Simple Science
151. Machines
Man's Way of Helping Himself:
A glance into our machine shops, our factories, and even our homes shows how widespread is the use of complex machinery. But all machines, however complicated in appearance, are in reality but modifications and combinations of one or more of four simple machines devised long ago by our remote ancestors. These simple devices are known to-day, as (1) the lever, represented by a crowbar, a pitchfork; (2) the inclined plane, represented by the plank upon which barrels are rolled into a wagon; (3) the pulley, represented by almost any contrivance for the raising of furniture to upper stories; (4) the wheel and axle, represented by cogwheels and coffee grinders.
Suppose a 600-pound bowlder which is embedded in the ground is needed for the tower of a building. The problem of the builder is to get the heavy bowlder out of the ground, to load it on a wagon for transportation, and finally to raise it to the tower. Obviously, he cannot do this alone; the greatest amount of force of which he is capable would not suffice to accomplish any one of these tasks. How then does he help himself and perform the impossible? Simply, by the use of some of the machine types mentioned above, illustrations of which are known in a general way to every schoolboy. The very knife with which a stick is whittled is a machine.
A glance into our machine shops, our factories, and even our homes shows how widespread is the use of complex machinery. But all machines, however complicated in appearance, are in reality but modifications and combinations of one or more of four simple machines devised long ago by our remote ancestors. These simple devices are known to-day, as (1) the lever, represented by a crowbar, a pitchfork; (2) the inclined plane, represented by the plank upon which barrels are rolled into a wagon; (3) the pulley, represented by almost any contrivance for the raising of furniture to upper stories; (4) the wheel and axle, represented by cogwheels and coffee grinders.
Suppose a 600-pound bowlder which is embedded in the ground is needed for the tower of a building. The problem of the builder is to get the heavy bowlder out of the ground, to load it on a wagon for transportation, and finally to raise it to the tower. Obviously, he cannot do this alone; the greatest amount of force of which he is capable would not suffice to accomplish any one of these tasks. How then does he help himself and perform the impossible? Simply, by the use of some of the machine types mentioned above, illustrations of which are known in a general way to every schoolboy. The very knife with which a stick is whittled is a machine.
152. The Lever
Man's Way of Helping Himself:
Balance a foot rule, containing a hole at its middle point F, as shown in Figure. If now a weight of 1 pound is suspended from the bar at some point, say 12, the balance is disturbed, and the bar swings about the point F as a center. The balance can be regained by suspending an equivalent weight at the opposite end of the bar, or by applying a 2-pound weight at a point 3 inches to the left of F. In the latter case a force of 1 pound actually balances a force of 2 pounds, but the 1-pound weight is twice as far from the point of suspension as is the 2-pound weight. The small weight makes up in distance what it lacks in magnitude.
Such an arrangement of a rod or bar is called a lever. In any form of lever there are only three things to be considered: the point where the weight rests, the point where the force acts, and the point called the fulcrum about which the rod rotates.
The distance from the force to the fulcrum is called the force arm. The distance from the weight to the fulcrum is called the weight arm; and it is a law of levers, as well as of all other machines, that the force multiplied by the length of the force arm must equal the weight multiplied by the length of the weight arm.
Force × force arm = weight × weight arm.
A force of 1 pound at a distance of 6, or with a force arm 6, will balance a weight of 2 pounds with a weight arm 3; that is,
1 × 6 = 2 × 3.
Similarly a force of 10 pounds may be made to sustain a weight of 100 pounds, providing the force arm is 10 times longer than the weight arm; and a force arm of 800 pounds, at a distance of 10 feet from the fulcrum, may be made to sustain a weight of 8000 pounds, providing the weight is 1 foot from the fulcrum. 
FIG. - The principle of the lever.
Balance a foot rule, containing a hole at its middle point F, as shown in Figure. If now a weight of 1 pound is suspended from the bar at some point, say 12, the balance is disturbed, and the bar swings about the point F as a center. The balance can be regained by suspending an equivalent weight at the opposite end of the bar, or by applying a 2-pound weight at a point 3 inches to the left of F. In the latter case a force of 1 pound actually balances a force of 2 pounds, but the 1-pound weight is twice as far from the point of suspension as is the 2-pound weight. The small weight makes up in distance what it lacks in magnitude.
Such an arrangement of a rod or bar is called a lever. In any form of lever there are only three things to be considered: the point where the weight rests, the point where the force acts, and the point called the fulcrum about which the rod rotates.
The distance from the force to the fulcrum is called the force arm. The distance from the weight to the fulcrum is called the weight arm; and it is a law of levers, as well as of all other machines, that the force multiplied by the length of the force arm must equal the weight multiplied by the length of the weight arm.
Force × force arm = weight × weight arm.
A force of 1 pound at a distance of 6, or with a force arm 6, will balance a weight of 2 pounds with a weight arm 3; that is,
1 × 6 = 2 × 3.
Similarly a force of 10 pounds may be made to sustain a weight of 100 pounds, providing the force arm is 10 times longer than the weight arm; and a force arm of 800 pounds, at a distance of 10 feet from the fulcrum, may be made to sustain a weight of 8000 pounds, providing the weight is 1 foot from the fulcrum.
FIG. - The principle of the lever.
153. Applications of the Lever
Man's Way of Helping Himself:
By means of a lever, a 600-pound bowlder can be easily pried out of the ground. Let the lever, any strong metal bar, be supported on a stone which serves as fulcrum; then if a man exerts his force at the end of the rod , the force arm will be the distance from the stone or fulcrum to the end of the bar, and the weight arm will be the distance from the fulcrum to the bowlder itself. The man pushes down with a force of 100 pounds, but with that amount succeeds in prying up the 600-pound bowlder. If, however, you look carefully, you will see that the force arm is 6 times as long as the weight arm, so that the smaller force is compensated for by the greater distance through which it acts.
At first sight it seems as though the man's work were done for him by the machine. But this is not so. The man must lower his end of the lever 3 feet in order to raise the bowlder 6 inches out of the ground. He does not at any time exert a large force, but he accomplishes his purpose by exerting a small force continuously through a correspondingly greater distance. He finds it easier to exert a force of 100 pounds continuously until his end has moved 3 feet rather than to exert a force of 600 pounds on the bowlder and move it 6 inches.
By the time the stone has been raised the man has done as much work as though the stone had been raised directly, but his inability to put forth sufficient muscular force to raise the bowlder directly would have rendered impossible a result which was easily accomplished when through the medium of the lever he could extend his small force through greater distance.
By means of a lever, a 600-pound bowlder can be easily pried out of the ground. Let the lever, any strong metal bar, be supported on a stone which serves as fulcrum; then if a man exerts his force at the end of the rod , the force arm will be the distance from the stone or fulcrum to the end of the bar, and the weight arm will be the distance from the fulcrum to the bowlder itself. The man pushes down with a force of 100 pounds, but with that amount succeeds in prying up the 600-pound bowlder. If, however, you look carefully, you will see that the force arm is 6 times as long as the weight arm, so that the smaller force is compensated for by the greater distance through which it acts.
At first sight it seems as though the man's work were done for him by the machine. But this is not so. The man must lower his end of the lever 3 feet in order to raise the bowlder 6 inches out of the ground. He does not at any time exert a large force, but he accomplishes his purpose by exerting a small force continuously through a correspondingly greater distance. He finds it easier to exert a force of 100 pounds continuously until his end has moved 3 feet rather than to exert a force of 600 pounds on the bowlder and move it 6 inches.
By the time the stone has been raised the man has done as much work as though the stone had been raised directly, but his inability to put forth sufficient muscular force to raise the bowlder directly would have rendered impossible a result which was easily accomplished when through the medium of the lever he could extend his small force through greater distance.
154. The Wheelbarrow as a Lever
Man's Way of Helping Himself:
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.
Suspend a stick with a hole at its center as in Figure, 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.
One of the most useful levers of this type is the wheelbarrow. 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.
This shows that in loading a wheelbarrow, it is important to arrange the load as near to the wheel as possible.
The nutcracker is an illustration of a double lever of the wheelbarrow kind; the nearer the nut is to the fulcrum, the easier the cracking.
Hammers, tack-lifters, scissors, forceps, are important levers, and if you will notice how many different levers are used by all classes of men, you will understand how valuable a machine this simple device is. 
FIG. - A slightly different form of lever. 
FIG. - The wheelbarrow lightend labor. 
FIG. - A modified wheelbarrow. 
FIG. - The nutcracker is a lever. 
FIG. - The hand exerts a small force over a long distance and draws out a nail. 
FIG. - Primitive man tried to lighten his task by balancing his burden.
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.
Suspend a stick with a hole at its center as in Figure, 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.
One of the most useful levers of this type is the wheelbarrow. 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.
This shows that in loading a wheelbarrow, it is important to arrange the load as near to the wheel as possible.
The nutcracker is an illustration of a double lever of the wheelbarrow kind; the nearer the nut is to the fulcrum, the easier the cracking.
Hammers, tack-lifters, scissors, forceps, are important levers, and if you will notice how many different levers are used by all classes of men, you will understand how valuable a machine this simple device is.
FIG. - A slightly different form of lever.
FIG. - The wheelbarrow lightend labor.
FIG. - A modified wheelbarrow.
FIG. - The nutcracker is a lever.
FIG. - The hand exerts a small force over a long distance and draws out a nail.
FIG. - Primitive man tried to lighten his task by balancing his burden.
155. The Inclined Plane
Man's Way of Helping Himself:
A man wishes to load the 600-pound bowlder on a wagon, and proceeds to do it by means of a plank. 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 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.
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.
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. 
FIG. - Less force is required to raise the roller along the incline than to raise it to A directly.
A man wishes to load the 600-pound bowlder on a wagon, and proceeds to do it by means of a plank. 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 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.
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.
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.
FIG. - Less force is required to raise the roller along the incline than to raise it to A directly.
156. Application
Man's Way of Helping Himself:
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.
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
Man's Way of Helping Himself:
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.
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.
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. 
FIG. - A well-graded railroad bed. 
FIG. - A long, gradual ascent is better than a shorter, steeper one.
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.
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.
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.
FIG. - A well-graded railroad bed.
FIG. - A long, gradual ascent is better than a shorter, steeper one.
158. The Wedge
Man's Way of Helping Himself:
If an inclined plane is pushed underneath or within an object, it serves as a wedge. Usually a wedge consists of two inclined planes.
A chisel and an ax are illustrations of wedges. Perhaps the most universal form of a wedge is our common pin. Can you explain how this is a wedge? 
FIG. - By means of a wedge, the stump is split.
If an inclined plane is pushed underneath or within an object, it serves as a wedge. Usually a wedge consists of two inclined planes.
A chisel and an ax are illustrations of wedges. Perhaps the most universal form of a wedge is our common pin. Can you explain how this is a wedge?
FIG. - By means of a wedge, the stump is split.
159. The Screw
Man's Way of Helping Himself:
Another valuable and indispensable form of the inclined plane is the screw. This consists of a metal rod around which passes a ridge, and Figure 108 shows clearly that a screw is simply a rod around which (in effect) an inclined plane has been wrapped.
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 illustration of the screw. 
FIG. - A screw as a simple machine.
Another valuable and indispensable form of the inclined plane is the screw. This consists of a metal rod around which passes a ridge, and Figure 108 shows clearly that a screw is simply a rod around which (in effect) an inclined plane has been wrapped.
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 illustration of the screw.
FIG. - A screw as a simple machine.
160. Pulleys
Man's Way of Helping Himself:
The pulley, another of the machines, is merely a grooved wheel around which a cord passes. 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. 
FIG. - By means of a pulley, a force in one direction produces motion in the opposite direction.
The pulley, another of the machines, is merely a grooved wheel around which a cord passes. 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.
FIG. - By means of a pulley, a force in one direction produces motion in the opposite direction.
