PROPOSITION VI. 29. The Spaces passed over by Bodies, urged by any Constant and Uniform Forces, acting during any Times, are in the Compaund Ratio of the Forcer and Squares of the Times directly, and the Body or Mass reciprocally. Or, the Spaces are as the Squares of the Times, when the Force and Body are given. let v denote the velocity acquired at the end of any time t, by any given body b, when it has passed over the space r. Then, because the velocity is as the time, by the last corol. therefore # v is the velocity at ; t, or at the middle point of the time; and as the increase of velocity is uniform, the same space's will be described in the same time t, by the velocity + v uniformly continued from beginning to end. But, in uniform motions, the space is in the compound ratio of the time and velocity; therefore f is as # tw, or indeed s = # tw. But, by the last corol, the velocity v is as #, or as the force and time directly, and as the body reciprocally. 2. Therefore s, or # tw, is as o, that is, the space is as the force. and square of the time directly, and as the body reciprocally. Or f is as to, the square of the time only, when b and fare given. 30. Corol. 1. The space f is also as tw, or in the comound ratio of the time and velocity; b and f being given. #. f = # twis the space actually described. But tv is the space which might be described in the same time t, with the last velocity v, if it were uniformly continued for the same or an equal time. Therefore the space f, or ; tv, which is actually described, is just half the space tw, which would be described with the last or greatest velocity, uniformly continued for an equal time t. 31. Corol. 2. The space f is also as vo, the square of the velocity; because the velocity v is as the time t. * Scholium. 32. Propositions 3, 4, 5, 6, give theorems for resolving all questicns relating to motions uniformly ageelerated. Thus, & put any body or quantity of matter, i Then, from these fundamental relations, m = by, m = fo, 33. And from these proportions those quantities are to be left out which are given, or which are proportional to each other. Thus, if the body or quantity of matter be always the same, then the space described is as the force and square of the time. And if the body be proporticial to the force, as all bodies are in respect to their gravity; then the space described is as the square of the time, or square of the velo city; and in this case, if F be put = % the accelerating The COMPOSITION AND RESOLUTION or 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. - . - --- 4. PRoposition v11. 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 Space; AB, AC, in an equal Time; then if both Forces act together, they will cause the Body to move, in £he same Time, through AD the Diagonal of the Parallelogram ABCD. - DRAw cd parallel to AB, and bá parallel to Ac. And while the body is carried over Ab or cd by the force in that direction, let it be carried over bal 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 A5 = time in Ac, and the time in AB = time in Ae; therefore A* : bā :: AB : LD by equality: hence the point d is in the diagonal AD. And as this is always the case in every point d, d, &c, #: the path of the body is the straight line Add, or the iagonal 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 bal or cd, t, and the time in AB or Ac, T; then it will be Ab or cal : AB or cD :: to : T’, and - bd or Ac : B D or Ac :: to : To, theref. by equality, A5 : bā :: AB : BD; 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. 87. 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 A D. | 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 equi- *valent to the double of AE, drawn to the middle of the line BC. And thus any 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 P. required to compound the three : - R #. AB, AC, AD; or to find `A −. the direction and quantity of one single force, which shall be equivalent to, and have the same effect, as if a body A were acted on by three forces in the directions AB, Aç, AD, and proportional to these three lines. First reduce the two Ac, AD to one AE, by completing the parallelogram ADEc. Then reduce 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 AP, AC, AD acted together. 41. Corol. 5. Hence also any single ^ -- 1: direct force AD, may be resolved into /* > two oblique forces, whose quantities / SF | and directions are AB, AC, having the same effect, by describing any paral- F. A lelogram whose diagonal may be / \, AD: and this is called the resolu- C I} tion of forces. So the force AD may be resolved into the two AB, Ac, by the parallelogram - ABDC ABD.c.; 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. 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, DR, 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 B, in that direction, as the oblique force expressed by, and acting in, the direction AB, produces in the same direction c5. And hence any given force AB, is to its effect in D3, 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 directlon S. 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 7 riangle, which are drawn Parallel to the Directions of the Forces AD, DB, cD. PRODUCE AD, BD, and draw cF, CE parallel to them. |