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Similarly, if h is the height in feet, and M the weight in pounds, P.E. Mh foot pounds.
150. The measure of kinetic energy. Since the force of the earth's attraction for M grams is Mg dynes, if we wish to express the potential energy in ergs instead of in gram centimeters, we have
Since this energy is all transformed into kinetic energy when the mass falls the distance h, the product Mgh also represents the number of ergs of kinetic energy which the moving weight has when it strikes the pile. If we wish to express this kinetic energy in terms of the velocity with which the weight strikes the pile, instead of the height from which it has fallen, we have only to substitute for h its value in terms of g and the velocity acquired (see equation (3), p. 76), namely h = v2/2 g. This gives the kinetic energy K.E. in the form
Since it makes no difference how a body has acquired its velocity, this represents the general formula for the kinetic energy in ergs of any moving body, in terms of its mass and its velocity.
Thus, the kinetic energy of a 100-gram bullet moving with a velocity of 10,000 cm. per second is
× 100 × (10,000)2 = 5,000,000,000 ergs.
Since 1 g. cm. is equivalent to 980 ergs, the energy of this bullet is 5,102,000 g. cm., or 51.02 kg. m.
We know, therefore, that the powder pushing on the bullet as it moved through the rifle barrel did 51.02 kg. m. of work upon the bullet in giving it the velocity of 100 m. per second.
In general terms, if M is in grams and v in centimeters per second, g. cm.; if M is in pounds and in feet per second,
2 × 980
QUESTIONS AND PROBLEMS
1. A stick of dynamite has great capacity for doing work. Before the explosion occurs, is the energy in the potential or the kinetic form? 2. Explain the use of the sand blast in cleaning castings, making frosted glass, cutting figures on glassware, cleaning off the walls of stone buildings, etc.
3. How much work is required to lift the 500-pound weight of a pile driver 30 ft.? How much potential energy is then stored in it? How much work does it do when it falls? If the falling mass drives the pile into the earth ft., what is its average force upon the pile?
4. A man weighing 198 lb. walked to the top of the stairway of the Washington Monument (500 ft. high) in 10 min. At what horse-power rate did he work?
5. A farm tractor drew a gang plow at the rate of 21 mi. per hour, maintaining an average drawbar pull of 1500 lb. At what average H.P. was the tractor working?
6. In the course of a stream there is a waterfall 22 ft. high. It is shown by measurement that 450 cu. ft. of water per second pours over it. How many foot pounds of energy per second could be obtained from it? What horse power?
7. How many gallons of water (8 lb. each) could a 10-horse-power engine raise in one hour to a height of 60 ft.?
8. A certain airplane using three 400-horse-power motors flew 80 mi. per hour. With how many pounds backward force did the propellers push against the air?
9. If a rifle bullet can just pass through a plank, how many planks will it pass through if its speed is doubled?
10. A steel ball dropped into a pail of moist clay from a height of a meter sinks to a depth of 2 cm. How far will it sink if dropped 4 m.?
11. Neglecting friction, find how much force a boy would have to exert to pull a 100-pound wagon up an incline which rises 5 ft. for every 100 ft. of length traversed on the incline. In addition to giving the numerical solution of the problem, state why you solve it as you do and how you know that your solution is correct.
151. Meaning of temperature. When a body feels hot to the touch we are accustomed to say that it has a high temperature, and when it feels cold that it has a low temperature. Thus the word "temperature" is used to denote the condition of hotness or coldness of the body whose state is being described.
152. Measurement of temperature. So far as we know, up to the time of Galileo no one had ever used any special instrument for the measurement of temperature. People knew how hot or how cold it was from their feelings only. But under some conditions this temperature sense is a very unreliable guide. For example, if the hand has been in hot water, tepid water will feel cold; while if it has been in cold. water, the same tepid water will feel warm; a room may feel hot to one who has been running, while it will feel cool to one who has been sitting still.
Difficulties of this sort have led to the introduction in modern times of mechanical devices, called thermometers, for measuring temperature. These instruments depend for their operation upon the fact that almost all bodies expand as they grow hot.
153. Galileo's thermometer. It was in 1592 that Galileo, at the University of Padua in Italy, constructed the first
* It is recommended that this chapter be preceded by laboratory measurements on the expansions of a gas and a solid. See, for example, Experiments 14 and 15 of the authors' Manual.
thermometer. He was familiar with the facts of expansion of solids, liquids, and gases; and since gases expand more than solids or liquids, he chose a gas as his expanding substance. His device was that shown in Fig. 143.
Let a bulb of air B be connected with a water manometer m, as in Fig. 143. If the bulb is warmed by holding a Bunsen burner beneath it, or even by placing the hand upon it, the water at m will at once begin to descend, showing that the pressure exerted by the air contained in the bulb has been increased by the increase in its temperature. If B is cooled with ice or ether, the water will rise at m.
FIG. 143. Expansion of air by heat
154. Significance of temperature from the standpoint of the kinetic theory. Now if, as was stated in § 64, gas pressure is due to the bombardment of the walls by the molecules of the gas, since the number of molecules in the bulb can scarcely have been changed by slightly heating it we are forced to conclude that the increase in pressure is due to an increase in the velocity of the molecules which are already there. From the standpoint of the kinetic theory the pressure exerted by a given number of molecules of a gas is determined by the kinetic energy of bombardment of these molecules against the containing walls. To increase the temperature is to increase the average kinetic energy of the molecules, and to diminish the temperature is to diminish this average kinetic energy. The kinetic theory thus furnishes a very simple and natural explanation of the fact of the expansion of gases with a rise in temperature.
155. The construction of a centigrade mercury thermometer. It was not until about 1700 that mercury thermometers were invented. On account of their extreme convenience these have now replaced all others for practical purposes.
The meaning of a degree of temperature change as measured by a mercury thermometer is best understood from a description of the method of making and graduating the thermometer.
A bulb is blown at one end of a piece of thick-walled glass tubing of small, uniform bore. Bulb and tube are filled with mercury, at a temperature slightly above the highest temperature for which the thermometer is to be used, and the tube is sealed
off in a hot flame. As the mercury cools, it contracts and falls away from the top of the tube, leaving a vacuum above it.
The bulb is next surrounded with melting snow or ice, as in Fig. 144, and the point at which the mercury stands in the tube is marked 0°. Then the bulb and tube are placed in the steam rising from boiling water under a pressure of 76 cm., as in Fig. 145, and the new position of the mercury is marked 100°.
FIG. 144. Method
point in calibrat
ing a thermometer
FIG. 145. Method of finding the 100°
point in calibrating a thermometer
The space between these two marks on the stem is then divided into 100 equal parts, and divisions of the same length are extended above the 100° mark and below the 0° mark.
One degree of change in temperature, measured on such a thermometer, means, then, such a temperature change as will cause the mercury in the stem to move over one of these divisions; that is, it is such a temperature change as will cause mercury contained in a glass bulb to expand 16 of the amount which it expands in passing from the temperature