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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 agaist 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 all be 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. Corol. 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 cr, as ce to cv.
49. Corol. 5. If three forces be in equilibrio by their mutual 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.
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 ▲, 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 вс perpendicular, and AC parallel to De.
For, by prob. 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 cв, 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 cв, 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 ex
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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 vemains still the same as it was before the stroke.
direction AB. Then A will have
only a momentum of 6 in that direction; which, together with the momentum of в, viz. 4, make up still the same momentum between them as before, namely, 10.
56. If в were in motion before the stroke with a momen tum 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 в 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 в is 6-5 or 1 in the same direction AB; therefore the sum 4+ 1, or 5, is still the same motion from 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
the motion of the body along the right line. For the same reason, the motions of all the other bodies, in their several directions, will still remain the same. Consequently their motions among themselves will continue the same, whether the including space be at rest, or be moved uniformly forward. And therefore their mutual actions on one another, must also remain the same in both cases.
59. If a Hard and Fixed Plane be struck by either a Soft or a Hard Unelastic Body, the Body will continue on it. But if the Plane be struck by a Perfectly Elastic Body, it will rebound from it again with the same. Velocity with which it struck the Plane.
FOR, since the parts which are struck, of the elastic body, suddenly yield and give way by the force of the blow, and as suddenly restore themselves again with a force equal to the force which impressed them, by the definition of elastic bodies; the intensity of the action of that restoring force on the plane, will be equal to the force or momentum with which the body struck the plane. And, as action and re-action are equal and contrary, the plane will act with the same force on the body, and so cause it to rebound or move back again with the same velocity as it had before the stroke.
But hard or soft bodies, being devoid of elasticity, by the definition, having no restoring force to throw them off again, they must necessarily adhere to the plane struck.
60. Corol. 1. The effect of the blow of the elastic body, on the plane, is double to that of the unelastic one, the velo city and mass being equal in each.
For the force of the blow from the unelastic body is as its mass and velocity, which is only destroyed by the resistance of the plane. But in the elastic body, that force is not only destroyed and sustained by the plane; but another also equal to it is sustained by the plane, in consequence of the restoring force, and by virtue of which the body is thrown back again with an equal velocity. And therefore the intensity of the blow is doubled.
61. Corol. 2. Hence unelastic bodies lose, by their collision, only half the motion lost by elastic bodies; their mass and velocities being equal. For the latter communicate double
the motion of the former.
62. If an Elastic Body A impinge on a Firm Plane DE at the Point B, it will rebound from it in an Angle equal to that in which it struck it; or the Angle of Incidence will be equal to the Angle of Reflection; namely, the Angle ABD equal to the Angle FBE.
LET AB express the force of the body a in the direction AB`; which let be resolved into the two AC, CB, parallel and perpendicular to the plane. Take BE and cr equal to ac, and draw BF. Now action and re-action being equal, the plane will resist the direct force CB by another BC equal to it, and in a contrary direction: whereas the other ac, being parallel to the plane, is not acted on or diminished by it, but still continues as before. The body is therefore reflected from the plane by two forces BC, BE, perpendicular and parallel to the plane, and therefore moves in the diagonal BF by composition. But, because ac is equal to be or CF, and that BC is common, the two triangles BCA, BCF are mutually similar and equal; and consequently the angles at a and rare equal, as also their equal alternate angles ABD, FBE, which are the angles of incidence and reflection.
63. To determine the Motion of Non-elastic Bodies when they strike each other Directly, or in the same Line of Direction.
LET the non-elastic body в, moving with the velocity v in the direction Bb, and the body b with the velocity, strike each other. Then because the momentum of any moving body is as the mass into the velocity, BVM is the momentum of the body, B, and bv = in the momentum of the body b, which let be the less powerful of the two motions. Then, by prop. 10, the bodies will both move together as one mass in the direction BC after the stroke, whether before the stroke the body b moved towards c or towards B. Now, according as that motion of b was from or towards B, that is whether the motions were in the same or contrary ways, the momentum after the stroke, in direction BC, will be the sum of difference