Mechanical Engineering Principles

FORCES ACTING AT A POINT 33

The space diagram is shown in Figure 3.19.
Since the system is in equilibrium, the vector
diagram must close. The vector diagram, shown in
Figure 3.20, is drawn as follows:

(i) The load of 200 N is drawn vertically as

shown byoa

0

35 °

200 N

40 °

a

Scale 0 40 80 120 N

b

c

d

Figure 3.20

(ii) The direction only of forceF 1 is known, so

from pointa,adis drawn at 40°to the vertical

(iii) The direction only of forceF 2 is known, so
from pointo,ocis drawn at 35°to the vertical

(iv) Linesadandoccross at pointb; hence the
vector diagram is given by triangleoab.By
measurement,abis 119 N andobis 133 N.

Thus the tensions in the ropes areF 1 =119 N and
F 2 =133 N.

Problem 9. Five coplanar forces are acting

on a body and the body is in equilibrium.

The forces are: 12 kN acting horizontally

to the right, 18 kN acting at an angle of

75 °, 7 kN acting at an angle of 165°,

16 kN acting from the nose of the 7 kN

force, and 15 kN acting from the nose of

the 16 kN force. Determine the directions

of the 16 kN and 15 kN forces relative to

the 12 kN force.

With reference to Figure 3.21,oais drawn 12 units
long horizontally to the right. From point a,ab
is drawn 18 units long at an angle of 75°.From
b,bcis drawn 7 units long at an angle of 165°.

The direction of the 16 kN force is not known, thus

arcpqis drawn with a compass, with centre atc,

radius 16 units. Since the forces are at equilibrium,

the polygon of forces must close. Using a compass

with centre at 0, arcrsis drawn having a radius

15 units. The point where the arcs intersect is

atd.

By measurement, angle φ= 198 °andα= 291 °

Scale 04812kN

12 kN

a

75 °

p

16 kN

s

q

d

c

b

7 kN

18 kN

165 °

15 kN

0

r

φ

α

Figure 3.21

Thus the 16 kN force acts at an angle of 198°(or

− 162 °) to the 12 kN force, and the 15 kN force

acts at an angle of 291°(or− 69 °) to the 12 kN

force.

Now try the following exercise

Exercise 16 Further problems on copla-

nar forces in equilibrium

1. A load of 12.5 N is lifted by two strings
   connected to the same point on the load,
   making angles of 22°and 31°on oppo-
   site sides of the vertical. Determine the
   tensions in the strings. [5.86 N, 8.06 N]


2. A two-legged sling and hoist chain used
   for lifting machine parts is shown in
   Figure 3.22. Determine the forces in each
   leg of the sling if parts exerting a down-
   ward force of 15 kN are lifted.
   [9.96 kN, 7.77 kN]