Equilibrium of Particles 2014

8/12/2019 Equilibrium of Particles 2014

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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

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Equilibrium of Particles 2014