[Ill.u.s.tration: _Fig. 122. Miter Gear Pitch_]

DIAMETRAL PITCH.--To ill.u.s.trate: If a gear has 40 teeth, and the pitch diameter of the wheel is 4 inches, there are 10 teeth to each inch of the pitch diameter, and the gear is then 10 _diametral pitch_.

CIRCULAR PITCH.--Now the term "circular pitch" grows out of the necessity of getting the measurement of the distance from the center of one tooth to the center of the next, and it is measured along the pitch line.

Supposing you wanted to know the number of teeth in a gear where the pitch diameter and the diametral pitch are given. You would proceed as follows: Let the diameter of the pitch circle be 10 inches, and the diameter of the diametral pitch be 4 inches. Multiplying these together the product is 40, thus giving the number of teeth.

[Ill.u.s.tration: _Fig. 123. Bevel Gears._]

It will thus be seen that if you have an idea of the diametral pitch and circular pitch, you can pretty fairly judge of the size that the teeth will be, and thus enable you to determine about what kind of teeth you should order.

HOW TO ORDER A GEAR.--In proceeding to order, therefore, you may give the pitch, or the diameter of the pitch circle, in which latter case the manufacturer of the gear will understand how to determine the number of the teeth. In case the intermeshing gears are of different diameters, state the number of teeth in the gear and also in the pinion, or indicate what the relative speed shall be.

[Ill.u.s.tration: _Fig. 124. Miter Gears._]

This should be followed by the diameter of the hole in the gear and also in the pinion; the backing of both gear and pinion; the width of the face; the diameter of the gear hub; diameter of the pinion hub; and, finally, whether the gears are to be fastened to the shafts by key-ways or set-screws.

Fig. 122 shows a sample pair of miter gears, with the measurements to indicate how to make the drawings. Fig. 123 shows the bevel gears.

BEVEL AND MITER GEARS.--When two intermeshing gears are on shafts which are at right angles to each other, they may be equal diametrically, or of different sizes. If both are of the same diameter, they are called bevel gears; if of different diameters, miter gears.

[Ill.u.s.tration: _Fig. 125. Sprocket Wheel._]

It is, in ordering gears of this character, that the novice finds it most difficult to know just what to do. In this case it is necessary to get the proper relation of speed between the two gears, and, for convenience, we shall, in the drawing, make the gears in the relation of 2 to 1.

DRAWING GEARS.--Draw two lines at right angles, Fig. 124, as 1 and 2, marking off the sizes of the two wheels at the points 3, 4. Then draw a vertical line (A) midway between the marks of the line 2, and this will be the center of the main pinion.

Also draw a horizontal line (B) midway between the marks on the vertical line (1), and this will represent the center of the small gear. These two cross lines (A, B) const.i.tute the intersecting axes of the two wheels, and a line (5), drawn from the mark (3 to 4), and another line (6), from the axes to the intersecting points of the lines (1, 2), will give the pitch line angles of the two wheels.

SPROCKET WHEELS.--For sprocket wheels the pitch line pa.s.ses centrally through the rollers (A) of the chain, as shown in Fig. 125, and the pitch of the chain is that distance between the centers of two adjacent rollers. In this case the cut of the teeth is determined by the chain.

CHAPTER XI

MECHANICAL POWERS

THE LEVER.--The lever is the most wonderful mechanical element in the world. The expression, _lever_, is not employed in the sense of a stick or a bar which is used against a fulcrum to lift or push something with, but as the type of numerous devices which employ the same principle.

Some of these devices are, the wedge, the screw, the pulley and the inclined plane. In some form or other, one or more of these are used in every piece of mechanism in the world.

Because the lever enables the user to raise or move an object hundreds of times heavier than is possible without it, has led thousands of people to misunderstand its meaning, because it has the appearance, to the ignorant, of being able to manufacture power.

WRONG INFERENCES FROM USE OF LEVER.--This lack of knowledge of first principles, has bred and is now breeding, so-called perpetual motion inventors (?) all over the civilized world. It is surprising how many men, to say nothing of boys, actually believe that power can be made without the expenditure of something which equalizes it.

The boy should not be led astray in this particular, and I shall try to make the matter plain by using the simple lever to ill.u.s.trate the fact that whenever power is exerted some form of energy is expended.

In Fig. 126 is a lever (A), resting on a fulcrum (B), the fulcrum being so placed that the lever is four times longer on one side than on the other. A weight (C) of 4 pounds is placed on the short end, and a 1-pound weight (D), called the _power_, on the short end. It will thus be seen that the lever is balanced by the two weights, or that the _weight_ and the _power_ are equal.

[Ill.u.s.tration: _Fig. 126. Simple Lever_]

THE LEVER PRINCIPLE.--Now, without stopping to inquire, the boy will say: "Certainly, I can understand that. As the lever is four times longer on one side of the fulcrum than on the other side, it requires only one-fourth of the weight to balance the four pounds. But suppose I push down the lever, at the point where the weight (D) is, then, for every pound I push down I can raise four pounds at C. In that case do I not produce four times the power?"

I answer, yes. But while I produce that power I am losing something which is equal to the power gained. What is that?

[Ill.u.s.tration: _Fig. 127. Lever Action_]

First: Look at Fig. 127; the distance traveled. The long end of the lever is at its highest point, which is A; and the short end of the lever is at its lowest point C. When the long end of the lever is pushed down, so it is at B, it moves four times farther than the short end moves upwardly, as the distance from C to D is just one-fourth that from A to B. The energy expended in moving four times the distance balances the power gained.

POWER VS. DISTANCE TRAVELED.--From this the following law is deduced: That whatever is gained in power is lost in the distance traveled.

Second: Using the same figure, supposing it was necessary to raise the short end of the lever, from C to D, in one second of time. In that case the hand pressing down the long end of the lever, would go from A to B in one second of time; or it would go four times as far as the short end, in the same time.

POWER VS. LOSS IN TIME.--This means another law: That what is gained in power is lost in time.

Distinguish clearly between these two motions. In the first case the long end of the lever is moved down from A to B in four seconds, and it had to travel four times the distance that the short end moves in going from C to D.

In the second case the long end is moved down, from A to B, in one second of time, and it had to go that distance in one-fourth of the time, so that four times as much energy was expended in the same time to raise the short end from C to D.

WRONGLY DIRECTED ENERGY.--More men have gone astray on the simple question of the power of the lever than on any other subject in mechanics. The writer has known instances where men knew the principles involved in the lever, who would still insist on trying to work out mechanical devices in which pulleys and gearing were involved, without seeming to understand that those mechanical devices are absolutely the same in principle.

This will be made plain by a few ill.u.s.trations. In Fig. 128, A is a pulley four times larger, diametrically, than B, and C is the pivot on which they turn. The pulleys are, of course, secured to each other. In this case we have the two weights, one of four pounds on the belt, which is on the small pulley (B), and a one-pound weight on the belt from the large pulley (A).

[Ill.u.s.tration: _Fig. 128. The Pulley_]

THE LEVER AND THE PULLEY.--If we should subst.i.tute a lever (D) for the pulleys, the similarity to the lever (Fig. 127) would be apparent at once. The pivot (C) in this case would act the same as the pivot (C) in the lever ill.u.s.tration.

In the same manner, and for like reasons, the wedge, the screw and the incline plane, are different structural applications of the principles set forth in the lever.

Whenever two gears are connected together, the lever principle is used, whether they are the same in size, diametrically, or not. If they are the same size then no change in power results; but instead, thereof, a change takes place in the direction of the motion.

[Ill.u.s.tration: _Fig. 129. Fig. 130. Change of Direction_]

When one end of the lever (A) goes down, the other end goes up, as shown in Fig. 129; and in Fig. 130, when the shaft (C) of one wheel turns in one direction, the shaft of the other wheel turns in the opposite direction.

It is plain that a gear, like a lever, may change direction as well as increase or decrease power. It is the thorough knowledge of these facts, and their application, which enables man to make the wonderful machinery we see on every hand.

SOURCES OF POWER.--Power is derived from a variety of sources, but what are called the _prime movers_ are derived from heat, through the various fuels, from water, from the winds and from the tides and waves of the ocean. In the case of water the power depends on the head, or height, of the surface of the water above the discharging orifice.

WATER POWER.--A column of water an inch square and 28 inches high gives a pressure at the base of one pound; and the pressure at the lower end is equal in all directions. If a tank of water 28 inches high has a single orifice in its bottom 1" x 1" in size, the pressure of water through that opening will be only one pound, and it will be one pound through every other orifice in the bottom of the same size.

CALCULATING FUEL ENERGY.--Power from fuels depends upon the expansion of the materials consumed, or upon the fact that heat expands some element, like water, which in turn produces the power. One cubic inch of water, when converted into steam, has a volume equal to one cubic foot, or about 1,700 times increase in bulk.

Advantage is taken of this in steam engine construction. If a cylinder has a piston in it with an area of 100 square inches, and a pipe one inch square supplies steam at 50 pounds pressure, the piston will have 50 pounds pressure on every square inch of its surface, equal to 5,000 pounds.

THE PRESSURE OR HEAD.--In addition to that there will also be 50 pounds pressure on each square inch of the head, as well as on the sides of the cylinder.