MECHANICS 731 Hence, if particular values be assigned to |, 77, we find four values of . Thus, in general, there are four positions of the w single resultant force passing through any point. But, without forming the biquadratic, we may easily obtain Minding s theorem. Suppose we seek the locus of all points in which the plane r; can be cut by the line of action of the single force. We have = 0, and the equations above are reduced to (0 - 7)2/2 -(0 + 7)^0: =0, - xz) + 2(j3 + y}wx = , From the last two we find so that fiaally, by the first, Had we put rj = 0, we should have found, by a similar process, (4) and (5) represent an hyperbola and an ellipse, or an ellipse and an hyperbola, respectively, according as /3 2 is greater or less than 7 2 . In either case the vertices of the hyperbola coincide with the foci of the ellipse ; so that the two curves are linked together. It is now easy to see that, from any assigned point of space, the two curves will appear to intersect one another in four points. Two, or all, of these may in special cases coincide. Lines drawn to these points give the four positions of the single force which can pass through the assigned point. adder aning wall.
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of Statical Methods and Theorems. 229. Suppose a ladder to be leaning against a vertical If there be no friction, what force, applied at the ^ W [U j us t su ffice to support it 1 In the treatment of all questions of this kind the student should commence by making a rough sketch of the situation, indicating all the forces con cerned, with the directions in which they act. As shown in fig. 62, the wall exerts an outward thrust S.on the upper end of the ladder, the ground an up ward thrust R on the lower end. The only other force is gravity, which may be supposed to produce a downward force at the middle of the ladder, equal to its whole weight. Unless there be some other horizontal force to balance S, -the ladder will obviously slide down. Suppose then a horizontal force F to be applied Fig. 62. at the lower end, and let the ladder be inclined at an angle a to the horizon. Then our conditions become horizontally S-F = 0, vertically W-R = 0, and for the couple in the plane of the figure, I being the length of the ladder, R [The last equation is obtained by taking moments about the lower end of the ladder, this point being chosen ( 217) because the directions of two of the forces pass through it.] From these equations we find at once F = S=Wcota. It is to be observed that the requisite force F is very small while the ladder is nearly vertical, but increases without limit as it becomes more nearly horizontal. Jse of 230. Next let us vary the question by supposing the riction. coefficient of friction on the ground to be /A. The equations are precisely the same as before, and the limiting value of a for which equilibrium is possible is now to be found by putting F-juR-juW. Thus 2 / u = cota gives the smallest value of a for which equilibrium is possible. For any larger value of a less friction is called into play. 231. If next we assume the wall also to be rough, a new friction force, G, comes in. The equations (for any given value of a) are S-F = 0, W-R-G = 0, Here there is a certain amount of indeterminateness which our formulas cannot escape (although of course it does not exist in nature) so long as we are not dealing with the limiting case in which motion is about to commence. In that case we have the additional conditions Thus, in all, there are five equations. These are requisite and necessary because there are four forces S, G, R, F to be determined, as well as the special value of the angle a. The result of eliminating the four forces is tana= ^ . 232. We may still further vary the question by sup- Man on posing a man of" weight w to ascend the ladder. Let e ladder, represent the fraction of the ladder s length which he has ascended. The equations are S-F-0, W + w-R-G-0, Introducing the condition that slipping is just about to commence, we obtain tan j tana where a has the value given in 231. Hence the limiting angle is increased or diminished by the load on the ladder according as i.e., The ratio w/W does not appear in this condition. But it shows its importance when e is either greater or less than . Hence, when the ladder is just about to slip, a man makes it more stable if he stands anywhere on the lower half of it, but brings it down if he mounts higher. We conclude that, so far as sliding is concerned, it is advan tageous to make the lower half of a ladder more massive than the upper half. 233. Suppose a ladder, with its lower end resting against a wall, to be supported by a horizontal rail parallel to the wall (fig. 63). This case is chosen because it illustrates definite limits within which stability is ensured. Let a be the half length of the ladder, o its inclination to the horizon, b the distance of the rail from thej wall. Suppose the ladder in such a position that if there were no friction it would slip downwards. Then the equations of equilibrium are Fig. 63. R + Gcosa-Ssina = 0. F + Scosa + Gsina-W-0, SSseca- Wacosa = 0. In the third of these equations the lower end of the ladder has been chosen as the point about which moments
are taken, because the lines of action of three of the forces