THE COMPOSITION AND RESOLUTION OF FORCES. 34. COMPOSITION of FORCES, is the uniting of two or more forces into one, which shall have the same effect; or the finding of one force that shall be equal to several others taken together, in any different directions. And the Resolution of Forces, is the finding of two or more forces which, acting in any different directions, shall have the same effect as any given single force. PROPOSITION VII. 35. If a Body at A be urged in the Directions AB and AC, by any two Similar Forces, such that they would separately cause the Body to pass over the Spaces AB, AC, in an equal Time; then if both Forces act together, they will cause the Body to move, in the same Time, through AD the Diagonal of the Parallelogram ABCD. DRAW cd parallel to AB, and bd parallel to AC. And while the body is carried over ab or cd by the force in that direction, let it be carried over bď by the force in that direction; by which means it will be found at d. Now, if the forces be impulsive or momentary, the motions will be uniform, and the spaces described will be as the times of description: theref. Ab or cd: AB or CD :: time in ab: time in AB, and bd or Ac: BD or AC :: time in Ac: time in AC; == but the time in ab time in Ac, and the time in AB = time in AC; therefore ab: bd:: AB: ED by equality hence the point d is in the diagonal AD. And as this is always the case in every point d, d, &c, therefore the path of the body is the straight line adD, or the diagonal of the parallelogram. But if the similar forces, by means of which the body is moved in the directions AB, AC, be uniformly accelerating ones, then the spaces will be as the squares of the times; in which case, call the time in bd or cd, t, and the time in AB or AC, T; then it will be Ab or cd: AB or CD::: T2, 1 and bd or Ac: BD or AC ::: T2, and so the body is always found in the diagonal, as before. 36. Corol. 36. Corol. 1. If the forces be not similar, by which the body is urged in the directions AB, AC, it will move in some curved line, depending on the nature of the forces. 37. Corol. 2. Hence it appears, that the body moves over the diagonal AD, by the compound motion, in the very same time that it would move over the side AB, by the single force impressed in that direction, or that it would move over the side Ac by the force impressed in that direction. 38. Coral. 3. The forces in the directions AB, AC, AD, are respectively proportional to the lines AB, AC, AD, and in these directions. 39. Corol. 4. The two oblique forces AB, AC, are equivalent to the single direct force AD, which may be compounded of these two, by drawing the diagonal of the parallelogram. Or they are equivalent to the double of AE, drawn to the middle of the line BC. And thus any E B. force may be compounded of two or more other forces; which is the meaning of the expression composition of forces. 40. Exam. Suppose it were required to compound the three forces AB, AC, AD; or to find the direction and quantity of one single force, which shall be equi- E valent to, and have the same effect, as if a body A were D. acted on by three forces in the directions AB, AC, AD, and proportional to these three lines. First reduce the two AC, AD to one AE, by completing the parallelogram ADEC. Then re duce the two AE, AB to one AF by the parallelogram AEFB. So shall the single force AF be the direction, and as the quantity, which shall of itself produce the same effect, as if all the three AB, AC, AD acted together. 41. Corol. 5. Hence also any single direct force AD, may be resolved into two oblique forces, whose quantities and directions are AB, AC, having the same effect, by describing any parallelogram whose diagonal may be AD: and this is called the resolution of forces. So the force AD E... T may be resolved into the two AB, AC, by the parallelogram ABDC ABDC; or into the two AE, AF, by the parallelogram AEDF; and so on, for any other two. And each of these may be resolved again into as many others as we please. B 42. Corol. 6. Hence too may be found the effect of any given force, in any other direction, besides that of the line in which it acts; as, of the force AB in any other given direction CB. For draw AD perpendicular to CB; then shall DB be the effect of the force AB in the direction CB. For, the given force AB is equivalent to the two AD, DB, or AE; of which the former AD, or EB, being perpendicular, does not alter the velocity in the direction CB; and therefore DB is the whole effect of AB in the direction CB. That is, a direct force expressed by the line DB acting in the direction DB, will produce the same effect or motion in a body в, in that direction, as the oblique force expressed by, and acting in, the direction AB, produces in the same direction CB. And hence any given force AB, is to its effect in DB, as AB to DB, or as radius to the cosine of the angle ABD of inclination of those directions. For the same reason, the force or effect in the direction AB, is to the force or effect in the direction AD or EB, as AB to AD; or as radius to sine of the same angle ABD, or cosine of the angle DAB of those directions. 43. Corol. 7. Hence also, if the two given forces, to be compounded, act in the same line, either both the same way, or the one directly opposite to the other; then their joint or compounded force will act in the same line also, and will be equal to the sum of the two when they act the same way, or to the difference of them when they act in opposite directions; and the compound force, whether it be the sum or difference, will always act in the direction of the greater of the two. PROPOSITION VIII. 44. If Three Forces A, B, C, acting all together in the same Plane, keep one another in Equilibrio; they will be Proportional to the Three Sides DE, EC, CD, of a Triangle, which are drawn Parallel to the Directions of the Forces AD, DB, CD. PRODUCE AD, BD, and draw CF, CE parallel to them. Then Then the force in CD is equivalent to the two AD, BD, by the supposition; but the force CD is also equivalent to the two ED and CE or FD; therefore, if CD represent the force c, then ED will represent its opposite force A, and CE, or FD, its opposite force B. Consequently the three forces A, B, C, are proportional to DE, CE, CD, the three lines parallel to the directions in which they act. A E 45. Corol. 1. Because the three sides CD, CE, DE, are proportional to the sines of their opposite angles E, D, c; therefore the three forces, when in equilibrio, are proportional to the sines of the angles of the triangle made of their lines of direction; namely, each force proportional to the sine of the angle made by the directions of the other two. 46. Corol. 2. The three forces, acting against, and keeping one another in equilibrio, are also proportional to the sides of any other triangle made by drawing lines either perpendicular to the directions of the forces, or forming any given angle with those directions. For such a triangle is always similar to the former, which is made by drawing lines parallel to the directions; and therefore their sides are in the same proportion to ane another, 47. Corol. 3. If any number of forces be kept in equilibrio by their actions against one another; they may be all reduced to two equal and opposite ones.-For, by cor. 4, prop. 7, any two of the forces may be reduced to one force acting in the same plane; then this last force and another may likewise be reduced to another force acting in their plane: and so on, till at last they be all reduced to the action of only two opposite forces; which will be equal, as well as opposite, because the whole are in equilibrio by the supposition. 48. Coral. 4. If one of the forces, as c, be a weight, which is sustained by two strings drawing in the directions DA, DB: then the force or tension of the string AD, is to the weight c, or tension of the string DC, as DE to DC; and the force or tension of the other string BD, is to the weight c, or tension of CD, as CE to CD. E 49. Cercl 49. Corol. 5. If three forces be in equilibrio by their mu tual actions; the line of direction of each force, as DC, passes through the opposite angle c of the parallelogram formed by the directions of the other two forces. 50. Remark. These properties, in this proposition and its corollaries, hold true of all similar forces whatever, whether they be instantaneous or continual, or whether they act by percussion, drawing, pushing, pressing, or weighing; and are of the utmost importance in mechanics and the doctrine of forces. ON THE COLLISION OF BODIES. PROPOSITION IX. 51. If a Body strike or act Obliquely on a Plain Surface, the Force or Energy of the Stroke, or Action, is as the Sine of the Angle of Incidence. Or, the Force on the Surface is to the same if it had acted Perpendicularly, as the Sine of Incidence is to Radius. LET AB express the direction and the absolute quantity of the oblique force on the plane DE; or let a given body A, moving with a certain velocity, impinge on the plane at B; then its force will be to the action on the plane, as radius to the sine of the angle ABD, or as AB to AD or BC, drawing AD and BC perpendicular, and AC parallel to DE. For, by prop. 7, the force AB is equivalent to the two forces AC, CB; of which the former AC does not act on the plane, because it is parallel to it. The plane is therefore only acted on by the direct force CB, which is to AB, as the 'sine of the angle BAC, or ABD, to radius. 52. Corol. 1. If a body act on another, in any direction, and by any kind of force, the action of that force on the second body, is made only in a direction perpendicular to the surface on which it acts. For the force in AB acts on DE Only by the force CB, and in that direction. 53. Corol. 2. If the plane DE be not absolutely fixed, it will move, after the stroke, in the direction perpendicular to its surface. For it is in that direction that the force is exerted. PROPOSITION |