# Students will be able to :

Mechanics

Oct 24, 2013 (4 years and 6 months ago)

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EQUILIBRIUM OF A PARTICLE, THE FREE
-
BODY
DIAGRAM & COPLANAR FORCE SYSTEMS

Ch. 3 Objectives
:

Students will be able to :

a) Explore the concept of equilibrium

b) Draw a free body diagram (FBD), and,

c) Apply equations of equilibrium to solve a 2D problem.

d) Apply equations of equilibrium to solve a 3D problem.

EQUILIBRIUM OF A PARTICLE, THE FREE
-
BODY DIAGRAM &
COPLANAR FORCE SYSTEMS

Equilibrium

A key concept in statics is that of
equilibrium
. If an object is at rest, we
will assume that it is in equilibrium and that the sum of the forces acting
on the object equal zero.

Resultant of all
forces acting on a
particle is zero.

Newton’s First Law of Physics
: If the resultant force on a
particle is zero, the particle will remain at rest or will continue
at constant speed in a straight line.

Equilibrium

EQUILIBRIUM OF A PARTICLE, THE FREE
-
BODY DIAGRAM &
COPLANAR FORCE SYSTEMS

Equilibrium

If an object is in equilibrium, then the resultant force acting on an object
equals zero. This is expressed as follows:

Some problems can be analyzed using only 2D, while others
require 3D.

Equations for 2D Equilibrium
:

If a problem is analyzed using
2D, then the vector equation
above can be expressed as:

Equations for 3D Equilibrium
:

If a problem is analyzed using 3D,
then the vector equation above can
be expressed as:

Since the forces involved in supporting the spool lie in a plane, this is
essentially a 2D equilibrium problem. How would you find the forces in
cables AB and AC?

2D Equilibrium
-

Applications

For a given force exerted on the boat’s towing pendant, what are
the forces in the bridle cables? What size of cable must you use?

This is again a 2D problem since the forces in cables AB, BC, and
BD all lie in the same plane.

2D Equilibrium
-

Applications

The crane is lifting a load. To decide if
the straps holding the load to the crane
hook will fail, you need to know the force
in the straps. How could you find the
forces?

Straps

3D Equilibrium
-

Applications

This is a 3D problem since the forces do
not lie in a single plane.

This shear leg derrick
is to be designed to lift
a maximum of 200 kg
of fish.

Finding the forces in
the cable and derrick
legs is a 3D problem.

3D Equilibrium
-

Applications

Coplanar Force Systems (2D Equilibrium)
-

(Section 3.3)

To determine the tensions in
the cables for a given weight
of the cylinder, you need to
learn how to draw a
free
body diagram

and apply the
equations of equilibrium.

This is an example of a 2
-
D or
coplanar force system.

If the whole assembly is in
equilibrium, then particle A is
also in equilibrium.

FREE BODY DIAGRAM (FBD)

Free Body Diagrams
are an important part of a course in Statics as
well as other courses in mechanics (Dynamics, Mechanics of
Materials, Fluid Mechanics, etc.,)

Free Body Diagram

-

A drawing that shows all
external forces acting on the particle.

Why
?

-

It is key to being able to write the
equations of equilibrium

which are used to
solve for the unknowns (usually forces or angles).

Procedure for drawing a Free Body Diagram (FBD)

Active forces: They want to move the particle.
Reactive forces: They tend to resist the motion.

Note : Cylinder mass = 40 Kg

1. Imagine the particle to be isolated or cut free from its
surroundings.

3. Identify each force and show all known magnitudes and
directions. Show all unknown magnitudes and / or directions
as variables .

F
C
= 392.4 N (What is this?)

2. Show all the forces that act on the particle.

A

F
B

F
D

30
˚

FBD at A

A

y

x

Area to be cut
or isolated

EQUATIONS OF 2
-
D EQUILIBRIUM

Or, written in a scalar form,

F
x

=
0

and

F
y

= 0

These are two scalar equations of equilibrium.

They can be used to solve for up to
two

unknowns.

Since particle A is in equilibrium, the
net force at A is zero.

So
F
B

+
F
C

+

F
D

= 0

or

F

= 0

FBD at A

A

In general, for a particle in equilibrium,

F

= 0

or

F
x

i

+

F
y

j

= 0 = 0
i

+

0
j

(a vector equation)

FBD at A

A

F
B

F
D

A

F
C
= 392.4 N

y

x

30
˚

EXAMPLE

Equations of equilibrium:

F
x

= F
B

cos 30º

F
D

= 0

F
y

= F
B

sin 30º

392.4 N = 0

Solving the second equation gives:
F
B

= 785 N

From the first equation, we get:
F
D

= 680 N

Note : Cylinder mass = 40 Kg

FBD at A

A

F
B

F
D

A

F
C
= 392.4 N

y

x

30
˚

Example
: Solve for the tensions in cables AB and AC.

Steps
: 1) Draw the FBD (at what point?)

2) Write and solve the 2D equations of
equilibrium

EXAMPLE
: Solve for the forces in cables CD, BC, and AB and the
weight in cylinder F. Discuss the approach. Include all necessary FBDs.

Pulleys

Ideal pulleys simply change the direction of a force.

The tension on each side of an ideal pulley is the same.

The tension is the same everywhere in a given rope or cable if ideal
pulleys are used.

In a later chapter non
-
ideal pulleys are introduced (belt friction and
bearing friction).

50 lb

Vertical

force

Horizontal

force

50 lb

T
1

T
2

T
2

For a frictionless pulley:
T
1

= T
2

Example

-

Determine the tension T required to support
the 100 lb block shown below.

Example
:
Determine the force P needed to support the
100
-
lb weight. Each pulley has a weight of 10 lb.
Also, what are the cord reactions at A and B?

Example:

A 350
-
lb load is supported by the
rope
-
and
-
pulley arrangement shown.
Knowing that

= 35

, determine the angle

and the force P.

SPRINGS

L

L
o

s

F = k|L

L
o
|

F = k

s

Springs can be used to apply
forces of tension (spring pulling)
or compression (spring pushing).

Hooke’s Law
:

Spring Force = (spring constant)

(
deformation)

or

Example
: A 20 lb weight is added
to a spring as shown. Determine
the spring constant, k.

12”

20 lb

16”

Example
:

Determine the mass of each cylinder if they cause a sag of
s

=
0.5 m when suspended from the rings at
A

and
B
. Note that
s

= 0 when
the cylinders are removed.

THREE
-
DIMENSIONAL FORCE SYSTEMS

Recall that with 3D problems we will use three equations of equilibrium.

Also recall from the last chapter that
3D forces may be specified in different
ways, including:

1) With coordinate direction angles
(
α
,
β
, and
γ
),

2)
With angles of projection onto a
plane,

3)
With distances. When distances
are specified, we typically express
the force in Cartesian vector form
using position vectors as follows:

Example

3D Equilibrium

1) Draw a free body diagram of Point A. Let the unknown force
magnitudes be F
B
, F
C
, F
D

.

2) Represent each force in the Cartesian vector form.

3) Apply equilibrium equations to solve for the three unknowns.

Given:

A 600 N load is supported
by three cords with the
geometry as shown.

Find:

The tension in cords AB,

Plan
:

EXAMPLE

(continued)

F
B

= F
B
(sin 30

i

+ cos 30

j
) N

= {0
.
5 F
B

i

+
0
.
866 F
B
j
} N

F
C

=

F
C
i

N

F
D

= F
D
(
r

/r
)

= F
D
{ (1

i

2
j

+ 2
k
)
/
(1
2

+ 2
2

+ 2
2
)
½

} N

= { 0
.
333 F
D

i

0
.
667 F
D

j

+ 0
.
667 F
D
k

} N

FBD at A

F
C

F
D

A

600 N

z

y

30
˚

F
B

x

1 m

2 m

2 m

EXAMPLE

(continued)

Solving the three simultaneous equations yields

F
C

= 646 N

F
D

= 900 N

F
B

= 693 N

y

Now equate the respective
i

,
j

,
k

components to zero.

F
x

= 0.5 F
B

F
C

+
0
.
333 F
D
= 0

F
y

= 0.866 F
B

0
.
667 F
D

= 0

F
z

= 0.667 F
D

600 = 0

FBD at A

F
C

F
D

A

600 N

z

30
˚

F
B

x

1 m

2 m

2 m

Example

3D Equilibrium

A 3500 lb motor and plate are supported by three
cables and d = 2 ft. Find the magnitude of the
tension in each of the cables.

Example

3D Equilibrium

Three cables are used to tether a balloon as shown.
Determine the vertical force P exerted by the balloon
at A knowing that the tension in cable AB is 60 lb.