PRELIMINARY CHAPTER MENSURATION AND CURVE CONSTRUCTION FOR the solution of many questions in Applied Mechanics, it is necessary that the student should be able to find the periphery or area of a plane figure of given dimensions, or the surface or volume of a solid body. Also the construction of certain curves is often necessary; and a knowledge of a few simple properties of these curves enables us to solve questions with greater ease and rapidity than could otherwise be attained. Hence we commence with the present chapter. Most of the forms we meet with are of simple character, or else can be readily resolved into such simple ones; we consider then these simple forms in order. The Parallelogram. RULE.—The area of a parallelogram is obtained by multiplying any side by the perpendicular distance between it and the opposite side. Thus in the figure we have two parallelograms, but the area of each is Fig. 1. ax b square inches or feet, according as a and b are inches or feet. Or, if it be more convenient, the area of the righthand figure is cx d. Or we can, as in the centre triangle, express it as cx d 2 There are other rules which may be used, generally involving the trigonometrical functions of the angles, depending on what data are given. But the engineer should generally, in such cases, use the given dimensions to construct the figure to scale; and then measure the necessary dimensions required for the preceding rule. EXAMPLE. Find the area of a triangle, the sides of which are 7, 8, and 9 inches respectively. If the triangle be constructed to scale, on the 8-inch side as base, and the perpendicular height is then measured, it will be found to be 6.7 inches. The Trapezium.-This figure has one pair of sides parallel, but not the other. RULE. The area of a trapezium is obtained by multiplying half the sum of the parallel sides by the perpendicular distance between them. a + c xb. 2 Fig. 3. We can see how this rule is obtained if we divide the figures into two triangles by the line BD. Any Figure bounded by Straight Lines.—The method just given can evidently be extended, and we thus obtain— RULE. To obtain the area of any rectilinear figure. Divide the figure into triangles, or parallelograms, and find the area of each separately. Finally add all the results together. We will apply this rule to the case of a trapezoid or quadrilateral; this is a four-sided figure having no parallel sides. Quadrilateral, however, includes all cases, parallel or not. ABCD is the given quadrilateral. Divide into triangles as shown. Then Area ABCD=ABD+BCD, = 1. BD × h1+ BD × h2, = BD (h1+h2). A 1 B C Fig. 4. More complex cases are solved in an exactly similar manner, the only difficulty being in deciding on the manner of dividing the area. For this no rules can be given; practice alone will enable the student to decide on the quickest method for any given case. We come now to areas enclosed by curved lines, of which the simplest is Fig. 5. The Circle. RULE.—The area of a circle is obtained by squaring the diameter, and multiplying the result by .7854. Or squaring the radius and multiplying by 3.1416. The number 3.1416, of which .7854 is one-fourth, is denoted by ; and hence in the figure [For most practical purposes π may be taken as 22.] In rectilinear figures the periphery or circumference can be directly measured; but this cannot be done in curved figures, hence we require a rule. RULE. The circumference of a circle is obtained by multiplying the diameter by π. Hence in figure 5 Circumference=πd or 2πr. The Ring. This may represent the cross-section of a hollow shaft, or column. If a table of squares be available, use (1), otherwise use (2). It is immaterial whether the hole be concentric with the outside or not. The equations are the same. But the B result in words would not be true unless by "thickness" mean thickness be understood. A Fig. 7. D Sector of a Circle.-ABCD is the sector. The angle BAD is its angle. RULE. The area of a sector is obtained by multiplying the area of the whole circle by the ratio of the angle of the sector to four right angles. |