shall come in contact with every tooth of the wheel in succession before it engages with the same tooth a second time. This is effected by making the number of teeth and the number of leaves such that no integer will divide them exactly; or, arithmetically speaking, by making them prime to each other. Thus the number of the teeth is made just one more than a number exactly divisible by the number of leaves. The extra cog is called the hunting cog. Suppose, for instance, there are eight leaves, and the diameter of the wheel is required to be about eight times the diameter of the pinion. If the wheel had 64 teeth, it would be exactly eight times the diameter of the pinion; but the teeth and leaves would not hunt. By cutting the wheel with 65 teeth, and supposing the leaf A to be first engaged with the tooth B, it will, after eight revolutions of the pinion, be engaged with the tooth before B, and after eight more revolutions of the pinion it will be engaged with the second tooth before B. Hence every revolution of the wheel is of a revolution behind, and the pinion must revolve 8 x 65, or 520 times before the leaf A again engages with the tooth B. Hunting cogs, however, are only necessary where the strain on the cogs is great, as in mill-work.

5

Lantern. Sometimes the pinion is furnished with a lantern (fig. 50), A, instead of ordinary teeth. This

Fig. 50.

consists of two circular discs, BB, with cylindrical teeth between them. This form of pinion is used where the pressure is so great that ordinary leaves would be in danger of breaking.

.

Rack and Pinion.-If a small wheel with cogs be caused to work with a bar, B (fig. 51), having cogs the same distance apart as those of the wheel, the motion of A round its axis will cause B to move longitudinally. This is called a rack 1 and pinion. It is evident in this case that the rack B is analogous to the rope in the wheel and axle.

To find the Mechanical Efficiency of a Crane.— Divide the number of the driven teeth by the number of the drivers, and the quotient will be the relative velocity, which, multiplied by the length of the winch and force in pounds, and Fig. 51. divided by the radius of the barrel, will give the weight the crane is capable of raising.

Example.-A force of 18 lbs. is applied to a winch, which is 8 inches long; the pinion has 6 teeth, and the wheel 72; and the barrel is 6 inches in diameter. What weight will be raised?

72

6

If w=winch, r-radius of barrel, P=power, v=velocity, and W weight; then

v w P

Wr

and P =

v w

CHAPTER IX.

THE MECHANICAL POWERS (continued).

3. THE PULLEY.

Preliminary. When we wish to apply force to a body, we can do so by the direct application of the hand; by pushing or pulling with a stick; or by pulling a rope attached to the body. The transmission of the power through the stick is due to its rigidity. The transmission of the power through the rope is due to

1 A.-Sax., ræccan, to reach, extend.

2 L., rigidus, stiff.

its inextensibility. A pulling force, therefore, applied to one end of a rope, has an equal effect at the other end. Owing also to the flexibility of a rope, this force in one direction may be made to balance an equal force in any other direction. Thus, if we have a power acting in the direction A P (fig. 52), it can be made to exert an equal power in the direction W A, by passing

the rope over a fixed point, A, situated at the point of junction of the lines PA and W A. Any appreciable excess of P over W, will cause the rope to move towards P. The point A being fixed, and the rope flexible and inextensible, if P moves any distance in the direction AP, W will move an equal distance in the direction W A.

This

A force applied by means of a cord is called tension.s Definitions and Description.-As no rope is perfectly flexible, the friction in passing over a fixed object at A, would be so great as to absorb a great portion of the power. It is usual, therefore, to pass the rope over a wheel, A (fig. 53), turning freely on an axle. wheel and axle is called a pulley. The wheel, usually called the sheave, has a groove cut on its circumference, to allow the rope to pass over it without slipping off. The wheel turns freely on an axle, or pin, within a case called the shell; and the whole apparatus of shell, pin, and sheave, is called a block.

1 L., in, not; and extensibilis, able to be stretched. flexibilitas, from flecto, to bend.

3 L., tensus, stretched.

2 L.,

The rope or iron band passing round the block, with a hook attached, for the purpose of fixing the block in any required position, is called the strap, or strop.

Varieties of Blocks.-Blocks are of various kinds. When they contain one sheave (fig. 54), they are called

1

Fig. 54.

Fig. 55.

Fig. 56.

single blocks; when two sheaves (fig. 55), double blocks; when three sheaves (fig. 56), threefold blocks, and so on. Large blocks are called purchase blocks.

8

Fig. 57

Fig. 58.

Fig. 59.

Sometimes blocks have one sheave above the other. When the sheaves are equal in size, the block is called a sister block (fig. 57); when one is larger than the other, it is called a fiddle block (fig. 58); and when the plane of one sheave is in a direction contrary to that of the other, it is called a shoe block (fig. 59).

The Rope. The rope passing over the sheave is called the fall. The part of the fall to which the power is applied, is called the hauling part; and the part to which the weight is applied, is called the standing part. A system of blocks is called a tackle; and when the blocks are very large, and are used for the purpose of raising heavy weights, the system is called a purchase.

When chain is used instead of rope, the block over which it passes is called a gin.

Use of the Pulley.-By means of the pulley and flexible cord, we are enabled to balance a force exerted in one direction by an equal force exerted in the same or another direction. The pulley is only made use of to change the direction of the power, and may be considered to be a lever of the first kind. Thus, fig. 60 is a section of the sheave of a pulley, in which A

पि N

Fig. 60.

Fig. 61.

is the axle or pin. A force, D, in the direction CD, acting on the arm AC of the lever BA C, is made to balance a force, E, acting in the direction EB, applied to the arm B of the lever. These arms being equal in length, equal forces applied to the arms will produce equilibrium. It is evident, therefore, that there is no power gained by the pulley ITSELF.

The rope possessing inextensibility, the power necessary to balance a weight, is the same, whether the two parts of the rope are parallel, as A B (fig. 61), or divergent, as AC, AD. Consequently the tension of every part is equal.

Omitted Resistances.-To ascertain the mechanical efficiency of a system of pulleys, it will be convenient to consider the ropes to be perfectly flexible, to have neither weight nor thickness, and the sheaves to move without friction (see p. 44).

To estimate the working efficiency of any system, the above detracting forces must be added to the power necessary to raise a given weight.

Fixed Pulleys.-Pulleys are either fixed or movable. In the use of fixed pulleys there is no advantage, since the power and weight are necessarily equal, but they