Mechanical Engineering Principles

FORCES ACTING AT A POINT 27

3.4 The resultant of two coplanar

forces

For two forces acting at a point, there are three

possibilities.

(a) For forces acting in the same direction and

having the same line of action, the single force

having the same effect as both of the forces,

called theresultant forceor just theresultant,

is the arithmetic sum of the separate forces.

Forces ofF 1 andF 2 acting at pointP, as shown

in Figure 3.5(a), have exactly the same effect

on pointPas forceFshown in Figure 3.5(b),

where F = F 1 +F 2 and acts in the same

direction asF 1 andF 2. ThusFis the resultant

ofF 1 andF 2

P

P

F 2

F

F 1

(F 1 +F 2 )

(a)

(b)

Figure 3.5

(b) For forces acting in opposite directions along
the same line of action, the resultant force is the
arithmetic difference between the two forces.
Forces ofF 1 andF 2 acting at pointPas shown
in Figure 3.6(a) have exactly the same effect
on pointPas forceFshown in Figure 3.6(b),
whereF =F 2 −F 1 and acts in the direction
ofF 2 ,sinceF 2 is greater thanF 1.

ThusFis the resultant ofF 1 andF 2

P

P

F 1

F

F 2

(F 2 – F 1 )

(a)

(b)

Figure 3.6

(c) When two forces do not have the same line

of action, the magnitude and direction of the

resultant force may be found by a procedure

called vector addition of forces. There are two

graphical methods of performingvector addi-

tion, known as thetriangle of forcesmethod

(see Section 3.5) and theparallelogram of

forcesmethod (see Section 3.6)

Problem 1. Determine the resultant force of

two forces of 5 kN and 8 kN,

(a) acting in the same direction and having

the same line of action,

(b) acting in opposite directions but having

the same line of action.

P

P

8 kN

8 kN

5 kN

0

Scale

5 10 kN (force)

(a)

(b)

5 kN

Figure 3.7

(a) The vector diagram of the two forces acting in

the same direction is shown in Figure 3.7(a),

which assumes that the line of action is hori-

zontal, although since it is not specified, could

be in any direction. From above, the resultant

forceFis given by:

F=F 1 +F 2 ,

i.e. F=( 5 + 8 )kN=13 kN

in the direction of the original forces.

(b) The vector diagram of the two forces acting in

opposite directions is shown in Figure 3.7(b),

again assuming that the line of action is in a

horizontal direction. From above, the resultant

forceFis given by:

F=F 2 −F 1 ,i.e.F=( 8 − 5 )kN

=3kN

in the direction of the 8 kN force.