dv by the differential of the time, or analytically F= all we know about it. which is dt 6. The direction of a force is the straight line in which it causes a body to move. This is known by experience only. 7. In dynamics, force is proportional to the indefinitely small space caused to be moved over in a given indefinitely small time. 8. Velocity is the space moved over in a given time, how small soever the parts may be into which the interval is divided. 9. The velocity of a body moving uniformly, is the straight line or space over which it moves in a given interval of time; hence if the velocity v be the space moved over in one second or unit of time, vt is the space moved over in t seconds or units of time; or representing the space by s, s=vt. 10. Thus it is proved that the space described with a uniform motion is proportional to the product of the time and the velocity. 11. Conversely, v, the space moved over in one second of time, is equal to s, the space moved over in t seconds of time, multiplied 12. Hence the velocity varies directly as the space, and inversely as the time; and because t = 13. The time varies directly as the space, and inversely as the velocity. 14. Forces are proportional to the velocities they generate in equal times. The intensity of forces can only be known by comparing their effects under precisely similar circumstances. Thus two forces are equal, which in a given time will generate equal velocities in bodies of the same magnitude; and one force is said to be double of another which, in a given time, will generate double the velocity in one body that it will do in another body of the same magnitude. 15. The intensity of a force may therefore be expressed by the ratios of numbers, or both its intensity and direction by the ratios of lines, since the direction of a force is the straight line in which it causes the body to move. 16. In general, a line expressing the intensity of a force is taken in the direction of the force, beginning from the point of application. 17. Since motion is the change of rectilinear distance between two points, it appears that force, velocity, and motion are expressed by the ratios of spaces; we are acquainted with the ratios of quantities only. Uniform Motion. 18. A body is said to move uniformly, when, in equal successive intervals of time, how short soever, it moves over equal intervals of space. 19. Hence in uniform motion the space is proportional to the time. 20. The only uniform motion that comes under our observation is the rotation of the earth upon its axis; all other motions in nature are accelerated or retarded. The rotation of the earth forms the only standard of time to which all recurring periods are referred. To be certain of the uniformity of its rotation is, therefore, of the greatest importance. The descent of materials from a higher to a lower level at its surface, or a change of internal temperature, would alter the length of the radius, and consequently the time of rotation: such causes of disturbance do take place; but it will be shown that their effects are so minute as to be insensible, and that the earth's rotation has suffered no sensible change from the earliest times recorded. 21. The equality of successive intervals of time may be measured by the recurrence of an event under circumstances as precisely similar as possible: for example, from the oscillations of a pendulum. When dissimilarity of circumstances takes place, we rectify our conclusions respecting the presumed equality of the intervals, by introducing an equation, which is a quantity to be added or taken away, in order to obtain the equality. Composition and Resolution of Forces. fig. 1. m A 22. Let m be a particle of mat.C ter which is free to move in every direction; if two forces, repre sented both in intensity and direction by the lines mA and mB, be applied to it, and urge it towards C, the particle will move by the combined action of these two forces, and it will require a force equal 7 to their sum, applied in a contrary direction, to keep it at rest. It is then said to be in a state of equilibrium. 23. If the forces mA, mB, be applied to a particle m in contrary A. fig.2. directions, and if mB be greater than mA, the particle m will be put in motion by the difference of these forces, and a force equal to their difference acting in a contrary direction will be required to keep the particle at rest. 24. When the forces mA, mB are equal, and in contrary directions, the particle will remain at rest. 25. It is usual to determine the position of points, lines, surfaces, and the motions of bodies in space, by means of three plane surfaces, oP, oQ, OR, fig. 3, intersecting at given angles. The intersecting or co-ordinate planes are generally assumed to be perpendicular to each other, so that xoy, roz, yoz, are right R B fig. 3. A 22 fig. 4. angles. The position of or, oy, oz, the axes of the co-ordinates, and their origin o, are arbitrary; that is, they may be placed where we please, and are therefore always assumed to be known. Hence the position of a point m in space is determined, if its distance from each co-ordinate plane be given; for by taking oA, oB, oC, fig. 4, respectively equal to the given distances, and drawing three planes through A, B, and C, parallel to the co-ordinate planes, they will intersect in m. 26. If a force applied to a particle of matter at m, (fig. 5,) make it approach to the plane oQ uniformly by the space mA, in a given time t; and if another force applied to m cause it to approach the plane oR uniformly by the space mB, in the same time t, the particle will move in the diagonal R R B m 10 B m fig. 5. e mo, by the simultaneous action of these two forces. For, since the forces are proportional to the spaces, if a be the space described in one second, at will be the space described in t seconds; hence if at be equal to the space mA, and bt equal to which is the equation to a straight line mo, passing through o, the origin of the co-ordinates. If the co-ordinates be rectangular, α is the tangent of the angle moA, for mB = oA, and oAm is a b right angle; hence oA: Am::1: tan Aom; whence mA = oA x tan Aom = mB. tan Aom. As this relation is the same for every point of the straight line mo, it is called its equation. Now since forces are proportional to the velocities they generate in equal times, mA, mB are proportional to the forces, and may be taken to represent them. The forces mA, mB are called component or partial forces, and mo is called the resulting force. The resulting force being that which, taken in a contrary direction, will keep the component forces in equilibrio. 27. Thus the resulting force is represented in magnitude and direction by the diagonal of a parallelogram, whose sides are mA, mB the partial ones. y m B fig. 6. x 28. Since the diagonal cm, fig. 6, is the resultant of the two forces mA, mB, whatever may be the angle they make with each other, so, conversely these two forces may be used in place of the single force mc. But mc may be resolved into any two forces whatever which form the sides of a parallelogram of which it is the diagonal; it may, therefore, be resolved into two forces ma, mb, which are at right angles to each other. Hence it is always possible to resolve a force me into two others which are parallel to two rectangular axes or, oy, situate in the same plane with the force; by drawing through m the lines ma, mb, respectively, parallel to or, oy, and completing the parallelogram macb. 29. If from any point C, fig. 7, of the direction of a resulting force mC, perpendiculars CD, CE, be drawn on the directions of the 'y P B C fig. 8. A R 30. Let BQ, fig. 8, be a figure formed by parallel planes seen in perspective, of which mo is the diagonal. If mo represent any force both in direction and intensity, acting on a material point m, it is evident from what has been said, that this force may be resolved into two other forces, mC, mR, because mo is the diagonal of the parallelogram mCoR. Again mC is the diagonal of the parallelogram mQCP, therefore it may be resolved into the two forces mQ, mP; and thus the force mo may be resolved into three forces, mP, mQ, and mR; and as this is independent of the angles of the figure, the force mo may be resolved into three forces at right angles to each other. It appears then, that any force mo may be resolved into three other forces parallel to three rectangular axes given in position: and conversely, three forces mP, mQ, mR, acting on a material point m, the resulting force mo may be obtained by constructing the figure BQ with sides proportional to these forces, and drawing the diagonal mo. 31. Therefore, if the directions and intensities with which any number of forces urge a material point be given, they may be reduced to one single force whose direction and intensity is known. For example, if there were four forces, mA, mB, mC, mD, fig. 9, acting on m, if the resulting force of mA and mB be found, and then that of mC and mD; these four forces would B fig. 9. y be reduced to two, and by finding the resulting force of these two, the four forces would be reduced to one. 32. Again, this single resulting force may be resolved into three |