J. A. COOLIDGE

Although there are nominally six simple machines, the lever, the pulley, the wheel and axle, the inclined plane, the wedge, and the screw; and although these machines are generally very different in appearance, they may be divided into two classes, according as the machine turns about a fixed point or pivot when the weight is moved, or the machine is made to slide under the weight, or the weight is made to slide or roll over the surface of the machine. Thus the lever may include the wheel and axle, and the pulley, and the laws of the lever: - Power x Power arm = Weight x

Elementary Mechanics IV The Wheel and Axle 200

Weight arm, may be seen to be true in these machines. The inclined plane, on the other hand, finds its modifications in the wedge, and in the screw which is a movable inclined plane in spiral form.

The wheel and axle is like a lever in this; first, that it turns about a pivot; and second, that the power and weight are attached to it at different distances from the pivot, thus making a long and a short arm. The lever is limited in its actions in that the weight can be lifted only a short distance, while in the wheel and axle, the weight may be lifted a much longer distance, in some cases many feet.

For experimenting with the wheel and axle much time and trouble may be saved, if one has an old bicycle wheel and its bearings. The large wheel with the tire removed furnishes an excellent wheel and the axle, or better a slightly larger axle made from a spool, may be made to furnish an attachment for the weight to be lifted. As comparatively few have such conveniences we will give directions for making a wheel that will serve our purpose. An axle may be made from a rod of brass or smooth iron 6" long, and 5-16" diameter. A wheel 3" diameter should be care" fully cut from some 5/8" stock, and two pieces 3 J" diameter cut from some 3/8" stock should be glued or nailed, one on each side. This will make a grooved wheel. A better one can be made if one has access to a turning lathe, or if one can cut one out of a roller of some kind. A wooden rolling pin costing five cents will furnish wheels for a dozen. A smaller wheel 1" in diameter and with the larger one must be centered arid fastened with shellac or cement to the metal axle. The axle should be mounted in a frame or box at least 15" high. Two stout linen threads smooth and flexible should be used. Fasten one to the rim of the wheel and wind it around the wheel four or five times so that it shall unwind left handed, as can be seen at C, Fig. 10. The other should be fastened to the smaller wheel or axle and should be wound in the opposite direction as in A of the same figure.

At A and B fasten two of the weights we have made and used in our former experiments making them balance. If any difficulty is found in doing this, some small additional weight may be added either to A or B. Compare the weight at A with that at B. Compare also the distances R H and P C. Using the hand as an extra force pull down on A and measure the distance B rises while A descends 6". Also notice the speed with which A moves as compared with that of B. As the diameter of the wheels are 3" and 1", the radii, or distances P H and P C are as 3 to 1; the distances moved by A and B should be 6" and 2"; and the speed of A should be three times that of B. If the weight B is not found to be three times the weight A the difference is due to friction.

To determine the friction in the experiment, let us use, instead of a weight at A, the spring balance and pull in a horizontal direction as seen in Fig. 11. The average of the force necessary to raise B and that required to let it descend slowly will be the true force A. Compare this with the weight used before at A. Try several cases of equilibruim between A and B, using forces at A, of 8, 16, 24 oz. as the force and record in tabular form.

A. 8 oz. B.....oz.

16 oz. .....oz.

24 oz. .....oz.

Does A x PH = B x P C ? If this is so the law of the lever is again verified, and may be called law of the wheel and axle. The power x the radius of wheel = the weight x the radius of axles. The ratio of the force needed, to the weight to be moved may be found by dividing the radius of the wheel by the radius of axle. The brakes on cars and the steering apparatus on board a vessel are about the only illustrations of the wheel and axle one is likely to see; but many modifications are in use and can be seeb. far more frequently. The crank and axle is the same machine with this exception that all of the wheel is removed except one or two spokes, and that the force is applied at the end of a spoke. The power in some cases may be applied anywhere along one of the spokes, thus allowing the power arms to be of different lengths. In many machines the crank is curved so that all semblance of a wheel is gone. The derrick, windlass, capstan, brakes on electric cars, and many other contriivances are examples of the wheel and axle.

Take two pieces 12" long, \" wide and 3/8" thick, and nail them to the wheel just used so as to make two arms as shown in Fig. 13. If we consider the original wheel as the hub of an enlarged wheel, and P H as a spoke of this wheel, we may experiment with this as a crank and axle. The pivot P is the same, the cord at C holds the weight B, and the power A is attached 12" from P. P H -- by P C = the ratio of weight to the power. As the power arm P H is so much larger than P C the weight is very much larger than the power.

On the bottom of our box directly under B fasten a screw eye. Make the cord C B shorter and hang on the lower end a spring balance, fastening the hook to the screw eye in the floor beneath. Hang on the end A just weight enough to hold the balance vertical. Place extra weights at A, in each case recording the pull exerted on the spring balance. The balance records for us the weights that can be lifted at B by the forces used at A. Try forces of 2, 4, and 6 oz. at A and again see whether the law holds true.

Some of the values of the crank and axle may be seen from the following examples: - Here are some men moving a house. A horse is attached to the end of a bar that turns an axle set vertically and around which a rope is being coiled as the horse turns the axle. The other end of the rope is fastened to a house set on rollers. If the bar to which the horse is attached is 10' long, and the pull of the horse after allowing for the loss by friction, is 1000 lbs. the value of the power x the radius of the wheel, i. e., the length of the bar, is 10 x 1000 = 10,000. As the weight moved x the radius of axle must equal this sum, if the radius of axle is \ a foot, 1/2 x the weight moved = 10,000 and the whole of the weight moved = 20,000 pounds. Of course the weight represents the pull exerted on the house, and as the house rests on rollers, this pull must be able to move a weight many times larger. With a loss from friction of 50°, the weight would be doubled, or 40,000 lbs. The force exerted by the horse, in a real case of house moving, is increased several times by a system of pulleys.

The sailors are hoisting an anchor. Each at the end of a capstan bar is pushing with a force of 50 lbs. If the bar is 8' long, the total force of 100 lbs. will make the value of power x length of crank, 100 x 8 = 800. If the radius of the capstan is 6", 1/2 x the weight moved as 800, and the whole weight = 1600 lbs. With a larger force from each man, with more men and less friction, much greater weights can be lifted. Sometimes the power exerted on the crank is not applied directly to the weight, but the circumference of the axle is a toothed wheel and fits or "meshes "into the circumference of a larger wheel. The second wheel moves more slowly than the first and may be attached in the same manner to a third. These gear wheels allow an immense gain in power although there is considerable loss from friction and the weight is moved very slowly. In transferring power from one set of shafting to another, the friction of the belt on the circumference of the wheels transfers the motion from one to the other. By making one wheel large and the second small a very great speed may be obtained. For example, suppose the large wheel have a diameter of five times the smaller, then the speed of the smaller will be five times the larger. By adjusting these differences any speed desired may be obtained.