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

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.

0F

F

This follows from Newton’s second law of

motion, which can be written as .

Since the force system is in equilibrium, then

and therefore the particle’s

acceleration . Consequently, the particle

indeed moves with constant velocity or remains

at rest.

amF

0am

0a

FREE BODY

DIAGRAM

To apply the equation of equilibrium, we must

account for all the known and unknown

forces ( ) 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.

F

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.

Procedure for Drawing a Free Body Diagram:

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 and components. In this case the equation of equilibrium,

i

j

0F

0

0

0

y

x

yx

F

F

jFiFF

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,

0F

0

0

0

0

z

y

x

zyx

F

F

F

kFjFiFF

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.

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Ff

N

Free Body Diagram Samples

Cable Arrangement

FBD TBC

TAC W

W

P

P P C

FDB

FAB

FCB

B

P TAC

W

C

TAB TCB

TCB TCD

TCD

TDE

W W

W

N1

N2 W

Fspring

TAB

TAC

W

N

TAB

F

TBC

TAD

TAC

TAB

W

TAB

TAC

TAD

F