933

THEORY AND PRACTICE OF ENGINEERING.

Book II.

them, side by side, but under one another; for it is stated that in the trochlea are two pulleysturning on an axis; through the upper pulley the drawing rope is passed ; then it descendsand passes round the under pulley of an inferior trochlea, and returns to the under pulleyof the superior trochlea, when it descends to the inferior, and in a hole therein the end ofthe rope is fixed, the other end of the rope passing to the bottom of the machine. Theblocks used in hoisting the various parts of the Menai bridge were of different forms, asshown above, and were admirably suited to their various purposes; they were well pro-portioned to their works, and the ropes could be taken out easily, as indicated in the singlepulley.

JFheel and Axle consists of a cylinder which revolves about its axis, having a wheel oflarger diameter affixed to it; the weight is suspended by a cord which is wound round theaxle ; the power applied acts on the periphery of the wheel. The equilibrium is establishedwhen the power is such a multiple of the weight that acts on the axle, as the diameter of theaxle is of the diameter of the wheel; thus, if the diameter of the axle be 1 foot, and that ofthe wheel 10 feet, a power of 1 cwt. will be an equipoise of 10 cwt. or A ton: when thepower has turned the wheel once round, the axle has revolved once also; the power hasgone over a space indicated by the circumference of the wheel, whilst the weight hasbeen raised a height equal to the circumference of the axle.

The wheel and axle bears so strong a resemblance to the lever,that it is not improperly called a continual one; for if two cordsperfectly flexible are w'ound round two cylinders in opposite directionsth<£e will be an equilibrium if the drums or rollers are of a similardiameter; and the forces which operate upon any cylinders areuniversally as their radii, or as their diameters, of which they are thehalves : weights in turning round a cylinder exert the same power asthe arms or spokes of a lever equal in length to their semi-diameter :the cylinder is often united to a winch or lever ; and in this case theradii of the cylinder must be compared with the length of the winch.

Let us consider the wheel and axle to consist of a cylindrical axisC D, of any diameter, to which is fixed a wheel A B of a largerdiameter, and having its plane perpendicular to the axis, as well asits centre corresponding: the pivots F and D are those on which theaxle moves when the wheel is made to revolve.

There will be perfect equilibrium when the power is to the weight, as the axle is to theradius of the wheel. Let A B be the axle, O the wheel, W the weight applied to thepoint E, and let the power P be applied to the point

M in the direction M C. \ V 'H'

Let O be the centre of the wheel, and K that ofthe axle; through E draw a plane perpendicular tothe axis A B, cutting it at K ; therefore the radiusK E of the cylinder will be perpendicular to EW:through O draw OF, parallel to KE, and thisradius will be horizontal and in the plane of thewheel: let two forces W', W", each of which is equalto W, be applied in opposite directions to the pointF, and in the vertical line G H, the one W' acting inthe direction F G, and the other W /x in the directionFII. These two opposite forces destroying eachother, the equilibrium of the machine is not altered.

Join FE, meeting in the axis D, and bisectingK O in D; then KE = FO. Let us consider thestate of the forces W acting at E, and W x actingat F. Since W' = W, the effect of these forces

will be the same as 2 W acting on D : but as the point D is in the axis of the cylinder,which axis is immovable, the force 2 W can have no effect in altering the equilibrium orturning the machine round its axis.

The other forces, therefore, are P applied at M, and W" applied at F, and acting in thedirection of FH. We may now consider MOF as a bent lever whose fulcrum is O ; andsince the forces at M and F, in the direction MC, FH, perpendicular to the arms of thelever, are in equilibrio, these forces will be inversely as the length of the arms, viz.P:W"=OF. OMj but since FO = EK and W' = W we have

P : W = E K : OM.

Fig.1282.

When, therefore, there is an equilibrium in a wheel and axle, the power is to the weight,as the radius of the axle is to the radius of the wheel.

In the windlass the axle is horizontal, and in the capstan it is vertical, and the advantagesgained on the latter application is evident, as it affords the means for a number of workmen