140. A train of gear wheels. A form of machine capable of very high mechanical advantage is the train of gear wheels shown in Fig. 132. Let the student show from the principle of work,

namely Es Rs', that the mechanical advantage, that is, R/E,

=

141. The worm wheel. Another device of high mechanical advantage is the worm wheel (Fig. 133). Show that if I is the length of the crank arm C, n the number of teeth in the cogwheel W, and r the radius of the axle, the

mechanical advantage is given by

(1)

R

142. The differential pulley. In the differential pulley (Fig. 134) FIG. 134. The differential pulley an endless chain passes first over

the fixed pulley A, then down and around the movable pulley C, then up again over the fixed pulley B, which is rigidly attached to A but differs slightly from it in diameter. On the circumference of all

the pulleys are projections which fit between the links and thus keep the chains from slipping. When the chain is pulled down at E, as in Fig. 134 (2) until the upper rigid system of pulleys has made one complete revolution, the chain between the upper and lower pulleys has been shortened by the difference between the circumferences of the pulleys A and B, for the chain has been pulled up a distance equal to the circumference of the larger pulley and let down a distance equal to the circumference of the smaller pulley. Hence the load R has been lifted by half the difference between the circumferences of A and B. The mechanical advantage is therefore equal to the circumference of A divided by half the difference between the circumferences of A and B.

SUMMARY. Newton's principle of work. In all mechanical devices of whatever sort, if only friction may be neglected, the work expended upon the machine is equal to the work accomplished by it.

QUESTIONS AND PROBLEMS

1. Name two or three household appliances whose mechanical advantage is less than 1.

2. Analyze several types of manual labor (such, for instance, as messenger service, digging in hard soil and soft soil, chopping or sawing wood, running a lawn mower, gathering in a farm crop, etc.) and see if the definition (W = Fs) holds for each. Is not FX s the thing paid for in every case?

3. A force of 80 kg. on a wheel whose diameter is 3 m. balances a weight of 150 kg. on the axle. Find the diameter of the axle.

4. A 1500-pound safe must be raised 5 ft. The force which can be applied is 250 lb. What is the shortest inclined plane which can be used for the purpose?

5. A 300-pound barrel was rolled up a plank 12 ft. long into a doorway 3 ft. high. What force was applied parallel to the plank?

FIG. 135. Windlass with gears

6. In the windlass of Fig. 135 the crank handle has a length of 2 ft., and the barrel a diameter of 8 in. There are 20 cogs in the

small cogwheel and 60 in the large one. What is the mechanical advantage of the arrangement?

7. A small jackscrew has 20 threads to the inch. Using a lever 31⁄2 in. long will give what mechanical advantage? (Use 3.1416.)

8. The screw of a lard press has 5 threads to the inch, and the length of each handle is 6 in. If there were no friction, what pressure would result from a rotating force of 10 lb. applied to the end of each handle? (See Fig. 130.)

POWER AND ENERGY

143. Definition of power. Let us suppose that a boy and a man were each sent to load a cart with sand. Although the boy may have been three times as many minutes in loading his cart as was the man, the boy nevertheless performed the same amount of work as did the man, for in loading his cart he raised an equal weight of sand the same height. Time, therefore, is not a factor which enters into the determination of work. In a minute, however, the man accomplished three times as much work as the boy. We say, therefore, that the man worked at three times the rate of the boy. The rate of doing work is called power, or activity. Thus, if P represents power, W the work done, and t the time required to do the work, then

W

P=

or P ==

Fs.

t

t

(10)

144. Horse power. James Watt (1736-1819), the inventor of the steam engine, considered that an average horse could do 33,000 foot pounds of work per minute, or 550 foot pounds per second. The metric equivalent is 76.05 kilogram meters per second. This number is probably considerably too high; but it has been taken ever since, in English-speaking countries, as the unit of power, and named the horse power (H.P.). The power of steam engines has usually been rated in horse power. The horse power of an ordinary railroad locomotive is from 500 to 1000. Stationary engines and steamboat engines of the largest size often run from 5000 to 20,000 H. P.

The power of an average horse is about H.P. and that of an ordinary man about H.P.

145. The kilowatt. In the metric system the erg is taken as the absolute unit of work. The corresponding unit of power is an erg per second. This is, however, so small that it is customary to take as the practical unit 10,000,000 ergs per second; that is, one joule per second (see § 126, p. 115). This unit is called the watt, in honor of James Watt. The power of dynamos and electric motors is almost always expressed in kilowatts, a kilowatt representing 1000 watts; and in modern practice even steam engines are being increasingly rated in kilowatts rather than in horse power. A horse power is equivalent to 746 watts, or about of a kilowatt.

H

A

ро

B

FIG. 136. Illustration of potential energy

146. Definition of energy. The energy of a body is defined as its capacity for doing work. In general, inanimate bodies possess energy only because of work which has been done upon them at some previous time. Thus, suppose a kilogram weight is lifted from the first position in Fig. 136 through a height of 1 meter and placed upon the hook H at the end of a cord which passes over a frictionless pulley p and is attached at the other end to a second kilogram weight B. The operation of lifting A from position 1 to position 2 has required an expenditure upon it of 1 kilogram meter (100,000 gram centimeters, or 98,000,000 ergs) of work. But in position 2 the weight A is itself possessed of a certain capacity for doing work that it did not have before; for if it is now started downward by the application of the slightest conceivable force, it will of its own accord return to position 1, and will in so doing raise the kilogram weight B through a height of 1 meter. In other words, it will do upon B exactly the same amount of work that was originally done upon it.

This picture shows the relative sizes of Stephenson's original locomotive, the Rocket, which ran in October, 1829, between Manchester and Liverpool, and the largest locomotive thus far built, the Virginian Mallet, constructed in the shops of the American Locomotive Company at Schenectady, New York, for use on the Virginian Railroad. The Rocket weighed 4 tons and won a £500 prize by drawing a coach containing 30 people at a rate of from 26 to 30 miles per hour. The Virginian Mallet weighs 450 tons and has a tractive power of 176,600 pounds. It has approximately 5100 H.P.