moment, i.e. everywhere be normal, and hence so also are the pressures of the pulley on the rope (page 106). These pressures then are everywhere at right angles to the motion of the piece of rope they act on, and can therefore have no effect on the motion. Hence then TA and TB are effort and resistance, and there are no other forces. But velocity of A= velocity of B, so the velocity ratio is unity, and therefore so is the force ratio, .. TA=TB. We have then the principle that in the absence of friction the tension of a rope is unaltered by passing round a pulley, and the work we have done will not be affected by any motion of the pulley, so long as no moment be applied to it, i.e. we may apply any force we please through its centre without affecting our equations. We see then at once that we cannot obtain any mechanical advantage by the use of a single fixed pulley, i.e. pulley with fixed centre, as Fig. 80, for we have by our principle P=W. But now in addition to a fixed pulley let us take a movable pulley.. W is not now fastened to the rope to which P is applied, but to the framework of the movable pulley, and the rope, after passing round both, is led up and fastened to the frame of the fixed pulley. Let P be drawn down say 2 ft., then W rises, shortening both ab and cd, and neglecting the little deviation from parallelism they shorten equally; so each shortens I foot, which is therefore the rise of W, Fig. 82. 2 ft. ... Velocity ratio= = I ft. 2 the mechanical advantage being 2. We will now verify this. W is supported by ac and bd. The tensions in these are equal, and each equal P (see previous principle), as above. .'. W=2P, The actual construction of a block is shown here (Fig. 83), the hooks being for the attachment of ropes, so that this may be hung up to a fixed point, forming the fixed pulley, or W be hung to the hook, and it can Fig. 83. Fig. 84. Fig. 85. form the movable pulley. A pair of such blocks with the rope which goes round the sheaves is called a tackle, or system of pulleys. If we desire to still further increase the mechanical advantage, we can do so by using more than one sheave, say for example three as here shown (Fig. 84). We use a pair of such blocks, and call the whole a pair of three-sheaved blocks. The rope would pass in turn round an upper and under pulley, being finally fastened to the lower hook of the top or fixed block. Fig. 85 shows a diagrammatic representation of the run of the rope. We will now examine the general case when there are n plies of rope supporting the lower block, [" may be odd or even; in a pair of three-sheaved blocks it is 6; but we may have 3 sheaves at top and 2 at bottom, then the final fastening would be to the top hook of the lower block, and there would be 5 plies.] By our principle the tension all through the rope is P, therefore nP supports W, and or W=nP, Force ratio=n. Also if P move a distance, all the n plies shorten equal amounts, so that each shortens x/n, the rope remaining of unaltered length. All sorts of combinations of pulleys can be used for various purposes, but the same principle applies to all, and so we shall not examine their working; moreover, the one we have considered is of far more importance than all the rest together. Belt Connection.-Next we will consider the connection of two turning pairs. We have generally so far considered the magnifying of the effort as the effect sought after, but inseparably connected with this we have seen there is a modification of the velocity. For in all cases In some cases it is the modification of velocity which is chiefly aimed at, the alteration of effort which necessarily follows being regarded as of subsidiary importance, or even in some cases, e.g. the mechanism of a watch, of no importance at all. In this case we should not call the watch a machine but a mechanism, its sole object being the production of a certain motion. We shall then treat the present case, in the first instance, from the view of modification of motion, and deduce, when necessary, the accompanying change of effort. And this method has the advantage that our main treatment, being purely geometrical, will be just as true when we have to take frictional forces into account as now when we omit them, which would not be the case if we based it on the modification of the effort. [This will be seen in the examples considered in the succeeding chapter.] We have now then this problem : Given two turning pairs, i.e. two shafts turning in bearings fixed to the earth or to some framework, it is required to connect them so as to have a given velocity ratio. Ist Case. Where the distance apart of the centres is large. A and B are the two shafts, the bearings are not shown. Fig. 86. Fix now pulleys CD and EF on A and B respectively, and connect these by B) an endless belt or rope passing round them as shown. Then as A turns the belt turns with the pulley CD, and so causes the rotation of EF, i.e. of B. Let now AA angular velocity of A, The motion being turning, the velocity ratio will be one of angular velocities. Let "A radius of CD, "Bradius of EF. The principle governing the connection is that the total length of the belt is constant, also the lengths of the parts CE, EF, FD and DC, and that the belt does not slip on the pulleys. or Since the belt does not slip on CD, Speed of point C of belt = speed of point C of pulley, Similarly from EF = speed of any point on periphery, =AA.A (page 31). Speed of point E of belt ABB. But CE is of constant length, whence That is, the angular velocities of the shafts vary inversely as the radii of the pulleys. We must then take r ̧ and B such as to satisfy the above relation. The shaft A may represent the main shaft of a factory, and we see that by fitting to it pulleys, which drive by belts other pulleys on shafts fitted to different machines, we can from the one shaft obtain any number of different angular velocities in the other shafts. Length of Belt.-There are two ways in which the belt may be put on, either as in Fig. 86, called an open belt, or in Fig. 87, called a crossed belt. |