154. The Wheelbarrow as a Lever. The principle of the lever is always the same; but the relative position of the important points may vary. For example, the fulcrum is sometimes at one end, the force at the opposite end, and the weight to be lifted between them.

[Ill.u.s.tration: FIG. 98.--A slightly different form of lever.]

Suspend a stick with a hole at its center as in Figure 98, and hang a 4-pound weight at a distance of 1 foot from the fulcrum, supporting the load by means of a spring balance 2 feet from the fulcrum. The pointer on the spring balance shows that the force required to balance the 4-pound load is but 2 pounds.

The force is 2 feet from the fulcrum, and the weight (4) is 1 foot from the fulcrum, so that

Force distance = Weight distance, or 2 2 = 4 1.

Move the 4-pound weight so that it is very near the fulcrum, say but 6 inches from it; then the spring balance registers a force only one fourth as great as the weight which it suspends. In other words a force of 1 at a distance of 24 inches (2 feet) is equivalent to a force of 4 at a distance of 6 inches.

[Ill.u.s.tration: FIG. 99.--The wheelbarrow lightened labor.]

One of the most useful levers of this type is the wheelbarrow (Fig.

99). The fulcrum is at the wheel, the force is at the handles, the weight is on the wheelbarrow. If the load is halfway from the fulcrum to the man's hands, the man will have to lift with a force equal to one half the load. If the load is one fourth as far from the fulcrum as the man's hands, he will need to lift with a force only one fourth as great as that of the load.

[Ill.u.s.tration: FIG. 100.--A modified wheelbarrow.]

This shows that in loading a wheelbarrow, it is important to arrange the load as near to the wheel as possible.

[Ill.u.s.tration: FIG. 101.--The nutcracker is a lever.]

The nutcracker (Fig. 101) is an ill.u.s.tration of a double lever of the wheelbarrow kind; the nearer the nut is to the fulcrum, the easier the cracking.

[Ill.u.s.tration: FIG. 102.--The hand exerts a small force over a long distance and draws out a nail.]

Hammers (Fig. 102), tack-lifters, scissors, forceps, are important levers, and if you will notice how many different levers (fig. 103) are used by all cla.s.ses of men, you will understand how valuable a machine this simple device is.

155. The Inclined Plane. A man wishes to load the 600-pound bowlder on a wagon, and proceeds to do it by means of a plank, as in Figure 93. Such an arrangement is called an inclined plane.

The advantage of an inclined plane can be seen by the following experiment. Select a smooth board 4 feet long and prop it so that the end _A_ (Fig. 104) is 1 foot above the level of the table; the length of the incline is then 4 times as great as its height. Fasten a metal roller to a spring balance and observe its weight. Then pull the roller uniformly upward along the plank and notice what the pull is on the balance, being careful always to hold the balance parallel to the incline.

When the roller is raised along the incline, the balance registers a pull only one fourth as great as the actual weight of the roller. That is, when the roller weighs 12, a force of 3 suffices to raise it to the height _A_ along the incline; but the smaller force must be applied throughout the entire length of the incline. In many cases, it is preferable to exert a force of 30 pounds, for example, over the distance _CA_ than a force of 120 pounds over the shorter distance _BA_.

[Ill.u.s.tration: FIG. 103.--Primitive man tried to lighten his task by balancing his burden.]

Prop the board so that the end _A_ is 2 feet above the table level; that is, arrange the inclined plane in such a way that its length is twice as great as its height. In that case the steady pull on the balance will be one half the weight of the roller; or a force of 6 pounds will suffice to raise the 12-pound roller.

[Ill.u.s.tration: FIG. 104.--Less force is required to raise the roller along the incline than to raise it to _A_ directly.]

The steeper the incline, the more force necessary to raise a weight; whereas if the incline is small, the necessary lifting force is greatly reduced. On an inclined plane whose length is ten times its height, the lifting force is reduced to one tenth the weight of the load. The advantage of an incline depends upon the relative length and height, or is equal to the ratio of the length to the height.

156. Application. By the use of an inclined plank a strong man can load the 600-pound bowlder on a wagon. Suppose the floor of the wagon is 2 feet above the ground, then if a 6-foot plank is used, 200 pounds of force will suffice to raise the bowlder; but the man will have to push with this force against the bowlder while it moves over the entire length of the plank.

Since work is equal to force multiplied by distance, the man has done work represented by 200 6, or 1200. This is exactly the amount of work which would have been necessary to raise the bowlder directly. A man of even enormous strength could not lift such a weight (600 lb.) even an inch directly, but a strong man can furnish the smaller force (200) over a distance of 6 feet; hence, while the machine does not lessen the total amount of work required of a man, it creates a new distribution of work and makes possible, and even easy, results which otherwise would be impossible by human agency.

157. Railroads and Highways. The problem of the incline is an important one to engineers who have under their direction the construction of our highways and the laying of our railroad tracks. It requires tremendous force to pull a load up grade, and most of us are familiar with the struggling horse and the puffing locomotive. For this reason engineers, wherever possible, level down the steep places, and reduce the strain as far as possible.

[Ill.u.s.tration: FIG. 105.--A well-graded railroad bed.]

The slope of the road is called its grade, and the grade itself is simply the number of feet the hill rises per mile. A road a mile long (5280 feet) has a grade of 132 if the crest of the hill is 132 feet above the level at which the road started.

[Ill.u.s.tration: FIG. 106.--A long, gradual ascent is better than a shorter, steeper one.]

In such an incline, the ratio of length to height is 5280 132, or 40; and hence in order to pull a train of cars to the summit, the engine would need to exert a continuous pull equal to one fortieth of the combined weight of the train.

If, on the other hand, the ascent had been gradual, so that the grade was 66 feet per mile, a pull from the engine of one eightieth of the combined weight would have sufficed to land the train of cars at the crest of the grade.

Because of these facts, engineers spend great sums in grading down railroad beds, making them as nearly level as possible. In mountainous regions, the topography of the land prevents the elimination of all steep grades, but nevertheless the attempt is always made to follow the easiest grades.

158. The Wedge. If an inclined plane is pushed underneath or within an object, it serves as a wedge. Usually a wedge consists of two inclined planes (Fig. 107).

[Ill.u.s.tration: FIG. 107.--By means of a wedge, the stump is split.]

A chisel and an ax are ill.u.s.trations of wedges. Perhaps the most universal form of a wedge is our common pin. Can you explain how this is a wedge?

159. The Screw. Another valuable and indispensable form of the inclined plane is the screw. This consists of a metal rod around which pa.s.ses a ridge, and Figure 108 shows clearly that a screw is simply a rod around which (in effect) an inclined plane has been wrapped.

[Ill.u.s.tration: FIG. 108--A screw as a simple machine.]

The ridge encircling the screw is called the thread, and the distance between two successive threads is called the pitch. It is easy to see that the closer the threads and the smaller the pitch, the greater the advantage of the screw, and hence the less force needed in overcoming resistance. A corkscrew is a familiar ill.u.s.tration of the screw.

160. Pulleys. The pulley, another of the machines, is merely a grooved wheel around which a cord pa.s.ses. It is sometimes more convenient to move a load in one direction rather than in another, and the pulley in its simplest form enables us to do this. In order to raise a flag to the top of a mast, it is not necessary to climb the mast, and so pull up the flag; the same result is accomplished much more easily by attaching the flag to a movable string, somewhat as in Figure 109, and pulling from below. As the string is pulled down, the flag rises and ultimately reaches the desired position.

If we employ a stationary pulley, as in Figure 109, we do not change the force, because the force required to balance the load is as large as the load itself. The only advantage is that a force in one direction may be used to produce motion in another direction. Such a pulley is known as a fixed pulley.

[Ill.u.s.tration: FIG. 109.--By means of a pulley, a force in one direction produces motion in the opposite direction.]

161. Movable Pulleys. By the use of a movable pulley, we are able to support a weight by a force equal to only one half the load. In Figure 109, the downward pull of the weight and the downward pull of the hand are equal; in Figure 110, the spring balance supports only one half the entire load, the remaining half being borne by the hook to which the string is attached. The weight is divided equally between the two parts of the string which pa.s.ses around the pulley, so that each strand bears only one half of the burden.

We have seen in our study of the lever and the inclined plane that an increase in force is always accompanied by a decrease in distance, and in the case of the pulley we naturally look for a similar result. If you raise the balance (Fig. 110) 12 feet, you will find that the weight rises only 6 feet; if you raise the balance 24 inches, you will find that the weight rises 12 inches. You must exercise a force of 100 pounds over 12 feet of s.p.a.ce in order to raise a weight of 200 pounds a distance of 6 feet. When we raise 100 pounds through 12 feet or 200 pounds through 6 feet the total work done is the same; but the pulley enables those who cannot furnish a force of 200 pounds for the s.p.a.ce of 6 feet to accomplish the task by furnishing 100 pounds for the s.p.a.ce of 12 feet.

[Ill.u.s.tration: FIG. 110.--A movable pulley lightens labor.]

162. Combination of Pulleys. A combination of pulleys called block and tackle is used where very heavy loads are to be moved. In Figure 111 the upper block of pulleys is fixed, the lower block is movable, and one continuous rope pa.s.ses around the various pulleys. The load is supported by 6 strands, and each strand bears one sixth of the load.

If the hand pulls with a force of 1 pound at _P_, it can raise a load of 6 pounds at _W_, but the hand will have to pull downward 6 feet at _P_ in order to raise the load at _W_ 1 foot. If 8 pulleys were used, a force equivalent to one eighth of the load would suffice to move _W_, but this force would have to be exerted over a distance 8 times as great as that through which _W_ was raised.

[Ill.u.s.tration: FIG. 111.--An effective arrangement of pulleys known as block and tackle.]

163. Practical Application. In our childhood many of us saw with wonder the appearance and disappearance of flags flying at the tops of high masts, but observation soon taught us that the flags were raised by pulleys. In tenements, where there is no yard for the family washing, clothes often appear flapping in mid-air. This seems most marvelous until we learn that the lines are pulled back and forth by pulleys at the window and at a distant support. By means of pulleys, awnings are raised and lowered, and the use of pulleys by furniture movers, etc., is familiar to every wide-awake observer on the streets.

164. Wheel and Axle. The wheel and axle consists of a large wheel and a small axle so fastened that they rotate together.

[Ill.u.s.tration: FIG. 112.--The wheel and axle.]

When the large wheel makes one revolution, _P_ falls a distance equal to the circ.u.mference of the wheel. While _P_ moves downward, _W_ likewise moves, but its motion is upward, and the distance it moves is small, being equal only to the circ.u.mference of the small axle. But a small force at _P_ will sustain a larger force at _W_; if the circ.u.mference of the large wheel is 40 inches, and that of the small wheel 10 inches, a load of 100 at _W_ can be sustained by a force of 25 at _P_. The increase in force of the wheel and axle depends upon the relative size of the two parts, that is, upon the circ.u.mference of wheel as compared with circ.u.mference of axle, and since the ratio between circ.u.mference and radius is constant, the ratio of the wheel and axle combination is the ratio of the long radius to the short radius.

For example, in a wheel and axle of radii 20 and 4, respectively, a given weight at _P_ would balance 5 times as great a load at _W_.