It is often convenient, especially when many forces are con cerned in a single problem, to use two lines instead of one to represent a force—one to represent the magnitude and one the line of action, the arrowhead being placed on either. Thus Fig. 2 also represents the force of the preceding example, AB (one-half inch long) representing the magnitude of the force and ab its line of action. The line AB might have been drawn anywhere in the figure, but its length is definite, being fixed by the scale.
The part of a drawing in which the body upon which forces act is represented, and in which the lines of action of the forces are drawn, is called the space diagram (Fig. 2a).
If the body were drawn to scale, the scale would be a cer tain number of inches or feet to the inch. The part of a drawing in which the force magnitudes are laid off (Fig. 2b) is called by various names; let us call it the force diagram. The scale of a force diagram is always a certain number of pounds or tons to the inch.
3. Notation. When forces are represented in two separate diagrams, it is convenient to use a special notation, namely: a capital letter at each end of the line representing the magnitude of the force, and the same small letters on opposite sides of the line representing the action line of the force (see Fig. 2). When we wish to refer to a force, we shall state the capital letters used in the notation of that force; thus "force AB" means the force whose magnitude, action line, and sense are represented by the lines AB and ab.
In the algebraic work we shall usually denote a force by the letter F.
4. Scales. In this subject, scales will always be expressed in feet or pounds to an inch, or thus, 1 inch = 10 feet, 1 inch = 100 pounds, etc. The number of feet or pounds represented by one inch on the drawing is called the scale number.
To find the length of the line to represent a certain distance or force, divide the distance or force by the scale number; the quotient is the length to be laid of in the drawing. To find the magnitude of a distance or a force represented by a certain line in a drawing, multiply the length of the line by the scale num ber; the product is the magnitude of the distance or force, as the case may be.
The scale to be used in making drawings depends, of course, upon how large the drawing is to be, and upon the size of the quantities which must be represented. In any case, it is con venient to select the scale number so that the quotients obtained by dividing the quantities to be represented may be easily laid off by means of the divided scale which is at hand.
Examples. 1. If one has a scale divided into 32nds, what is the convenient scale for representing 40 pounds, 32 pounds, 56 pounds, and 70 pounds ? According to the scale, 1 inch = 32 pounds, the lengths representing the forces are respectively : 4032 56 70 = 1± ; 1 ' = ; 2 = inches.
32 Since all of these distances can be easily laid off by means of the "sixteenths scale," 1 inch = 32 pounds is convenient.
2. What are the forces represented by three lines, 1.20, 2.11, and 0.75 inches long, the scale being 1 inch = 200 pounds ? According to the rule given in the foregoing, we multiply each of the lengths by 200, thus : 1. To a scale of 1 inch = 500 pounds, how long are the lines to represent forces of 1,250, 675, and 900 pounds ? Ans. 2.5, 1.35, and 1.8 inches 2. To a scale of 1 inch = 80 pounds, how large are the forces represented by 1± and 1.6 inches ? Ans. 100 and 128 pounds. 5. Concurrent and Non=concurrent Forces. If the lines of action of several forces intersect in a point they are called rent forces, or a concurrent system, and the point of intersection is called the point of concurrence of the forces. If the lines of action of several forces do not intersect in the same point, they are called non-concurrent, or a non-concurrent system.
We shall deal only with forces whose lines of action lie in the same plane. It is true that one meets with problems in which there are forces whose lines of action do not lie in a plane, but such problems can usually be solved by means of the principles herein explained.
6. Equilibrium and Equilibrant. When a number of forces act upon a body which is at rest, each tends to move it ; but the effects of all the forces acting upon that body may counteract or neutralize one another, and the forces are said to be balanced or in equilibrium. Any one of the forces of a system in equilibrium balances all the others. A single force which balances a number of forces is called the equilibrant of those forces.
7. Resultant and Composition. Any force which would pro euce the same effect (so far as balancing other forces is concerned) as that of any system, is called the resultant of that system. Evidently the resultant and the equilibrant of a system of forces must be equal in magnitude, opposite in sense, and act along the same line.
The process of determining the resultant of a system of forces is called composition.
8. Components and Resolution. Any number of forces whose combined effect is the same as that of a single force are called components of that force. The process of determining the components of a force is called resolution. The most important case of this is the resolution of a force into two components.