So the equation of energy becomes

Turning moment on shaft × 2′′=thrust × pitch of propeller.

Other Examples.—As an example of a fixed screw and moving nut we may take an arrangement used by Whitworth in making compressed steel shafts.

B

A is the mould, BB are screws, and the cross-piece

с

B

C is run down by turning the nuts DD at about 500 revolutions per minute. The projection on the cap fits into the top of the mould, and when the mould has been filled and C run down, the compression is produced by hydraulic pressure in the space inside the vessel E, forcing the mould up against C. The mould shown is for a piece of hollow shaft.

In the motion of the saddle of a screw-cutting lathe we have each piece partially fixed, for the leading screw can turn in its bearings but not move lengthways, while the saddle can slide along the bed of the lathe but not turn. The effort is applied to the screw, and the resistance to the point of the tool which is rigidly attached to, and therefore is for the time a part of, the saddle.

Fig. 78.

The Screw as an Inclined Plane.-Looking back to page 103 we have the equation

Px 2πr=Rxp.

Now this equation is the same as that for an inclined plane of height p, base 2πr, the effort acting horizontally (page 95).

Hence the velocity ratio also must be the same, which is easily seen to be the case, and in fact the cases are identical, except that the horizontal motion of the resist

ing body, ie, the slider in one case and nut in the other, is in the plane all in one direction, while in the screw it is in a circular path. Now this alteration can be effected by coiling the plane round in a cylinder of radius (Fig. 79), and then its slant side takes the form of a thread.

It is usual to consider the screw as derived in this

way, but we have preferred to obtain its properties by consideration of the actual mode of constructing it in practice. This mode, however, we must notice, presupposes the existence of another screw, viz. the leading screw of the lathe, while the present gives a screw independently of any previous screws.

We have here considered a screw as being virtually an inclined plane, in order to show the identity of the two ideas, but we shall not now use this new conception, since we do not gain by it. It will, however, be required in chap. vii.

EXAMPLES.

1. The draught of a waggon is 40 lbs. per ton. Assuming this to be constant irrespective of slope, compare the speed with which a horse could draw the waggon on a level with that with which he could pull it up a slope of 1 in 50; exerting energy at the same rate in each case. Ans. 2.12: I.

2. If the waggon above weigh 2 tons, find the greatest slope

up which two horses could pull it, supposing they can each exert a pull of 150 lbs. Ans. 1 in 28.

3. Why does a horse zigzag when pulling a load up a steep hill?

A loaded cart weighs 1 ton, constant_resistance 45 lbs. The horse can only exert a pull of 112 lbs.; how many times must he cross the road in going up a hill 150 yards long, rising 1 in 25, width of road 35 ft.? Ans. 12.

4. A truck weighing 24 tons rests on an incline at 30° to the horizontal. It is fastened up by a rope 6 ft. long, fastened to a hook in the truck 3 ft. from the ground, and connected at the other end to, Ist, a fixed point 3 ft. from the ground; 2d, a point on the ground. Find in each case the tension of the rope, and the pressure on the incline.

Ans. Tensions, 1, 1.3 tons; pressures, 1.97, 2.6 tons.

5. In question 10, page 75, the capstan is 3 ft. diameter. How many men would be required to turn it, each exerting a push of 40 lbs., and the distance of the resultant push on each bar from the centre of the capstan being 8 ft.? Ans. 75.

6. The pitch of a screw propeller is 14 ft., and the twisting moment applied to it is 120 tons-inches. Find the thrust.

Ans. 4 tons.

7. In question 9, p. 38, what force applied to the handle will lift I ton? Ans. 8 lbs.

CHAPTER VI

PULLEYS, BELTS, AND WHEEL GEARS

THE simple machines already considered have consisted practically of one pair. We will now consider some cases of the connection of two pairs.

Pulley Blocks.-The pairs here connected are not real but virtual pairs. Taking the case of a small weight lifting a large one, each weight forms a virtual sliding pair with the earth, and the pairs are connected by the pulley blocks and ropes, so that motion of the one causes a certain motion of the other. If the end of the fall, i.e. the part to which the effort is applied, be pulled in some other way than by a weight, there are some means generally by which it is guided in a straight path, and then any piece of it may be considered as forming, with the earth, a sliding pair.

[By the above manner of consideration the wheel and axle and screw are also connections of pairs. There is, however, a

further difficulty in pulleys, due to the rope connection, hence we place them in this chapter.]

Thus in Fig. 80, which is the simplest of all pulleys, the piece between P and the pulley may be taken as forming a sliding pair with the earth, being connected where it meets the pulley to the rope, and the effort applied by the hand say which is applying the effort P.

W

Fig. 80.

[It may seem strange to describe the piece of rope as being connected to the rope, because it is a part of the

latter, but it certainly is connected, and in fact by the closest of all possible connections.]

A set of pulleys, or of blocks, as they are usually called, consists of a rope or ropes passing round small wheels called sheaves, which rotate on pins. Now we must inquire why these sheaves are fitted, and if their motion relative to the pins forming a turning pair has any effect on the energy or work.

(a) B

It is a very common use for one pulley or sheave to place it as in Fig. 81 (a), in order to change the direction of a rope which passes over it. Now this effect could equally be obtained, as in (b), by passing the rope over a rounded surface; but then there would be considerable friction as the rope slid over the surface. The sheave then is fitted to avoid this friction, and now there is no slipping between the rope and sheave, but all the relative motion takes place at the surface of the pin, and thus the friction is very much reduced. The reason for fitting the sheave then is to change the direction of the rope without undue friction; but, being fitted, has it any effect in modifying the tension of the rope?

Fig. 81.

We are not considering friction at present, so we suppose the motion of the sheave on the pin to be frictionless, and in this case the answer to the question just asked is No. For let the motion be in the direction of the arrow, and consider the piece of rope AB as a body acted on by tensions TA, Tв at A and B respectively, and by the pressures of the pulley. These latter are everywhere normal to the pulley, because, since the pulley turns uniformly, and the pin being frictionless can exert no moment on it, it follows that the rope can exert no moment on it, so that the pressure of the rope on the pulley must have no