Mechanical Engineering Principles

FORCES ACTING AT A POINT 29

3. 13 N at 0°and 25 N at 30°
   [36.8 N at 20°]


4. 5 N at 60°and 8 N at 90°
   [12.6 N at 79°]


5. 1.3 kN at 45°and 2.8 kN at− 30 °
   [3.4 kN at− 8 °]

3.6 The parallelogram of forces

method

A simple procedure for the parallelogram of forces
method of vector addition is as follows:

(i) Draw a vector representing one of the forces,

using an appropriate scale and in the direction

of its line of action.

(ii) From thetailof this vector and using the same

scale draw a vector representing the second

force in the direction of its line of action.

(iii) Complete the parallelogram using the two
vectors drawn in (i) and (ii) as two sides of
the parallelogram.

(iv) The resultant force is represented in both mag-
nitude and direction by the vector correspond-
ing to the diagonal of the parallelogram drawn
from the tail of the vectors in (i) and (ii).

Problem 4. Use the parallelogram of forces

method to find the magnitude and direction

of the resultant of a force of 250 N acting at

an angle of 135°and a force of 400 N acting

at an angle of− 120 °.

From the procedure given above and with reference
to Figure 3.10:

(i) abis drawn at an angle of 135°and 250 units

in length

(ii) acis drawn at an angle of− 120 °and 400

units in length

(iii) bdandcdare drawn to complete the parallel-
ogram

(iv) adis drawn. By measurement,adis 413 units
long at an angle of− 156 °.

That is, the resultant force is413 Nat an angle of
− 156 °

Scale

0 100 200 300 400 500 N (force)

a

b

d

c400 N

250 N

413 N

156 °

120 °

135 °

Figure 3.10

Now try the following exercise

Exercise 13 Further problems on the par-

allelogram of forces method

In questions 1 to 5, use the parallelogram of

forces method to determine the magnitude and

direction of the resultant of the forces given.

1. 1.7 N at 45°and 2.4 N at− 60 °
   [2.6 N at− 20 °]


2. 9 N at 126°and 14 N at 223°
   [15.7 N at− 172 °]


3. 23.8 N at− 50 °and 14.4 N at 215°
   [26.7 N at− 82 °]


4. 0.7 kN at 147°and 1.3 kN at− 71 °
   [0.86 kN at− 101 °]


5. 47 N at 79°and 58 N at 247°
   [15.5 N at− 152 °]

3.7 Resultant of coplanar forces by

calculation

An alternative to the graphical methods of deter-

mining the resultant of two coplanar forces is by

calculation. This can be achieved bytrigonometry

using thecosine ruleand thesine rule, as shown

in Problem 5 following, or byresolution of forces

(see Section 3.10).