Page:The American Cyclopædia (1879) Volume XI.djvu/333

MECHANICS 321 the revolving cylinder, which, being marked with equidistant vertical and transverse lines, indicates at a glance the direction of the curve. When the paper is flattened out, this curve is found to be parabolic, thus showing that the spaces through which the weight falls are in proportion to the squares of the times. The other laws may also be verified by Morin's ap- paratus ; but such verification is unnecessary, for the other laws are natural consequences of the law of squares. The following formulas are of frequent use in mechanical calculations. If the number of seconds during which a body falls from rest is represented by , and the space fallen through in one second by ^, the entire space fallen through will be expressed by the following equation : s=^gt* (1). Now, as the velocity acquired in falling during one second is g t and as velocity is proportional to time of fall, we derive the following equa- tion : v=gt, and 2 =<7 2 a (2). Dividing this by (1), we have y = 2gr, whence 0=V20s (3). Ex- ample : What velocity will a body acquire in falling 1,000 ft., and what time will it occupy in falling? Taking (3), =V2^s = V^4 : 32q = 253-6 ft. per second. Again, taking equation (1), s=$gt wederive t=y=^^ = 7'88 seconds. A body does not fall in a perfect- ly vertical direction, because the point from which it falls, in consequence of its greater distance from the earth's centre, describes a greater circle than the point to which it falls. It will therefore strike a point somewhat to the east about one fourth of an inch for a fall of 150 ft. The motion of an ascending body is retarded by the same law as that by which a descending body is accelerated. II. COMPOSITION AND EESOLUTION OF FOECES. The union of two or more forces to produce a mechanical effect is called a composition of forces. Conversely, when a single force is re- placed by two or more forces which produce the same effect, or when it is resolved into components for the purpose of mathemati- cal analysis, such operation is called a resolu- tion of forces. Analyses of cases must have regard to : 1, the quantity or intensity of the force or power ; 2, the direction in which the force acts ; and 3, the part of the body or load to which it is applied, and which is called the point of application. The quantity of force or power is usually expressed by assigning it a value in weight. It may also be represented by a straight line of proportionate length. Two or more forces acting in the same direction are equal to their sum ; acting in opposite direc- tions, they are equal to their difference. When two forces act together to produce a third, they may be represented by two sides of a triangle, while the resultant is represented by the third side. If a point is kept at rest by the action of three forces, these forces may be represented in quantity and direction by the sides of a tri- angle. Thus, the point a, fig. 6, will be kept at rest when acted upon by three forces in the direction of the arrows &, c, and d, where the forces are represented respectively in quantity and direction by the sides &', c', and d'. If the adjacent sides of a parallelogram represent two forces in quantity and di- rection, the resultant forces will always be represented by the diagonal contained between them. Thus, if c a and c d, fig. V, represent two forces equal in quantity, having the direction shown by the arrows, their resultant will be represented by the longer diagonal c 5 ; but if a & and 5 d represent two forces acting in the direction of the arrows, the resultant will be represented I' FIG. 6. Fia 7. by the shorter diagonal a d. These proposi- tions may be experimentally verified by the method of Gravesande. Let two weights of 8 and 10 Ibs. be suspended over two friction pulleys by a string, as shown in fig. 8, and let a third weight of say 14 Ibs. be suspended from this string, between the pulleys. After a time the system will come to rest. If now the string supporting the middle weight be ex- tended upward vertical- ly to some point as d, and da and de be drawn parallel to the strings a 5 and & c, a parallelogram will be formed whose adjacent sides a~b and 5c, and whose diagonal T>d will have the re- spective values 8, 10, and 14. The point & is acted upon by three forces, represented by the respective sides of the triangle a d ~b in quantity and direction, the weight 8 acting in the direction 5c, the weight 10 acting in the direction 5 , and the weight 14 acting in the direction d &. The resultant of a number of forces acting upon the same point of a body may be determined by findiDg the resultant of the first two, and of this with the third, &o. This will be ob- vious by supposing four equal forces, ae, ~be, c e, and d e, fig. 9, acting at right angles to each other upon the point e. The resultant of ae and ~b e will be the diagonal / e of the parallelogram e If; the resultant of / and c e will be I e ; and that of I e and d e will be FIG. 8.