The motion in machines may be of two kinds. On the application of force to a machine previously at rest a certain movement is induced, and this movement for a time is accelerative ; but in some machines, after a while, the resisting power and the friction of the materials destroy the acceleration, when, unless the machine is subject to varia tions of force, as is the case with those Which are impelled by the wind or by the force of men or animals, the movement will become uniform. On the other hand, there are machines which are acted on by a con stantly accelerative power, as when a weight at one end of a rope pass ing over a wheel descends from au elevated place and raises a weight attached to the other extremity.
If the velocities of the points of application of the equivalent forces are uniform, a simple equation will express the dynamical equilibrium of the machine; for, F representing the moving power, and v the velocity with which it moves, f the force of resistance and v its velo city, we have in the case of equilibrium FV=fV; the first member of the equation is frequently designated the momentum of impulse, and the second the effect produced by the machine.
But the effect of a moving power on a machine in motion is different from that of an equal power on a machine at rest ; for the effect pro duced by any constant power in the former case depends upon its relative velocity, or the difference between its own velocity and that of the machine, and, by dynamics, it varies with the square of the rela tive velocity. Therefore, in order to introduce the absolute effect of a force into the equation of equilibrum in place of the efficient force, there must be given the velocity which would render the force quite ineffectual, as well as the actual velocity of the point of application : let the former be represented by v', and the latter by v; than F' repre senting the absolute force when the velocity is zero, and F the actual force when the velocity is v (F' being determined by the weight or resistance which is just sufficient to prevent the power from communi cating motion to the machine, and v' by the velocity with which the machine can move when the resistance is zero), : F : : : 2 whence 1r= V" • Then the first member of the equation F v =f v becomes v V" or, putting v' for v'—v, which gives v---=V—v', it becomes (v'—v') v' Now, in order to find the velocity which is consistent with the pro duction of the greatest effect by the machine, this expression, which represents the equivalent of f v, the efficient action of the machine, is io to 0 maximum; therefore, differentiating that expression, r' being the variable, and making the result sere, we have 2?-3e= 0; whence des —v 3 and, by substitution, v 10 Hence, if the resistance oppowd to the machine is susceptible of being varied, it should be rendered such that the velocity v of the point of application of the equivalent force is of the greatest velocity which the power can produce if unreeisted. Substituting
4 this value of v in the above equation for r we get v 9 r' ; therefore r v, the momentum of impulse, or the effect of the insehine, becomes 4 when that effect Is a maximum, the resistance remaining 27 unaltered.
If two bodies are connected together by a flexible line (supposed to be without weight) passing over a pulley at the common summit of a doubly inclined plane, the parts of the line being parallel to the suf fices of the two the relation between the weights may be determined so that the momentum of that which is to be raised by the descent of the other may be a maximum. Let p and se be the weights of the bodies. or the forces of gravity acting on them vertically, and let and I be the respective inclinations of the planes on which they are placed, to the horizon; then p sin. e and w sin. r are the forces of gravity on the planes, and consequently gig 8—w sin 6' is the ae p + te eelerative force by which p descends.
Now, by dynamics, the velocity of a body varies with the force and Cute ; therefore, v representing the velocity of p or so, and f the time of motion, p Fin sin 0' p+ w end consequently the momentum ter varies with this expression is to be a maximum; therefore, differentiating it, so being the variable, and making the result equal to zero, the value of re may be found in terms of p by a quadratic equation: thus, the required relation may be obtained.
If it were required to find, in any machine which when reduced to its most simple state may be considered as a lever or a wheel and axle, the ratio of the velocity of the moving power to that of the resistance to be overcome when the latter is a maximum, tho following process may be used.