= = and 0, which We arrive therefore at But since the angle FPf is, in this case, therefore less than π, the resultant cannot be is absurd, and consequently c = 1. the general result, that if F, ƒ be on a particle, and inclined to each other at the angle 24, their resultant R is inclined to each of them at the angle , and its magnitude is determined by the equation R = 2F cos 0. two equal forces acting 5. It will be immediately obvious that, since the forces F and ƒ are perfectly equal and similarly situated with respect to PR, they contribute equally to the resultant R {; and, consequently, the efficiency of each in the direction PR is equal to R, or F cos p. 6. To determine the magnitude and direction of the resultant of any two forces acting on a particle. Let F, f (fig. 109) be the two forces acting on the particle P; R their resultant, perpendicular to which draw LPM; let a, ẞ denote the angles FPR, ƒPR respectively, and the angle FPf between the forces. Then the efficiencies of F and f, in the direction PR, are respectively F cos a, f cos ẞ, the sum of which must be equal to R, since the efficiency of R is equivalent to the united efficiencies of F and f in any proposed direction, because R is their resultant; .. RF cos a + fcos ẞ......(1). Now the efficiency of R in the direction PL perpendicular to itself = R cos 90° = 0; and the efficiency of Fin the direction PL = F cos FPL, and that of ƒ in the same direction = ƒ cos ƒPL; and by squaring equations (1) and (2), we have R2 = F2 cos2 a + 2 Ff cos a cos ß +f2 cos2 ß, 0 = F2 sin3 a - 2 Ff sin a sin ẞ+f2 sin2 ß; and adding these together, R2 = F2 + 2 Ff(cos a cos ß – sin a sin ß) + ƒa But because o = a + ß; .. cos o = cos a cos ẞ sin a sin ß; and, consequently, R2 = F2 + 2 Ff cos +ƒ3." This equation shews that the diagonal of a parallelogram represents the magnitude of the resultant of two forces, which are themselves represented in magnitude and direction by the sides and equation (2) shews that the same diagonal also represents the direction of the resultant. MISCELLANEOUS PROBLEMS. 1. Two given weights are suspended from the ends of a bent lever, the arms of which are given, and include a given angle; find the position of equilibrium. 2. A bent lever of uniform thickness rests with its shorter arm horizontal. But if the length of this arm were doubled the lever would rest with the other arm horizontal. Compare the lengths of the arms, and find their inclination. Shew 3. Two forces act at angles aß upon the arms a, b of a straight lever which is not attached to its fulcrum. that if there be equilibrium ab :: tan ß: tan ɑ. 4. The beam of a false balance being uniform, shew that the lengths of the arms are respectively proportional to the differences between the true and apparent weights of a given substance. 5. A beam of oak 30 feet long balances upon a point 10 feet from one end: but when a weight of 10 lbs. is suspended at the thin end, the prop must be moved 2 feet to preserve equilibrium. Find the weight of the oak. 6. Two equal forces act in opposite directions along two opposite sides of a parallelogram, and a third force along the diagonal. Find the force which will keep them in equili brium. 7. If forces proportional to the sides of a polygon be applied in the plane of the figure at the middle points of the sides and perpendicular thereto, they will balance. 8. A given body is supported on an inclined plane, first by a power parallel to the base, and then by a power parallel to the plane. Compare the pressures on the plane in the two cases. 9. A rope of given length is used to pull down a vertical pillar; at what height from the base of the pillar must it be fastened that a given force pulling it may be most effica cious? 10. A weight P hangs vertically by a string from a fixed point A; a string PBW being now fastened to P is passed over a fixed pulley B (so that BP is horizontal) and supports a weight W. Find how much this will draw AP from the vertical. 11. C, D are two smooth pegs, and ACDB is a heavy circular arc, which passes over one peg and under the other : find the position of equilibrium. 12. A given sphere rests between two given inclined planes, find the pressure upon each. 13. Two weights support each other on two given inclined planes which have a common vertex, by means of a string passing over the vertex; find the proportion of the weights. 14. A given cone is placed with its base on an inclined plane, the coefficient of friction for which is known: determine whether, upon increasing the inclination of the plane, the cone will tumble or slide. 15. A weight is suspended from one extremity of a string which passes over two fixed pulleys and through a ring at its other extremity; find the position of equilibrium. 16. A given beam rests with its lower end on a smooth horizontal plane, and its upper end on a given inclined plane; find the force which must act at the foot of the beam to prevent sliding. 17. Two given heavy particles being connected by an inflexible rod of given length are placed within a hemispherical bowl; find the position of equilibrium, and the compressing force upon the rod. 18. A rigid rod AB is moveable in a vertical plane about a fixed hinge A, the end B leans against a smooth vertical wall. Find the pressures on the wall and hinge. 19. A beam of given length and weight is placed with one end on a vertical, and the other on a horizontal plane; find the force necessary to keep it at rest, and the pressures on the two planes. 20. A and B are two given points in a horizontal line, to which are fastened two strings AC, BCW of given lengths; the string BCW passes through a ring at C, and to it is fastened a given weight W; find the position in which the ring will rest. 21. AC, BCD are two given beams moveable in a vertical plane about hinges A, B in a horizontal plane. BD the longer leans upon the end C of AC the shorter. Find the position of equilibrium. 22. If a rod rest in equilibrium with its ends on two smooth inclined planes, the intersection of the planes must be a horizontal line. 23. A beam has a ring at one extremity which moves up and down a vertical rod. Find the position of the beam when it rests upon the arc of a circle a diameter of which coincides with the rod. 24. The upper end of a given rod rests against a smooth vertical plane, and the lower end is suspended by a given string fastened to a point in the plane; find the position of equilibrium. 25. A given uniform rod passing freely through an orifice in a vertical plane rests in equilibrium with one end upon a given inclined plane; find its position. 26. A heavy beam leans against an upright prop; the lower end of the beam rests upon the horizontal plane and |