5. Why is sand often placed on a track in starting a heavy train? 6. In what way is friction an advantage in lifting buildings with a jackscrew? In what way is it a disadvantage?

7. A smooth block is 10 × 8 × 3 in. Compare the distances which it will slide when given a certain initial velocity on smooth ice if resting first, on a 10 x 8 face; second, on a 10 × 3 face; third, on an 8x3 face.

8. What is the coefficient of friction of brass on brass if a force of 25 lb. is required to maintain uniform motion in a brass block weighing 200 lb. when it slides horizontally on a brass bed?

9. The coefficient of friction between a block and a table is .3. What force will be required to keep a 500-gram block in uniform motion?

EFFICIENCY

176. Definition of efficiency. Since it is only in an ideal machine that there is no friction, in all actual machines the work done by the acting force always exceeds, by the amount of the work done against friction, the amount of potential and kinetic energy stored up. We have seen that the former is wasted work in the sense that it can never be regained. Since the energy stored up represents work which can be regained, it is termed useful work. In most machines an effort is made to have the useful work as large a fraction of the total work expended as possible. The ratio of the useful work to the total work done by the acting force is called the EFFICIENCY of the machine. Thus

Efficiency

=

Useful work accomplished
Total work expended

(1)

Thus, if in the system of pulleys shown in Fig. 116 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 1 is 240 × g. cm. Hence the efficiency is equal to 6

240 × 10
50 × 1

=

4

80%.

5

177. 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%. When

inclined planes are used as machines, the friction is also small, so that the efficiency generally lies between 90% and 100%. The efficiency of the commercial block and tackle (Fig. 116), with several movable pulleys, is usually considerably less, varying between 40% and 60%. 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%. The differential pulley of Fig. 136 has also a very high mechanical advantage with a very small efficiency. Gear wheels such as those shown in Fig. 134, or chain gears such as those used in bicycles, are machines of comparatively high efficiency, often utilizing between 90% and 100% of the energy expended upon them.

178. Efficiency of overshot water wheels. The overshot water wheel (Fig. 163) 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 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% and 90%. 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.

179. Efficiency of undershot water wheels. The old-style undershot wheel (Fig. 164) - 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 1. It seldom transforms into useful work more than 25% or 30% 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. 165), developed since 1880 and now very commonly used for small-power purposes in cities supplied with waterworks, sometimes has an efficiency as high as 83%. The water is delivered from a nozzle O against cup-shaped buckets arranged as in the figure. At the Big Creek development in California, Pelton wheels 94 inches in diameter are driven by water coming with a velocity of 350 feet per second (how many miles per hour?) through nozzles 6 inches in diameter. The head of water is here 1900 ft.

FIG. 164. The undershot wheel

180. 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, rotating in a horizontal plane. Fig. 166 shows the method of installing a turbine at Niagara. Cis the outer case into which the water enters from the penstock p. Fig. 167, (1), shows the outer case with contained turbine; Fig. 167, (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. 167, (3), is the wheel itself; and Fig. 167, (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 T (Fig. 166). The amount of water which passes through the turbine can be controlled by means of the rod P (Fig. 167, (1)), which can be turned so as to increase or decrease the size of the openings between the guides G (Fig. 167, (2)). The energy expended upon the turbine per second is the product of the mass of water which passes through it by the height of the turbine pit. Efficiencies as high as 90% have been attained with such wheels.

One of the largest turbines in existence is operated by the Puget Sound Power Co. It develops 25,000 horse power under a 440-foot head of water.

1. Why is the efficiency of the jackscrew low and that of the lever high? 2. Find the efficiency of a machine in which an effort of 12 lb. moving 5 ft. raises a weight of 25 lb. 2 ft.

3. What amount of work was done on a block and tackle having an efficiency of 60% when by means of it a weight of 750 lb. was raised 50 ft.?

4. A force pump driven by a 1-horse-power engine lifted 4 cu. ft. of water per minute to a height of 100 ft. What was the efficiency of the pump?

5. If it is necessary to pull on a block and tackle with a force of 100 lb. in order to lift a weight of 300 lb., and if the force must move 6 ft. to raise the weight 1 ft., what is the efficiency of the system?

6. If the efficiency had been 65%, what force would have been necessary in the preceding problem?

7. The Niagara turbine pits are 136 ft. deep, and their average horse power is 5000. Their efficiency is 85%. How much water does each turbine discharge per minute?

MECHANICAL EQUIVALENT OF HEAT*

181. What becomes of wasted work? In all the devices for transforming work which we have considered we have found that on account of frictional resistances a certain per cent of the work expended upon the machine is wasted. The question which at once suggests itself is, What becomes of this wasted work? The following familiar facts suggest an answer. When two sticks are vigorously rubbed together, they become hot; augers and drills often become too hot to hold; matches are ignited by friction; if a strip of lead is struck a few sharp blows with a hammer, it is appreciably warmed. Now, since we learned in Chapter VIII that, according to modern notions, increasing the temperature of a body means simply increasing the average velocity of its molecules, and therefore their average kinetic energy, the above facts point strongly to the conclusion that in each case the mechanical energy expended has been simply transformed into the energy of molecular motion. This view was first brought into prominence in 1798 by Benjamin Thompson, Count Rumford, an American by birth, who was led to it by observing that in the boring of cannon heat was continuously developed. It was first carefully tested by the English physicist James Prescott Joule (see opposite p. 122) (1818-1889) in a series of epoch-making experiments extending from 1842 to 1870. In order to understand these experiments we must first learn how heat quantities are measured.

*This subject should be preceded by a laboratory experiment upon the "law of mixtures," and either preceded or accompanied by experiments upon specific heat and mechanical equivalent. See authors' Manual, Experiments 18, 19, and 20.