TWO AND THREE
DIMENSIONAL
EQUILIBRIUM OF
PARTICLES
Only concurrent forces can act on a
particle whose shape and dimensions are
neglected and its whole mass is assumed to
be concentrated at a single point, its mass
center.
Equilibrium can be thought of as an unchanging
– stable condition. All the bodies that are at
rest are in equilibrium.
A particle acted upon by balanced forces is in
equilibrium provided it is at rest if originally
at rest or has a constant velocity (moving
along a straight path with constant speed) if
originally in motion.
Therefore moving objects can also be in
equilibrium. Such bodies are said to be in
“steady translation”. Most often, however, the
term “equilibrium”, or more specifically “static
equilibrium” is used to describe an object at
rest.
To maintain equilibrium, it is necessary to
satisfy Newton’s first law of motion, which
requires the resultant force acting on a
particle to be equal to zero. This condition
may be stated mathematically as

F  0

where F 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.
This follows from Newton’s second law of


motion, which can be written as F  ma .
Since the force system is in equilibrium, then

ma  0 and therefore the particle’s

acceleration a  0 . Consequently, the particle
indeed moves with constant velocity or remains
at rest.
FREE BODY
DIAGRAM
To apply the equation of equilibrium, we must
account for all the known and unknown

forces ( F ) which act on the particle. The
best way to do this is to draw the particle’s
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.
Procedure for Drawing a Free Body Diagram:
1) Draw 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 comprised of the mass center of the
particle.
2) Set up the Reference System If not
indicated, set up a reference system in
accordance with the geometry of the
problem.
3) Indicate Forces On the sketch,
indicate all the forces that act on the
particle. 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.
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.
Coplanar Force 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 i and j
components. In this case the equation of
equilibrium,

F  0



F  Fx i  Fy j  0
Fx  0
Fy  0
Note that both the x and y components
must be equal to zero separately. These
scalar equations of equilibrium require that
the algebraic sum of the x and y components
of all the forces acting on the particle be
equal to zero.
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 particle’s
free body diagram.
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.
Forces can be represented only by their
magnitudes. When doing this, the sense of
direction (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.
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 the opposite direction to that
assumed initially.
Three Dimensional Force Systems
If a particle is under the effect of spatial
forces then each force can be resolved into
its x, y and z components. In this case,

F  0




F  Fx i  Fy j  Fz k  0
Fx  0
Fy  0
Fz  0
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.
In the three dimensional case, the forces
must be represented in vector form.
Some common
supports and
reactions in two
dimensional
particle
equilibrium
problems. F1, F2
and F3 are
forces applied to
the particle by
cables and/or
bars that might
be attached to
the particle. Rx
and Ry are
reaction forces.
Some common
supports and
reactions in
three
dimensional
particle
equilibrium
problems. F1, F2
and F3 are
forces applied to
the particle by
cables and/or
bars that might
be attached to
the particle. Rx,
Ry and Rz are
reaction forces.
next page
Collar on rod
FBD of rod
Ff
N
Free Body Diagram
Samples
Cable Arrangement
TBC
FBD
P
P
C
P
TAC
W
W
FAB
B
FCB
FDB
TAB
TCB
W
TDE
TCD TCD
TCB
W
W
P
TAC
C
W
N1
N2
W
TAC
W
TAB
N
Fspring
TBC
TAB
F
TAD
TAC
TAB
W
F
TAB
TAD
TAC