A and B; join AB and divide it in c in such a manner that AC:CB::Q:P then the resultant (R') equals P+Q and acts through c along a line parallel to AP Or BQ and in the same direction as P and Q. If c rests on a fixed point P and Q will balance about c and the fixed point will sustain a pressure R'. FIG. 15. Secondly, let P and Q (fig. 15) be the two parallel forces acting at a and Bin opposite directions. Suppose q to be the greater. In AB produced take a point ç such that P AC:CB::Q:P then the resultant (R') equals Q-P and acts through c along a line parallel to AP and BQ and in the same direction as Q. If c rests on a fixed point P and Q will balance about c, and the fixed point will sustain a pressure R'. Ex. 168. If weights of 12 lbs. and 8 lbs. are hung from a and b respectively, the ends of a rod 5 ft. long, and if the weight of the rod is neglected, determine the distance from a of the point round which these forces balance, and the pressure on that point. Ans. (1) 2 ft. (2) 20 lbs. Ex. 169. Let AB be a rod 12 ft. long (whose weight is neglected), from a a weight of 20 lbs. is hung, and an unknown weight (P) from B, it is found that the two balance about a point 3 ft. from A; determine P. Ans. 6 lbs. Ex. 170.-If a weight of 16 lbs. is hung from the end A, and 12 lbs. from the end B of a rod (whose weight is neglected), and if they balance about a point c, whose distance from a is 4 ft., what is the length of the rod ? Ans. 10 ft. Ex. 171. Draw a straight line AB, 8 ft. long; forces of 5 lbs. and 7 lbs. act at a and b respectively at right angles to AB and in opposite directions. Determine their resultant. 30. Conditions of Equilibrium of Three Parallel Forces. In the last article we saw that the forces P and Q acting severally at A and B are equivalent to the force R' acting at c; now R' will clearly be balanced by an equal opposite force R; and therefore P and Q acting at a and B will be balanced by the force R acting at c. Hence H the following conditions must be fulfilled by three parallel forces that are in equilibrium on a given body : (a) Two of the forces (P and q) must act in the same direction, and the remaining force (R) in the opposite direction, the line along which the latter acts lying between those along which the former severally act. (b) The sum of the former forces (P and q) must equal the latter force (R). (c) If any line be drawn cutting the lines of action of the forces (P, Q, R in A, B, C, respectively) the portion of the line between any two forces is proportional to the remaining force, i.e. BC:CA::P:Q AB: BC::R:P 31. Centre of Gravity. - Since each part of a body is heavy, it follows that the weight of a body is distributed throughout it; there exists, however, in every body a certain point called its centre of gravity, through which we may suppose the whole weight of the body to act, whenever that weight is one of the forces to be considered in a mechanical question. It admits of proof that the centre of gravity of any uniform prism or cylinder is the middle point of its geometrical axis: and as a uniform rod is merely a thin cylinder its centre of gravity will be at its middle point. Ex. 172. Two men, A and B, carry a weight of 3 cwt. slung on a pole, the ends of which rest on their shoulders; the distance of the weight from a is 6 ft., and from bis 4 ft. Find the pressure sustained by each If p is the pressure sustained by a and that sustained by B man. and therefore P+Q=3cwt. P-1 cwt.and Q=1cwt. Ex. 173. There is a beam of oak 30 ft. long and 2 ft. square; at a distance of 1 ft. from one end is hung a weight of 1 ton; how far from that end must the point of support be on which the beam when horizontal will rest, and what will be the pressure on that point? Ans. (1) 11.61 ft. (2) 9245 lbs. Ex. 174. If a mass of granite 30 ft. long, 1 ft. high, and 3 ft. wide is supported in a horizontal position on two points each 3 inches within the ends (and therefore 295 feet apart), find the pressure on each point of support. Ans. 7383 lbs. Ex. 175. If in the last Ex. another mass of granite with the same section and half as long is laid lengthwise on the former, their ends being square with each other; determine the single force to which their two weights are equivalent, and the line along which it acts, and hence the pressure on the two points of support. Ans. (1) Resultant acts 17.5 feet from one end. (2) Pressures on points of support respectively 9197 and 12,950 lbs. Ex. 176. If in the last case the upper block is shifted round through a right angle in such a manner that the middle point of the upper block is exactly over a point in the axis of the lower, and the end of the lower in the same plane with one face of the latter, determine the pressures on the points of support. Ans. 7695 lbs. and 14,452 lbs. Ex. 177.-A ladder AB, 50 ft. long, weighs 120 lbs. ; its centre of gravity is 10 ft. from A; if two men carry it so that its ends rest on their shoulders, determine how much of the weight each must support. If the one of them nearer to the end в is to support a weight of 40 lbs., where must he stand? Ans. (1) 96 lbs. and 24 lbs. (2) 20 ft. from в. 32. The Parallelogram of Forces. When two forces act on a point along different lines, their resultant is determined by the following rule, which is called the principle of the parallelogram of forces: - If two forces act on a point, and if lines be drawn representing those forces, and on them as sides a parallelogram be constructed, that diagonal which passes through the point will represent the resultant of the forces. The student, when applying this principle to any particular case, must bear in mind the meaning of the words a line represents a force (Art. 25). |