilibrium
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Chapter 12
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Static Equilibrium
and
Elasticity
Equilibrium Summary
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There are two necessary conditions for equilibrium
The resultant external force must equal zero:
ΣF = 0
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This is a statement of translational equilibrium
The acceleration of the center of mass of the object must be
zero when viewed from an inertial frame of reference
The resultant external torque about any axis must
be zero:
Στ = 0
• This is a statement of rotational equilibrium
• The angular acceleration must equal zero
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Equilibrium implies the object is at rest (static)
or its center of mass moves with a constant
velocity (dynamic)
Static equilibrium is a common situation in
engineering
Principles involved are of particular interest to
civil engineers, architects, and mechanical
engineers
Static vs. Dynamic Equilibrium
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In this chapter, we will concentrate on static
equilibrium
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The object will not be moving
Dynamic equilibrium is also possible
The object would be rotating with a constant
angular velocity
In either case, the Στ = 0
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Equilibrium Equations
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We will restrict the applications to situations in
which all the forces lie in the xy plane
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Ch 12: Question 3
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These are called coplanar forces since they lie in
the same plane
There are three resulting equations
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ΣFx = 0
ΣFy = 0
Στz = 0
Ch 12: Question 4
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Can an object be in equilibrium when only
one force acts upon it?
(a) Give an example in which the net force
acting on an object is zero and yet the net
torque is nonzero.
(b) Give an example in which the net torque
acting on an object is zero and yet the net
force is nonzero.
Ch 12: Question 6
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If you measure the net force and the net
torque on a system to be zero,
(a) could the system still be rotating with
respect to you?
(b) Could it be translating with respect to you?
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Axis of Rotation for Torque
Equation
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The net torque is about an axis through any
point in the xy plane
The choice of an axis is arbitrary
If an object is in translational equilibrium and
the net torque is zero about one axis, then the
net torque must be zero about any other axis
Center of Gravity, cont
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The torque due to the gravitational force on
an object of mass M is the force Mg acting at
the center of gravity of the object
If g is uniform over the object, then the center
of gravity of the object coincides with its
center of mass
If the object is homogeneous and
symmetrical, the center of gravity coincides
with its geometric center
Center of Gravity
All the various
gravitational forces acting
on all the various mass
elements are equivalent
to a single gravitational
force acting through a
single point called the
center of gravity (CG)
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Weighted Hand Example
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Model the forearm as a
rigid bar
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The weight of the
forearm is ignored
There are no forces in
the x-direction
Apply the first condition
for equilibrium (ΣFy = 0)
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Weighted Hand Example, cont
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Apply the second
condition for equilibrium
using the joint O as the
axis of rotation (Στ =0)
Generate the
equilibrium conditions
from the free-body
diagram
Solve for the unknown
forces (F and R)
Ladder Example, 2
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Draw a free-body
diagram for the ladder
The frictional force is
ƒ = µn
Let O be the axis of
rotation
Apply the equations for
the two conditions of
equilibrium
Solve the equations
Ladder Example
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The ladder is uniform
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So the weight of the
ladder acts through its
geometric center (its
center of gravity)
There is static friction
between the ladder and
the ground
Ladder Example, Extended
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Add a person of mass
M at a distance d from
the base of the ladder
The higher the person
climbs, the larger the
angle at the base
needs to be
Eventually, the ladder
may slip
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Ch 12: Question 12
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A ladder stands on the ground, leaning
against a wall. Would you feel safer climbing
up the ladder if you were told that the ground
is frictionless but the wall is rough, or that the
wall is frictionless but the ground is rough?
Justify your answer.
Ch 12: Problem 3
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For what value of x will the beam be
balanced at P such that the normal force at
O is zero?
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