THE SCREW.

101

256. In practice the forms of the threads of screws may vary, as we see exemplified in the annexed diagrams.

M

G

F

257. We may obtain in the following way some notion of the most essential characteristic of the Screw, namely its making at every point the same angle with the straight lines parallel to the axis of the cylinder. Let ABNM be any rectangle. Take any point Cin BN, and make CD, DE, EF,... all equal to BC. Join CA, and through D, E, F,...draw straight lines parallel to CA, f meeting AM at the points c, d, e,...respectively. Then if we conceive ABNM to be formed into the convex surface of a right cylinder, the straight lines AC, CD, dE, eF,...will compose a connected curve which takes the shape of a thread of a Screw, supposing the thread to be excessively fine. In this diagram BC represents what is called the distance between two consecutive threads of the Screw.

e

d

с

B

E

258. Suppose the axis of the Screw to be vertical, and let a Weight W be placed on the Screw. Then the Weight, by its tendency to descend, would cause the Screw to turn round in the block unless this motion were prevented by some Power. We will suppose this Power P to act at the end of a horizontal arm perpendicular to the lever, and horizontally; the arm is firmly attached to the cylinder, as is shewn in the diagram of Art. 255, in which the arm is represented as attached to the cylinder at A. The distance between the axis of the cylinder and the point of application of the Power we shall call the Power-arm.

It is found that when there is equilibrium P is to Win the same proportion as the distance between two consecutive threads of the Screw is to the circumference of the circle having the Power-arm for radius. The reasoning on which this depends is not simple enough to find a place here, so that this may be taken as an experimental fact. It is interesting however to observe its agreement with the principle of Art. 208, as illustrated in Arts. 248 and 251. For suppose the Power to be a little greater than is necessary for equilibrium, then the Weight will be moved; by turning the Screw once round the Weight will be raised through a space equal to the distance between two consecutive threads. Also the whole space passed over by the end of the Power-arm, estimated in the direction of the Power, consists of a multitude of small spaces which are together equivalent to the circumference of the circle having the Power-arm for radius.

259. The most common use of a screw is not to support a Weight, but to exert a Pressure. Thus suppose a fixed horizontal_board above the body denoted by W in the diagram of Art. 255; then by turning the Screw the body will be compressed between the head of the Screw and the fixed board. A bookbinder's press is an example of this mode of using the Screw. The proportion between P and W will be that stated in Art. 258, where W now denotes the whole force exerted parallel to the axis of the Screw by the body which is compressed; a force arising partly from the weight of the body, but mainly from the resistance which it offers to compression. The Screw-pile is another exemplification of the same thing; it is used for foundations which are to be laid under water. The thread of a Screw is cut out in the lower part of a wooden or metal pile; and by means of a capstan the pile is gradually screwed down to the depth which it is required to take: this process is found to succeed where it would be practically impossible to drive a pile down by blows.

260. In practice there will be much friction in the use of a screw; in the familiar case to which we allude at the beginning of Art. 255 this friction is in fact the Weight which the Power has to overcome.

XVII. COMPOUND MACHINES.

261. We have already spoken of the mechanical advantage of a machine, and have defined it to be the proportion of the Weight to the Power when the machine is în equilibrium: see Art. 207. Now we might theoretically obtain any amount of mechanical advantage by the use of any of the Mechanical Powers. For example, in the Wheel and Axle the advantage is the proportion of the radius of the Wheel to the radius of the Axle, and this proportion can be made theoretically as great as we please; but practically if the radius of the Axle is very small the machine is not strong enough for use, and if the radius of the Wheel is very great the machine becomes of an inconvenient size. Hence it is found advisable to employ various compound machines, by which great mechanical advantage may be obtained, combined with due strength and convenient size. We will now consider a few of these compound machines.

262. Combination of Levers. Let AB, BC, CD be three Levers, having fulcrums at K, L, M respectively.

Suppose all the Levers to be horizontal, and let the middle Lever have each end in contact with an end of one of the other Levers. Suppose the system in equilibrium with a Power P acting downwards at A, and a weight W acting downwards at D. It is easy to see that equilibrium can be secured by a proper adjustment of P and W; for P tends to raise the right-hand end of the Lever which has K for its fulcrum; thus the left-hand end of the Lever which has L for its fulcrum is pressed upwards, and therefore the right-hand end of the same Lever is pressed downwards: then the left-hand end of the Lever having M for its fulcrum is pressed downwards, and therefore the right-hand end of the same Lever is pressed

upwards, and if this upward pressure is sufficient W will be supported. It is found by theory and by trial that the advantage of this combination of Levers is expressed by the product of the numbers which express the advantages of the separate Levers. For example, suppose that AK is 3 times KB, that BL is 4 times LC, and that CM is 5 times MD; then the advantages of the Levers separately are expressed by 3, 4, and 5 respectively, and the advantage of the combination is expressed by 3 x 4 x 5, that is by 60. Hence any Power at A will support a Weight of 60 times that amount at D. If we suppose the Power to be a little greater than is necessary for equilibrium the Weight will be moved, but in order to raise the Weight through any space the Power must descend through 60 times that space.

263. Combinations of Wheels and Axles are often used. The Wheel of each of the pieces which form the combination is made to act on the Axle of the next by means of teeth or of a strap. It is found by theory and by trial that the advantage of this combination is expressed by the product of the numbers which express the advantages of the separate pieces.

264. The Differential Axle, or Chinese Wheel.

P

W

This machine may be considered as a combination of the Wheel and Axle with a single moveable Pully. Two

cylinders of different radii have a common axis with which they are firmly connected; the axis is supported in a horizontal position so that the two cylinders can turn as one body round the axis. A string has one end fastened to the larger cylinder, is coiled several times round the cylinder, then leaves it, passes under a moveable Pully and is coiled round the smaller cylinder to which the other end is fastened. The string is coiled in opposite ways round the two cylinders, so that as it winds off one it winds on the other. A Weight W is hung from the moveable Pully; and the equilibrium is maintained by a Power P applied at the end of a handle attached to the axis. It is found by theory and by trial that there is equilibrium on this machine when the Power is to the Weight in the same proportion as half the difference of the radii of the two cylinders is to the length of the arm at which the Power acts. Thus by making the difference of the radii of the two cylinders sufficiently small we can secure any amount of mechanical advantage.

265. It is not difficult to shew that the preceding statement is consistent with the principle of Art. 208. For suppose the Power to be a little greater than is necessary for equilibrium; thus the Weight will be raised. Let the Power describe the circumference of the circle of which the Power-arm is the radius. Then from the smaller cylinder a piece of the string is unwound equal in length to the circumference of the cylinder; and on the larger cylinder a piece of the string is wound equal in length to the circumference of the cylinder. Thus the excess of the circumference of the larger cylinder over the circumference of the smaller is equal to the whole length of string which is removed from the hanging position; so that each of the two vertical portions is shortened by half this length, which is therefore the space through which the Weight is raised. Thus the same number which expresses the proportion of the Weight to the Power when there is equilibrium, expresses also the proportion of the space passed over by the Power to the space passed over by the Weight when there is motion.