Mechanical Engineering Principles

FORCES ACTING AT A POINT 31

(iii) From the nose ofab,drawbcto represent
forceF 3

(iv) The resultant vector is given by lengthocin
Figure 3.13. The direction of resultantocis
from where we started, i.e. pointo,towhere
we finished, i.e. point c. When acting by
itself, the resultant force, given byoc,hasthe
same effect on the point as forcesF 1 ,F 2 and
F 3 have when acting together. The resulting
vector diagram of Figure 3.13 is called the
polygon of forces.

Problem 6. Determine graphically the

magnitude and direction of the resultant of

these three coplanar forces, which may be

considered as acting at a point:

ForceA, 12 N acting horizontally to the

right; forceB, 7 N inclined at 60°to force

A; forceC, 15 N inclined at 150°to forceA

FC= 15 N
FB= 7 N

FA= 12 N

150 ° 60 °

Figure 3.14

FC= 15 N

FB= 7 N

FA= 12 N

150 °

60 °

a

b

c

0

0 4 8 12

Resultant

Scale Newtons

φ

Figure 3.15

The space diagram is shown in Figure 3.14. The
vector diagram shown in Figure 3.15, is produced
as follows:

(i) oarepresents the 12 N force in magnitude and

direction

(ii) From the nose ofoa,abis drawn inclined at

60 °tooaand 7 units long

(iii) From the nose ofab,bcis drawn 15 units

long inclined at 150°tooa(i.e. 150°to the

horizontal)

(iv) ocrepresents the resultant; by measurement,

the resultant is 13.8 N inclined atφ= 80 °to

the horizontal.

Thus the resultant of the three forces,FA,FBand

FCis a force of 13.8 N at 80°to the horizontal.

Problem 7. The following coplanar forces

are acting at a point, the given angles being

measured from the horizontal: 100 N at 30°,

200 N at 80°,40Nat− 150 °, 120 N at

− 100 °and70Nat− 60 °. Determine

graphically the magnitude and direction of

the resultant of the five forces.

The five forces are shown in the space diagram

of Figure 3.16. Since the 200 N and 120 N forces

have the same line of action but are in opposite

sense, they can be represented by a single force of

200 −120, i.e. 80 N acting at 80°to the horizontal.

Similarly, the 100 N and 40 N forces can be repre-

sented by a force of 100−40, i.e. 60 N acting at

30 °to the horizontal. Hence the space diagram of

Figure 3.16 may be represented by the space dia-

gram of Figure 3.17. Such a simplification of the

vectors is not essential but it is easier to construct

the vector diagram from a space diagram having

three forces, than one with five.

100 N

70 N

120 N

200 N

40 N

150 °

80 °

30 °

60 °

100 °

Figure 3.16