ENGINEERING SCIENCE (CHAPTER 2)

2.1 Introduction • Statics is concerned with the equilibrium of bodies under the action of forces. • This chapter deals with situations where forces are in equilibrium and the determination of resultant forces and their moments.

2.1.1 Scalar and vector quantities • Scalar quantities can be fully defined by just a number; mass is an example of a scalar quantity. • To specify a scalar quantity all we need to do is give a single number to represent its size. • Quantities for which both the size and direction have to be specified are termed vectors.

2.1.2 Internal and external forces • The term external forces is used for the forces applied to an object from outside (by some other object). • The term internal forces is used for the forces induced in the object to counteract the externally applied forces.

2.2 Forces in equilibrium • If when two or more forces act on an object there is no resultant force (the object remains at rest or moving with a constant velocity), then the forces are said to be in equilibrium. 2.2.1 Two forces in equilibrium For two forces to be in equilibrium they have to: 1 Be equal in size. 2 Have lines of action which pass through the same point (such forces are said to be concurrent). 3 Act in exactly opposite directions.



2.2.2 Three forces in equilibrium For three forces to be in equilibrium (Figure 2.4) they have to : 1. All lie in the same plane (such forces are said to be coplanar). 2. Have lines of action which pass through the same point, i.e. they are concurrent. 3. Give no resultant force



Drawing the triangle of forces involves the following steps: 1 Select a suitable scale to represent the sizes of the forces. 2 Draw an arrow-headed line to represent one of the forces. 3 Take the forces in the sequence they occur when going, say, clockwise. Draw the arrow-headed line for the next force and draw it so that its line starts with its tail end from the arrowed end of the first force. 4 Draw arrow-headed line for the third force, starting with its tail end from the arrow end of the second force. 5 If the forces are in equilibrium, the arrow end of the third force will coincide with the tail end of the first arrow-headed line to give a closed triangle.

2.3 Resultant forces • A single force which is used to replace a number of forces and has the same effect is called a resultant force. • We can use the triangle rule to add two forces; this is because if we replace the two forces by a single force it must be equal in size and in the opposite direction to the force needed to give equilibrium. • For determining the resultant, the triangle rule can be stated as: to add two forces FI and F2 we place the tail of the arrow representing one vector at the head of the arrow representing the other and then the line that forms the third side of the triangle represents the vector which is the resultant of F1 and FZ (Figure 2.7(a)). Note that the directions of F1 and F2 go in one sense round the triangle and the resultant, goes in the opposite direction. Figure 2.7(b) shows the triangle rule with F1 and F2 in equilibrium with a third force and Figure 2.7(c) compares the equilibrium force with the resultant force.

An alternative and equivalent rule to the triangle rule for determining the sum of two vectors is the parallelogram rule. This can be stated as: if we place the tails of the arrows representing the two vectors F1 and Fz together and complete a parallelogram, then the diagonal of that parallelogram drawn from the junction of the two tails represents the sum of the vectors F1 and F2. Figure 2.8 illustrates this. The triangle rule is just the triangle formed between the diagonal and two adjacent sides of the parallelogram.

The procedure for drawing the parallelogram is as follows: 1 Select a suitable scale for drawing lines to represent the forces. 2 Draw an mowed line to represent the first force. 3 From the start of the first line, i.e. its tail end, draw an mowed line to represent the second force. 4 Complete the parallelogram by drawing lines parallel to these force lines. 5 The resultant is the line drawn as the diagonal from the start point, the direction of the resultant being outwards from the start point.

2.4 Resolving forces • A single force can be replaced by two forces at right angles to each other. • This is known as resolving a force into its components. • It is done by using the parallelogram of forces in reverse.





2.5 Moment of a force • The moment of a force about an axis is the product of the force F and its perpendicular distance r from the axis to the line of action of the force(Figure 2.15). • An alternative, but equivalent, way of defining the moment of a force about an axis is as the product of the force F and the radius r of its potential rotation about the axis. Thus: Moment = Fr







2.5.1 Couples • A couple is two coplanar parallel forces of the same size with their lines of action separated by some distance and acting in opposite directions (Figure2.20). • The moment of a couple about any axis is the algebraic sum of the moments due to each of the forces. • Thus, taking moments about the axis through A in Figure 2.21 gives clockwise moment = F(d + X) and anticlockwise moment = Fx. Hence: Moment of couple = F(d+x) – Fx = Fd • Hence, the moment of a couple is the product of the force size and the perpendicular distance between the forces.

2.6 Centre of gravity • The weight of an object is made up of the weights of each particle, each atom, of the object and so we have a multitude of forces which do not act at a single point. However, it is possible to have the same effect by replacing all the forces of an object by a single weight force acting at a particular point; this point is termed the centre of gravity.





2.7 Static equilibrium • An object is said to be in static equilibrium when there is no movement or tendency to movement in any direction. This requires that, for coplanar but not necessarily concurrent forces: 1. There must be no resultant force in any direction, i.e. the total of the upward components of forces must equal the downward components and the total of the rightward components of forces must equal the total of leftward components. 2. The sum of the anticlockwise moments about any axis must equal the sum of the clockwise moments about the same axis.



2.8 Measurement of force Methods that are commonly used for the measurement of forces are: 1 Elastic element methods which depend on the force causing some element to stretch, or become compressed, and so change in length. This change then becomes a measure of the force. The simplest example of such a method is the spring balance the extension of the spring being proportional to the force. Direct reading spring balances are not, however, capable of high accuracy since the extensions produced are relatively small

2 Hydraulic pressure methods use the change in pressure of hydraulic fluid that is produced by the application of a force as a measure of the force. A chamber containing hydraulic fluid (Figure 2.30) is connected to a pressure gauge, possibly a Bourdon tube pressure gauge. The chamber has a diaphragm to which the force is applied. The force causes the diaphragm to move and produces a change in pressure in the fluid which then shows up on the pressure gauge. Such methods tend to be used for forces up to 5 MN and typically have an accuracy of the order of ± l %.



Examples of activities and problems can refer to the book. THANK YOU