CA EL value of p, we have w.E. — c a . —. And in like manner may the relation between the power and resistance be found, in the case of equilibrium, whatever be the number of wheels and axles.
It is to be understood, in the above description, that the axles of the two wheels M N and as are supposed to be parallel to one another and to the horizon ; and that the parts of the string b a df are in a vertical plane perpendicular to those axes, in order to avoid the reductions which would be necessary on account of a loss of power resulting from an oblique action of the forces at a and b. The forces acting in AP and SW, or sc, are also supposed to be exactly or very nearly in one vertical plane, in order to avoid the strain on the axle which would otherwise take place. [MATERIALS, STRENGTH OF.) If the string passing over the circumference of the wheel Rs and the axle cn were to cross itself, as represented by the lines bdaf, the relation between the powers would be the same as before, but the weight w' would be raised in the direction ter' instead of ter.
It is easy to perceive that (as in the lever and other mechanical powers) the spaces described by the weights n and w, in a given time, when in motion, are to one another in the inverse ratio of those weights ; for the spaces described are respectively equal to the lengths of the strings which pass over the circumferences of the wheel-and-axle in the given time; and these lengths are proportional to the circum ferences, or radii, that is, inversely as the weights acting at the circumferences.
Hence the advantage in the wheel-and-axle may be increased either by increasing the radius of the wheel, or by diminishing that of the axle. In the latter case, of course, the axle would soon become too weak to sustain the weight. This is beautifully avoided by the use of a composrul axle, one part of which is of smaller radius than the other. One end of the cord carrying w is wound round the thicker, and the other, in a contrary direction, round the thinner part. As r descends, some of the cord unwinds from the thinner axle, while another part is wound up round the thicker ; hut as the latter part, of course, exceeds the former in length, the weight is raised in this proportion. Thus we may have an axle of virtually vanishing radius, and may, con sequently, almost indefinitely increase the power, but, of course, only at the expense of time.
Taking the measurements as in fig. I, and representing the radius of the thicker axle by ire, and that of the thinner by a quantity a, less than ; since the whole weight w is supported by the two parts of the cord, the tension of the cord = 4w.
Hence, by mechanics, taking moments about c, we get,