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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. Cord. 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
B AB, AC, are equivalent to the single direct force AD, which may be compounded of these two, by drawing the diagonal
E of the parallelogram. Or they are equivalent 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 1). 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 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 reduce the two AE, AB to one Af by the parallelogram A EFB. 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 AD, 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 paral
E. lelogram whose diagonal may be AD; and this is called the resolution of forces. So the force AD may be resolved into the two AB, AC, by the parallelogram
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.
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 as 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 CB. And hence any given force AB, is to its effect in de, 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.
44. If Tl:ree 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 Pai rallel to the Directions of the Forces AD, DB, CD. PRODUCE AN, BD, and draw CF, ce parallel to them.
Then the force in cd is equivalent to the two AD, BD, by the supposition; but the force cp is also equivalent to the two ED and ce or FD; therefore, if cd represent the force C, then it will represent its opposite
F force A, and ce, or F., its opposite force B.
Consequency the three forces A, B, C, are proportional to DE, CE, CD, the three lires parallel to the directions in which they act,
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 one 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. 79 any two of the forces may be reduced to one force acting in the same plane; then this last force and another
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. Corel. 4. If one of the forces, a 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 Ic; and the force or tension of the other string Bd, is to the weight c, or tension of cD, as CE 10 CD.
49. Corol. 5. If three forces be in equilibrio by their mus tual actions; the line of direction of each force, as no, pa ses 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.
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 Argle
of Incidence. Or, the Force on the Surface is to the same if it had acted Perpen
dicularly, 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 ce, 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 X. 54. If one Body A, strike another Body B, which is either at Rest
or moving towards the Body A, or moving from it, but with a less Velocity than that of A; then the Momenta, or Quantities of Motion, of the two Bodies, estimated in any one Direction, will be the very same after the Stroke that they were before it.
For, because action and re-action are always equal, and in contrary directions, whatever momentum the one body gains one way by the stroke, the other must just lose as much in the same direction; and therefore the quantity of motion in that direction, resulting from the motions of both the bodies, remains still the same as it was before the stroke.
55. Thus, if a with a momentum of 10, strike B at rest, and commu
с nicate to it a momentum of 4, in the direction AB. Then A will have only a momentum of 6 in that direction; which, together with the momentum of B, viz. 4, make up still the same momentum between them as before, namely 10.
56. If B were in motion before the stroke, with a momentum of 5, in the same direction, and receive from A an additional momentum of 2. Then the motion of A after the stroke will be 8, and that of B, 7; which between them make 15, the same as 10 and 5, the motions before, the stroke.
57. Lastly, if the bodies move in opposite directions, and meet one another, namely, A with a motion of 10, and B, of 5; and a communicate to B a motion of 6 in the direction AB of its motion. Then, before the stroke, the whole motion from both, in the direction of AB, is 10 – 5 or 5. But, after the stroke, the motion of A is 4 in the direction AB, and the motion of B is 6 5 or 1 in the same direction AB; therefore the sum 4 + 1, or 5, is still the same motion froin both, as it was before.
58. The Motion of Bodies included in a Given Space, is the same
with regard to each other, whether that Space be at Rest, or move uniformly in a Right Line.
For, if any force be equally impressed both on the body and the line in which it moves, this will cause no change in