WHEEL-AND-AXLE, is a machine consisting usually of a cylinder to which a wheel is firmly united, so that the mathematical axes of both are coincident. The wheel and cylinder are of wood or metal, and the diameter of the former is greater than that of the latter.
A cylinder on the circumference of which are fixed exteriorly boards whose planes, if produced, would pass through the axis, and which (being turned by the force of running water, or by the weight of men in the act of stepping from one board to the next above it) is employed to raise a heavy body by means of a rope passing over a smaller cylinder on the same axis, as in the treadmill, is a simple machine of this kind : the same may be said of a hollow cylinder which, with its axle, is made to revolve by men or animals walking in the direction of its circumference, in its interior surface. The capstan, the windlass, and the helm-wheel of a ship are only so many different forms of the same class of machines. Frequently also the axle is made to carry a wheel with teeth on its circumference, in order that, by revolving, motion may be communicated to machinery : such are the wind and water mills which are employed for grinding corn.
When it is required to exhibit the mechanical properties of the wheel-and-axle, a weight representing the moving power is applied at cite extremity of a string which at the other extremity is attached to and passes round the circumference of the wheel ; and a weight, repre senting the resistance to be overcome, is applied in like manner at one end of a string which pasaws round the axle or cylinder. Let MN (ill h. 1) be a section passing through the wheel and cylinder perpen dicularly to their common axis, and let CA, or ca', and CB be the semi-diameters of the circles in that section : let r represent the moving power and w a weight to be raised, or held in equilibrio ; A r or AY, and Bw, being the directions of the strings to which those weights are attached; and for simplicity, let these lints be in one plane and coincident with tangents to the circles at A, or A', and at n. Hero it is evident that the mechanical power of the wheel-and-axle is the same as that of a lever of the first kind ; for (the thickness of the ropes and the weight and inertia of the materials being disregarded) the forces r and w acting perpendicularly to the arms CA and ca, the effect is the same as if those forces were applied immediately at the extremities of the straight line An, or of the bent line A'CB, and a being the fulcrum or point of support, we have, by the nature of the lever, in the case of equilibrium, The wheel-and-axle has manifestly however a great advantage over the simple lever, since the weight w may be raised to any height which is consistent with the lengths of the ropes, by winding the rope round the axle.
If the power r or r' do not act in the direction of a tangent to the circle, but in some other, as A r"; then letting fall CD perpen dicularly on eA, produced if necessary, we have, by the lever, If the ropes to which the weights are attached have sensible thick nesses, and it is thought proper to take those thicknesses into consideration, the ropes may be conceived to be reduced to their mathematical axes, and these to pass over the circumferences of the wheel and cylinder at distances equal to the semidianieters : thus, if r and ii be the seinidiameters of the ropes passing over those circum ferences, respectively, we obtain, in the case first supposed, If it be required to determine the pressures on the supports of a wheel-and-axle when the weights applied to it are in equilihrio, and the whole inachine is at rest, the investigation may be conducted in the following manlier :—Let the weight of the wheel be represented by A and that of the cylinder by H; also let at and a (in fig. 2) be the points on which the two pivots rest ; then 4a is evidently the pressure supported on each of the points at and sr, arising from the weight of the cylinder alone. Let the weight A be suppoaed to act at c, the centre of the wheel, and let cat =in, ca=n; then, by mechanics, a in like manner, A expresses the pressure at at; each of these pressures arising from the weight of the wheel alone.