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 I 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 I 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.

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, TB 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

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 T1 and TB are effort and resistance, and there are no other forces.

A

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,

W

Fig. 82.

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.