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Equilibrium is an unchanging stablecondition.
All the bodies that are at rest are in
equilibrium.
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The bodies which move along a straight path
with constant speed can also be in equilibrium.Such bodies are said to be in steady
translation .
Most often, equilibrium , or more specifically
static equilibrium is used to describe an
object at rest.
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0 F
To maintain equilibrium, it is necessary to satisfy
Newtons first law, which requires the resul tant force
acting on a particle to be equal to zero . This condition
may be stated mathematically as
where is the vector sum of all the forces acting on
the particle. This equation is not only a necessary
condition for equilibrium; it is also a sufficient
condition .
F
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To apply the equation of equilibrium, we must
account for al l the known and unknown forces
which act on the particle.
The best way to do this is to draw the particles
Free Body Diagram (FBD). This diagram is
simply a sketch which shows the particle free from its surroundings with all the forces that act
on it.
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1) Dr aw Outlined Shape
Imagine the particle to be isolated or cut free from
its surroundings by drawing its outlined shape. A
simplified but accurate drawing is sufficient.
Particles will be drawn as unique points comprisedof the mass center of the particle.
Procedur e for Drawing a F ree Body Diagram:
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2) Set up the Reference System If not indicated, set up
a reference system in accordance with the geometry ofthe problem.
3) Indicate Forces On the sketch, indicate al l the forceson the par ticle . These forces can be active forces , which
tend to set the particle in motion, or they can be reactive
forces which are the result of the constraints or supports
that tend to prevent motion.
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4) Label Force Magnitudes The forces that are
known should be labeled with their proper magnitudes
and directions. Letters are used to represent the
magnitudes and directions of forces that are unknown.
5) Employ Equation of Equilibrium Finally,
equation of equilibrium must be employed to
determine the desired quantities. Care must be given to
the consistency of units used.
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Coplanar F orce Systems
If a particle is subjected to a system of coplanar
forces that lie in the x-y plane, then each force can be
resolved into its and components. In this case the
equation of equilibrium,
i j
0 F
0
0
0
y
x
y x
F
F j F i F F
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Note that both the x and y components must
be equal to zero separately . Since there are
only two scalar equations to be used, at most
two unknowns can be determined, which are
generally angles or magnitudes of forces
shown on the particles free body diagram.
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Scalar Notation
Since each of the two equilibrium equations
requires the resolution of vector components
along a specified x or y axis, scalar notation
can be used to represent the components when
applying these equations.
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Forces can be represented only by their
magnitudes. When doing this, the sense ofdirection (direction of arrowhead) of each force is
shown by using + or signs with respect to the
axes. If a force has an unknown magnitude, then
the arrowhead sense of the force on the free body
diagram can be assumed.
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Since the magnitude of a force is always positive,
if the solution yields a negative scalar, this
indicates that the sense of the force acts in theopposite direction to that assumed initially.
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Three Dimensional Force Systems
If a particle is under the effect of spatial forces then eachforce can be resolved into its x, y and z components. In thiscase,
0 F
0
00
0
z
y
x
z y x
F
F F
k F j F i F F
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Since there are three scalar equations to be used, at
most three unknowns can be determined.
These may again be angles, dimensions or
magnitudes of forces.
I n the three dimensional case, the forces must be
represented in vector form .
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F1, F
2 and F
3 are
forces applied to the
particle by cables
and/or bars that might be attached to the
particle.
R x and R y are reaction
forces.
Some common supports and reactions in two dimensional particle equilibrium problems
d h d l l l b bl
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F1, F 2 and F 3 are
forces applied to
the particle by
cables and/or bars
that might be
attached to the
particle.
R x, R y and R z are
reaction forces.
Some common supports and reactions in thr ee dimensional particle equi l ibri um problems
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A block on an incline with spring
Forces on block Free Body Diagram
NW=mg
F friction
F spring =kx
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Cables and Bars
Tension
Tension
Tension
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FBD T BC
T ACW
W
P P P C
a 90-b
Cable ArrangementA block and tacklesystem
FBD
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FDB
FAB
FCB
B
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P T AC
W
C
T AB T CB T CBT CD T CD
T DE
W W
W
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Fspring
TAB
TAC
W
N
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1 F 2 F (Compression)
(Compression)
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T AB
F
T BC