Contents
12 Equilibrium and Elasticity
1
12.1 Equilibrium
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
12.2 The Conditions for Equilibrium
. . . . . . . . . . . . . . . . .
2
12.3 The Center of Gravity (cog)
. . . . . . . . . . . . . . . . . . .
3
12.4 Examples
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4
12.4.1 Sample Problem 12  1
. . . . . . . . . . . . . . . . . .
4
12.4.2 Sample Problem 12  2
. . . . . . . . . . . . . . . . . .
5
12.4.3 Sample Problem 12  3
. . . . . . . . . . . . . . . . . .
6
12.4.4 Sample Problem 12  4
. . . . . . . . . . . . . . . . . .
7
12.5 Indeterminate Structures
. . . . . . . . . . . . . . . . . . . . .
8
12.6 Elasticity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
12
Equilibrium and Elasticity
In this chapter we will define equilibrium and find the conditions needed so
that an object is at equilibrium.
We will then apply these conditions to a variety of practical engineering
problems of static equilibrium.
We will also examine how a “rigid” body can be deformed by an external
force. In this section we will introduce the following concepts:
Stress and strain Young’s modulus (in connection with tension and com
pression) Shear modulus (in connection with shearing) Bulk modulus (in
connection to hydraulic stress)
12.1
Equilibrium
For an object,
1. The linear momentum
vector
P
of its center of mass is constant.
2.
Its angular momentum
vector
L
about its center of mass, or about any other
point is also constant.
We say such an object is in equilibrium. The two requirements for equi
librium are then
vector
P
= a constant and
vector
L
= a constant.
1
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Our concern in this chapter is with situations in which
vector
P
= 0 and
vector
L
= 0.
That is we are interested in objects that are not moving in any way (this
includes translational as well as rotational motion) in the reference frame
from which we observe them. Such objects are said to be in static equilibrium.
In chapter 8 we differentiated between stable and unstable static equilibrium.
If a body that is in static equilibrium is displaced slightly from this position,
the forces on it may return it to its old position. In this case we say that the
equilibrium is stable. If the body does not return to its old position, then
the equilibrium is unstable.
An example of unstable equilibrium is shown in the figures above. In fig.
a we balance a domino with the domino’s center of mass vertically above
the supporting edge.
The torque of the gravitational force
vector
F
g
about the
supporting edge is zero because the line of action of
vector
F
g
passes through the
edge. Thus the domino is in equilibrium. Even a slight force on the domino
ends the equilibrium.
As the line of action of
vector
F
g
moves to one side of the
supporting edge (see fig. b) the torque due to
vector
F
g
is nonzero and the domino
rotates in the clockwise direction away from its equilibrium position of fig. a.
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 Spring '11
 asdaf
 Equilibrium, Force, Gravitational forces

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