work to the total work done by the acting force is called the EFFICIENCY of the machine. Thus,

Thus, if in the system of pulleys shown in Fig. 114 it is necessary to add a weight of 50 g. at E in order to pull up slowly an added weight of 240 g. at R, the work done by the 50 g. while E is moving over 1 cm. will be 50 × 1 g. cm. The useful work accomplished in the same time is 240 × 1/6 g. cm. Hence the efficiency is equal to

157. Efficiencies of some simple machines. In simple levers the friction is generally so small as to be negligible; hence the efficiency of such machines is approximately 100 per cent. When inclined planes are used as machines the friction is also small, so that the efficiency generally lies between 90 per cent and 100 per cent. The efficiency of the commercial block and tackle (Fig. 114), with several movable pulleys, is usually considerably less, varying between 40 per cent and 60 per cent. In the jackscrew there is necessarily a very large amount of friction, so that although the mechanical advantage is enormous, the efficiency is often as low as 25 per cent. The differential pulley of Fig. 134 has also a very high mechanical advantage with a very small efficiency. Gear wheels such as those shown in Fig. 132, or chain gears such as those used in bicycles, are machines of comparatively high efficiency, often utilizing between 90 per cent and 100 per cent of the energy expended upon them.

158. Efficiency of overshot water wheels. The overshot water wheel (Fig. 144) utilizes chiefly the potential energy of the water at S, for the wheel is turned by the weight of the water in the buckets. The work expended on the wheel per second, in foot pounds or in gram centimeters, is the product of the weight of the water which passes over it per second by the distance through which it falls. The efficiency is the work which the wheel can

accomplish in a second divided by this quantity. Such wheels are very common in mountainous regions, where it is easy to obtain considerable fall but where the streams carry a small volume of water. The efficiency is high, being often between 80 per cent and 90 per cent. The loss is due not only to the friction in the bearings and gears (see C) but also to the fact that some of the water is spilled from the buckets or passes over without entering them at all. This may still be regarded as a frictional loss, since the energy disappears in internal friction when the water strikes the ground.

FIG. 144. Overshot water wheel

159. Efficiency of undershot water wheels. The old-style undershot wheel (Fig. 145), so common in flat countries, where there is little fall but an abundance of water, utilizes only the kinetic energy of the water running through the race from A. It seldom transforms into useful work more than 25 per cent or 30 per cent

A

FIG. 145. The undershot wheel

of the potential energy of the water above the dam. There are, however, certain modern forms of undershot wheel which are extremely efficient. For example, the Pelton wheel (Fig. 146), developed since 1880 and now very commonly used for small-power purposes in cities supplied with waterworks, as well as for operating huge power plants, sometimes has an efficiency as high

(1)

T

(4)

as 87 per cent. The water is delivered from a nozzle 0 against cup-shaped buckets arranged as in the figure. At the Big Creek development in California, Pelton wheels 121 in. in diameter are driven by water coming with a velocity of 350 ft. per second (how many miles per hour?) through nozzles 6 in. in diameter. The head of water (drop) is here 2130 ft. (See opposite page 145.) 160. Efficiency of water turbines. The turbine wheel was invented in France in 1833 and is now used more than any other form of water wheel. It stands completely under water in a case at the bottom of a turbine pit and rotates in a horizontal plane. Fig. 147 (1) shows one form of outer case with contained turbine; Fig. 147 (2) is the inner case, in which are the fixed guides G, which direct the water at the most advantageous angle against the blades of the wheel inside; Fig. 147 (3) is the wheel itself; and Fig. 147 (4) is a section of wheel and inner case, showing how the water enters through the guides and impinges upon the blades W. The spent water simply falls down from the blades into the tailrace below T (Fig. 147 (1)). The amount of water which passes through the turbine can be controlled by means of a valve P (Fig. 147 (1)), which can be turned so as to increase or decrease the size of the openings between the guides G (Fig. 147 (2)). On the opposite page is shown a huge modern turbine installation. The energy expended upon the turbine per second is the product of the weight of water which passes through it by the height of the turbine pit. Efficiencies as high as 93 per cent have been attained with such wheels.

FIG. 147. The turbine wheel: (1) outer case; (2) inner case; (3) rotating part; (4) section

On the American side of the Niagara River are installed three turbines, similar to the one shown in the picture, of 70,000 H.P., each operating under a head of 213.5 feet. They are the highest-powered turbines in the world. The picture shows clearly how the water enters a volute casing surrounding the whole wheel, from which it is evenly distributed against the blades of the huge runner 15 feet in diameter. Each of these complete turbines, not including the generators above, weighs 1,250,000 pounds, makes 107 R.P.M., and has an efficiency of 93 per cent. The generator shaft is of solid steel 34 inches in diameter. (Courtesy of the Westinghouse Electric and Manufacturing Company)

A HYDROELECTRIC PLANT IN THE HIGH SIERRAS

This is one of the Southern California Edison Company's hydroelectric power plants. Large water storage at an elevation of 7000 feet is obtained by two huge dams and a 13-mile-long tunnel through granite rock. The vertical drop of this water through more than a mile is fully utilized by a series of power plants. The picture shows one of these in which are installed Pelton wheels fed from a penstock 4333 feet long (seen on the mountain side) and having a vertical drop of 2130 feet. Each of these Pelton wheels has a maximum output of 43,000 H.P. and an efficiency of 88 per cent. The power is transmitted 241 miles at 220,000 volts