Forces are analyzed in a number of ways;
it is common approach to establish a coordinate
system to quantify the forces and their effects in a
system or body. Since it is customary to assign the axes,
the analysis may be coplanar (two-dimensional) or
non-coplanar (three-dimensional).
A system of forces may be represented by a resultant
force which has the same effect as the system.
Forces on
an object
2N
4N
6N
1N
Equivalent
Resultant
Force
3N
2N
6N
3N
3N
Resultant =0
The resultant force, much like any other force, has
magnitude and direction. The geometric sum of the
forces will yield the resultant.
Coplanar Force Systems analyze forces acting
on a body by taking their components along two
designated axes.
A force system can be identified into two main types:
 concurrent
 non-concurrent.
Concurrent Forces are forces whose lines of action
intersect at a common point. The resultant of
concurrent forces originates from the intersection.
Point of
Intersection
The resultant of concurrent forces must be defined
by magnitude and direction. Magnitude represents the
length of the vector while the direction is referred
from the defined axis.
Resultant
Force
Example 1:
Compute the value of the resultant of the
concurrent system of forces shown.
300lb
60○
1
2
100lb
45○
60○
200lb
400lb
Example 2:
In the concurrent force system shown in the figure.
Determine the value of P and the resultant force R
acting at 30o to the left from the negative y-axis.
Example 3:
The triangular block shown in the figure is subjected
to the loads P= 1600lbs and F = 600lbs. If AB= 8in and
BC= 6in, resolve each load into normal and tangential
components to AC.
A
P=1600lb
C
θ
B
F=600lb
Example 4:
The block shown in the figure is acted upon by
its weight W=200lb, a horizontal force Q=600lb,
and the pressure P exerted by the inclined
plane. The resultant R of these forces is up and
parallel to the incline thereby
W
sliding the block up it.
Q
Determine P and R.
(Hint: Take one axis
P
parallel to the incline)
18○
30○
Example 5:
If the resultant force
is required to act along
the positive u-axis and
have a magnitude of
5 KN, determine the
required magnitude
of FB and its
direction θ.
Example 6:
If the tension in rope AB is 100N, what is the
tension in rope BC? (Hint: The resultant of forces
AB and BC is in the direction of the boat.)
Example 7:
Two horses on opposite banks of canal pull a barge
moving parallel to the banks by means of two ropes. The
tension in these ropes are 200lb and 240lb while the
angle between them is 60 degrees. Find the
resultant pull on the
barge and the angle
between each of the
ropes and the sides of
θ
α
the canal.
Non-concurrent Forces are forces whose lines of
action are parallel. The resultant of parallel force has a
magnitude equal to the algebraic sum of the forces and
is located somewhere between them.
Resultant
Force
25KN/m
10KN/m
5KN
A
3m
3m
1m
B
10KN
The equivalent load of load diagrams – rectangular or
uniform, triangular or uniformly varying – is equal to
its area which is located at its geometric centroid.
To determine the location of the resultant, apply
Varignon’s Theorem – which states that the effect of
the whole is equal to the sum of the effects of it
components.
Using Varignon’s Theorem on taking moment about point
A, the location can be found. Recall that the moment of
a force about a point or axis is simply the magnitude of
the force multiplied by its level arm or perpendicular
distance to the point.
Also, remember that,
Moment = O if:
1. The Force intersects
the axis
2. The force is parallel to
the axis
Example 8:
Assuming clockwise moments to be positive,
compute for the moment of force F=450 lb and of
force P=361 lb about points A,B,C, and D.
A
F
C
D
D
P
B
Example 9:
Compute the resultant of the three forces shown in
the figure. Locate its intersection with the X and Y
axes. (Hint: take also the moment of three forces in
point O)
y-axis
390 lb
5
300 lb
12
30○
O
x-axis
722 lb
Example 10:
Find the resultant of the force system and its
location from point A.
Example 11:
Find the resultant of the non-concurrent force
system, its direction and location from point A.
10KN
10KN/m
5KN
θ=60
B
A
2m
1m
10KN
1m
3m
Example 12:
Find the value of P and F so that the four forces
shown in the figure produce an upward resultant
force of 300 lb acting at 4 ft from the left end of the
bar.
100lb
P
2 ft
F
3 ft
200lb
2 ft
Example 13:
The resultant of three parallel loads (one load is
missing) is 30 lb acting up at 10 ft to the right of A.
Compute the magnitude and position of the missing
load.
40lb
A
2 ft
60 lb
11 ft
Example 14:
The 16-ft wing of an airplane is subjected to a lift
which varies from zero at the tip to 360lb/ft at the
fuselage according to y=90x1/2, where x is measured
from the tip. Compute the resultant and its
location from the wing tip.
Example 15:
Determine the resultant of three forces acting on the
dam shown and
locate its intersection
24000 lbs
with the base AB. For
good design, this
intersection should
occur within the
middle third of
the base. Does it?
Example 16:
Serious neck injuries can occur when a football player
is struck in the face guard of his helmet in the manner
shown, giving rise to a guillotine mechanism.
Determine the moment
of the knee force P=50lb
about a point A. What
would be the magnitude
of the neck force F so
that it gives the counterbalancing moment
about point A.
Example 17:
Find the resultant of the non-concurrent force
system, its direction and location from point A.
20KN/m
10KN/m
10KN/m
10KN/m
A
1m
50KN
B
2m
1m
1.5m
80KN
1.5m
Example 18:
Find the resultant of the non-concurrent force
system, its direction and location from point A.
30 lb/ft
30 lb/ft
15 lb/ft
B
A
40lb
6 ft
4ft
3 ft
50lb
A couple is consists of two parallel, non-collinear
forces that are equal in magnitude but
oppositely directed.
Moment of a Couple = F*d = C
F
d
F
X
O
The moment of a couple is constant
and independent of the moment
center
Characteristics of a Couple:
1. The resultant force of a couple is zero.
2. The moment of a couple is the product of one
of the forces and the perpendicular distance
between their lines of action.
3. The moment of a couple is the same for all
points in the plane of the couple.
A force can be resolved into another force and a couple.
10KN
2m
2m
10KN
10KN
10KN
20KN-m
Example 19:
Assuming clockwise moments to be positive,
compute for the value of moment at point A.
80lb
80lb
100lb
100lb
100lb
200lb
1ft
A
1ft
Example 20:
Replace the System of
forces acting on the
frame by a resultant,
R, acting at Point A
and a couple acting
horizontally through B
and C.
20 lb
1 ft
A
B
3 ft
2 ft
4 ft
30 lb
C
60 lb
Example 21:
A couple consists of two vertical forces of 60 lb each.
One force acts up through A
A
and the other acts down
4 in
2 in
through D. Transform
D
B
the couple into an
equivalent couple
E
3 in
having horizontal
forces acting through C
G
E and F.
F
Example 22:
The three-step pulley shown in the figure is subjected
to the given couples. Compute the value
of the resultant couple. 40 lb
Also determine the
30 lb
forces acting at the rim
8’’
12’’
of the middle pulley
60 lb
60 lb
that are required to
balance the given system.
30 lb
16’’
40 lb