It is also a maxim assented to by engineers that the impelled point of a machine should not be allowed to move with a greater velocity than that with which the motive power can act upon it ; since in this case the excess of velocity in the machine will be employed in accele rating the motion of the power, and thus the general acceleration of the machine will suffer a corresponding diminution. The velocities of the impelled and working points should therefore be properly adjusted to the pressures, the inertia, and the friction, in order that all possible advantage may be derived from the machine.
A just estimate of the power of a machine ought to include the effects of all the momentary accelerations and retardations of motion to which it is subject, and all the losses arising from inertia and friction ; but as the introduction of these circumstances would excessively com plicate the investigation, it is usual to make the measure of the power depend on the Condition that the impelled and working points shall be in a state of uniform motion. For then, agreeably to the property of the simple lever, the velocities of those extreme points will be in versely proportional to the forces which would be in equilibrio at the same points ; and the rule propounded by Euler is, that in every machine, simple or complex, the pressure at the impelled point, multiplied by the velocity of that point, is equal to the product of the resistance at the working point by the velocity of the same point ; or the momentum of resistance (commonly called the performance of the machine) is equal to the momentum of impulse. Whatever objection may be made to this rule with respect to the measure of the power in action, no doubt can exist that it affords a correct value of the useful effect; and the latter may therefore be measured by the weight which might be raised by the machine to a given height vertically in a given time. The fact is 'sufficiently evident when a mass of any material is to be conveyed from one place to another, or when a body is let fall on any object from a given height. It follows that, if an algebraical expression be obtained for the momentum of the resistance in terms involving that resistance, the motive power and the distances of their points of application from the axis of motion, on making the diftrential of that expression equal to zero, the ratio Of the resistance to the moving power, When the useful effect of the machine is a maximum, may be found from the resulting equation.
If st represent the mass of any body moved, w its weight, which is equal to mg,g feet) expressing the force of gravity ; also, if n be the height to which the body may be raised in one second of time, and v the velocity which a body would acquire by falling vertically through a height equal to n, we shall have, by the theory of motions, 2g n ; whence w it (the momentum of resistance, or the useful effect of a machine) sl. This last expression is designated the
firing, or attire, force of the body moved; and it expresses the force of a body in motion, in contradistinction to the simple pressure exer cised by a body at rest It is commonly asserted that, in the employment of machinery, as much is lost in time as is gained in power, or that the momentum of resistance is proportional to the power employed; but this rule requires some modification. It can be shown to hold good in a well constructed machine when the object moved resists by its inertia only ; but if the inertia is but a small part of the resistance, the momentum of the latter, or the work done, is found to increase nearly as the square of the power employed.
The various ingenious contrivances 'which have been adopted in machines for regulating the velocities, and for converting oue species of motion into another, are noticed in the article WHEELS.
In investigating the relations between the motive powers and resist ances to be overcome, which render the effect produced a maximum with respect to quantity of motion or velocity, or which render the time of the performdnce a minimum, it is usual to consider that in every machine there is a certain point at which, if the moving power were immediately applied, and a certain point at which, if the resist ance to be overcome were immediately applied, the effect produced would be the same as that which is produced by the machine in its actual state. Thus, in a machine consisting of several wheels and axles with which weights are raised by means of ropes passing over their circumferences, the points at which the ropes immediately con nected with the moving power and resistance are tangents to the cir cumferences are those at which the forces are ,conceived to be applied. Also, if several forces act at once as moving powers, and resistances are to be overcome at once at various points, the resultant of all the forces and that of all the resistances must be taken for the effective moving power and the effective resistance. The points of application of these resultant forces are to be found, and at these points such resultant forces are conceived to be applied : the effects of friction, the rigidity of ropes, and every other impediment to the action of the machine, are also to be estimated and applied as additions to the resistance which is to be overcome; and thus a complex machine is reduced to an equivalent mechanical power of a simple form. The velocities of the points at which these resultant forces are con ceived to be applied are equal to the velocities of the power and resistance.