Mechanical Engineering Principles

30 MECHANICAL ENGINEERING PRINCIPLES

Problem 5. Use the cosine and sine rules to

determine the magnitude and direction of the

resultant of a force of 8 kN acting at an angle

of 50°to the horizontal and a force of 5 kN

actingatanangleof− 30 °to the horizontal.

50 °

30 °

50 °

50 ° 30 °

8 kN

5 kN

8 kN 5 kN

b

a

0

φ

(a) space diagram (b) vector diagram

Figure 3.11

The space diagram is shown in Figure 3.11(a). A
sketch is made of the vector diagram,oarepresent-
ing the 8 kN force in magnitude and direction and
abrepresenting the 5 kN force in magnitude and
direction. The resultant is given by lengthob.By
the cosine rule,

ob^2 =oa^2 +ab^2 − 2 (oa)(ab)cosoab

= 82 + 52 − 2 ( 8 )( 5 )cos 100°

(since oab= 180 °− 50 °− 30 °= 100 °)

= 64 + 25 −(− 13. 892 )= 102. 892

Hence ob=

√

102. 892 = 10 .14 kN

By the sine rule,

5

sin aob

=

10. 14

sin 100°

from which, sin aob=

5 sin 100°

10. 14

= 0. 4856

Hence aob=sin−^1 ( 0. 4856 )= 29. 05 °. Thus angle
φin Figure 3.11(b) is 50°− 29. 05 °= 20. 95 °

Hence the resultant of the two forces is 10.14 kN
acting at an angle of 20.95°to the horizontal

Now try the following exercise

Exercise 14 Further problems on the re-

sultant of coplanar forces by

calculation

1. Forces of 7.6 kN at 32°and 11.8 kN at
   143 ° act at a point. Use the cosine and

sine rules to calculate the magnitude and

direction of their resultant.

[11.52 kN at 105°]

In questions 2 to 5, calculate the resultant

of the given forces by using the cosine

and sine rules

2. 13 N at 0°and25Nat30°
   [36.8 N at 20°]


3. 1.3 kN at 45°and 2.8 kN at− 30 °
   [3.4 kN at− 8 °]


4. 9 N at 126°and 14 N at 223°
   [15.7 N at− 172 °]


5. 0.7 kN at 147°and 1.3 kN at− 71 °
   [0.86 kN at− 101 °]

3.8 Resultant of more than two

coplanar forces

For the three coplanar forcesF 1 ,F 2 andF 3 act-

ing at a point as shown in Figure 3.12, the vector

diagram is drawn using the nose-to-tail method of

Section 3.5. The procedure is:

F 2

F 3

F 1

Figure 3.12

(i) Drawoato scale to represent forceF 1 in both

magnitude and direction (see Figure 3.13)

F 2

b

a

0

c

F 3

F 1

Resultant

Figure 3.13

(ii) From the nose ofoa,drawab to represent

forceF 2