Mechanical Engineering Principles

3

Forces acting at a point

At the end of this chapter you should be

able to:

• distinguish between scalar and vector qu-
  antities


• define ‘centre of gravity’ of an object


• define ‘equilibrium’ of an object


• understand the terms ‘coplanar’ and ‘con-
  current’


• determine the resultant of two coplanar
  forces using

(a) the triangle of forces method

(b) the parallelogram of forces method

• calculate the resultant of two coplanar
  forces using

(a) the cosine and sine rules

(b) resolution of forces

• determine the resultant of more than two
  coplanar forces using

(a) the polygon of forces method

(b) calculation by resolution of forces

• determine unknown forces when three or
  more coplanar forces are in equilibrium

3.1 Scalar and vector quantities

Quantities used in engineering and science can be
divided into two groups:

(a) Scalar quantitieshave a size (or magnitude)
only and need no other information to spec-
ify them. Thus, 10 centimetres, 50 seconds,
7 litres and 3 kilograms are all examples of
scalar quantities.

(b) Vector quantities have both a size or
magnitude and a direction, called the line of

action of the quantity. Thus, a velocity of

50 kilometres per hour due east, an acceler-

ation of 9.81 metres per second squared ver-

tically downwards and a force of 15 Newtons

at an angle of 30 degrees are all examples of

vector quantities.

3.2 Centre of gravity and equilibrium

Thecentre of gravityof an object is a point where

the resultant gravitational force acting on the body

may be taken to act. For objects of uniform thickness

lying in a horizontal plane, the centre of gravity is

vertically in line with the point of balance of the

object. For a thin uniform rod the point of balance

and hence the centre of gravity is halfway along the

rod as shown in Figure 3.1(a).

G

L

(a)

G

(b)

G

(c)

L

2

Figure 3.1

A thin flat sheet of a material of uniform thickness

is called alaminaand the centre of gravity of a

rectangular lamina lies at the point of intersection of

its diagonals, as shown in Figure 3.1(b). The centre

of gravity of a circular lamina is at the centre of the

circle, as shown in Figure 3.1(c).

An object is inequilibriumwhen the forces act-

ing on the object are such that there is no tendency

for the object to move. The state of equilibrium of

an object can be divided into three groups.

(i) If an object is instable equilibriumand it

is slightly disturbed by pushing or pulling

(i.e. a disturbing force is applied), the centre

of gravity is raised and when the disturbing

force is removed, the object returns to its