UNITS OF MEASUREMENT
In the preceding lesson we have discussed some of the properties of matter. We have noted that all the materials on the surface of the earth are composed of various combinations of the 92 chemical elements. We have observed that matter can be transformed from one physical state to another or from one chemical combination to another, and that such processes are occurring continuously on the earth, but that in none of them is the matter destroyed; it is merely reshuffled.
Our next problem is to investigate the circumstances under which matter moves, or undergoes physical and chemical transformations. Before we can do this, however, it is necessary that we become familiar with our systems of measurement.
Mass, Length and Time. The three quantities that we deal with most frequently and hence are obliged to measure most often are mass, length and time.
The mass of a body is that property which gives it weight, or, more generally, causes it to have inertia or a resistance to any change of motion. A body has weight because of the attraction of gravity upon its mass. If gravity were reduced by one-half, the weight of a body, as measured by a spring balance, would also be reduced by one-half. For example, the weight of a given body on the earth is less by about one part in 200 at the equator than at the poles. Its mass, however, remains the same.
If gravity were zero, bodies would weigh nothing at all. Suppose under this condition that we had two hollow spheres identical in outward appearance, one filled with air and the other with lead. Neither would have any weight. How could we tell them apart? All we would need to do would be to shake them. The lead ball would feel ‘heavy’ and the one filled with air ‘light’. If we kicked the lead ball it would break our foot just as readily as if it had weight because it would still have the same inertia and resistance to change of motion, and hence the same mass.
Length is an already familiar concept which needs no explanation.
Time is measured in terms of the motion of some material system which is changing at a uniform speed. Mechanically oscillating systems like pendulums and tuning forks are the basis for most of our time measurements and form the control mechanisms of our clocks. Our master clock is the rotating earth whose hands are the stars which appear to go around the earth with uniform angular velocity once per sidereal or stellar day.
Units of Measurement.
The way we measure a quantity of any kind is to compare it with another quantity of the same kind which we employ as a unit of measurement. Thus we measure a mass by determining how many times greater it is than some standard mass; we measure a length by the number of multiples it contains of a standard length; and an interval of time by the multiples of some standard time interval. The choice of these standards is entirely arbitrary but if confusion is to be avoided two conditions must be rigidly observed: Different people performing a measurement of the same thing must use standards which either are the same or else the two standards must have a known ratio to each other; the other condition necessary is that the standard of measurement must not change. Unintelligibility results when either of these conditions is violated. The first type of unintelligibility would result if one man measured all of his lengths with a measuring stick of one length and another man used a measuring stick of a different length without the two ever having been compared. The second type of confusion would result if we attempted to measure lengths with a rubber band without specifying the tautness with which it is to be stretched.
In the early days almost endless confusion in the units of measurement existed due to the failure to observe one or both of these conditions. All sorts of units of measurement sprang up spontaneously and were in general use. Such units of length as that of a barley corn, the breadth of a hand, and the length of King John’s foot were not uncommon. Thus, it was customary to employ as units things like a barley corn which bear a single name but may vary considerably in size. The type of confusion that this could cause is illustrated by an apple dealer who advertised his apples at 25 cents per bucketful. He had on display several large size buckets filled with apples but when filling the customer’s order he used a bucket much smaller in size; yet no one could say that he had not received a ‘bucketful’ of apples. The trick of course lies in the fact that there is no standard size of ‘bucket’. The same liberties with a bushel measure would have landed our merchant in jail.
To eliminate this kind of confusion governments have had to establish standards of measurement so that today in the whole world only two systems of units are extensively used. These are the Metric system and the English system. It is to be hoped that soon there will be one only.
The Metric System. The Metric system was established by the French government immediately following the French Revolution. For the standard of length a bar composed of an alloy of platinum and iridium was constructed and is preserved at the Bureau of Weights and Measures near Paris. Near each end of this bar there are engraved transversely three fine parallel lines. The distance from the middle line at one end of the bar to the middle line at the other end when the bar is at the temperature of melting ice is defined to be 1 meter. This is the prototype of all the other meters in the world. Exact copies of this bar made by direct comparison have been constructed and distributed to the governments of the various countries of the world. In the United States this duplicate is kept at the Bureau of Standards in Washington. From this, additional copies are made and are obtained by manufacturers of tapes, meter sticks and other measuring scales from which these latter are graduated. Hence the meter stick that one uses in his laboratory is probably not more than three or four times removed from the original bar in Paris.
For units smaller and larger than a meter a decimal system of graduation is employed. Thus the centimeter is a hundredth part of a meter; a millimeter is a thousandth part of a meter; and a micron is a millionth part of a meter. Going up the scale a kilometer is 1,000 meters. There are other multiples and submultiples but the above are the ones most extensively used.
Similarly, the unit of mass is that of a platinum weight kept at the Bureau of Weights and Measures and defined to have a mass of 1 kilogram. The gram is accordingly a thousandth part of the mass of this standard kilogram. Just as in the case of the meter, duplicates of the standard kilogram in Paris have been constructed and distributed to the various countries.
While both the meter and the kilogram are entirely arbitrary, when they were constructed an effort was made to satisfy two useful conditions. The original meter was constructed as accurately as possible to be one ten-millionth part of the distance along the earth’s surface from the equator to the pole. This result of course was not achieved exactly so that by later measurements the earth’s quadrant is found to be 10,000,856 meters. Still, however, we can say with considerable exactness that the circumference of the earth is 40,000 kilometers.
In a similar manner an attempt was made to have the mass of 1 gram be that of a cubic centimeter of water at 4° Centigrade (the temperature at which water has its greatest density). Hence the kilogram is very nearly the mass of 1,000 cubic centimeters of water and for most purposes the mass of water can be taken to be 1 gram per cubic centimeter.
The unit of time is the second which is defined to be ^{1}/_{86,400}^{th} part of a mean solar day or ^{1}/_{86,164.09}^{th} of a stellar day. In addition to the second we have the familiar multiples, minutes and hours.
The English System. The unit of length in the English system of measurement is the distance between the centers of two transverse lines in two gold plugs in a bronze bar deposited at the Office of the Exchequer, when the bar is at a temperature of of 62 degrees Fahrenheit. This distance is the standard yard. A foot is defined to be one-third of a yard, and an inch one thirty-sixth of a yard.
The unit of mass in the English system is that of a certain piece of platinum marked ‘P. S., 1844, 1 lb.’, which is deposited at the same place as the standard yard. This is known as the standard pound avoirdupois. The unit of time in the English system is the same as in the Metric. Conversion Between Metric And English Units
CONVERSION BETWEEN METRIC AND ENGLISH UNITS. These two systems of measurement are inter-convertible when we know the magnitude of a standard in one system as measured in terms of the corresponding standard unit of the other system. By very exact measurement it has been established that
TABLE OF CONVERSION FACTORS | |
IMPERIAL | METRIC |
Standard Gravity: | |
Gravity | = 980.665 cm./sec.^{2} |
= 32.174 ft./sec.^{2} | |
Force: | |
1 dyne | = 1 gm. cm./sec.^{2} |
1 pound weight | = 2.2481×10^{−6} pound weight |
= 4.4493X10^{5 } dynes | |
Work: | |
1 erg | = 1 dyne-centimeter |
= 1 × 10^{−7} joules | |
1 joule | = 1 × 10^{7} ergs |
= 0.73756 foot-pound | |
1 foot-pound | = 1.35582 joules |
= 1.35582 × 10^{7} ergs | |
1 kilowatt-hour | = 3.6000 × 10^{6} joules |
= 2.6552 × 10^{6} foot-pounds | |
= 1.3410 horsepower-hours | |
1 horsepower-hour | = 1.9800 × 10^{6} foot-pounds |
= 2.6845 × 10^{6} joules | |
= 0.7457 kilowatt-hour | |
= 745.7 watt-hours | |
Power: | |
1 watt | = 1 joule per second |
= 0.001 kilowatt | |
= 1 × 10^{7} ergs per second | |
= 0.73756 foot-pound per second | |
= 1.3410 × 10^{-3} horsepower | |
1 kilowatt | = 1 × 10^{10} ergs per second |
= 1,000 joules per second | |
= 737.56 foot-pounds per second | |
= 1.3410 horsepower | |
1 horsepower | = 550 foot-pounds per second |
= 33,000 foot-pounds per minute | |
= 0.7456 kilowatt | |
= 745.7 watts | |
1 foot-pound/sec. | = 1.35582 watts |
= 1.8182 × 10^{-3} horsepower |
Except for purposes of exact measurement one will rarely need to employ more than the first three or four of the figures of the above conversion factors. Hence, approximately,
1 meter = 39.37 inches
1 kilogram = 2.20 pounds
Derived Units. The foregoing units of mass, length, and time are said to be fundamental. By means of these we can also measure a large number of other secondary quantities which are accordingly said to be derived quantities. For example, area is a derived quantity depending upon length, and a rational unit of area is a square whose length of side is the unit of length. Similarly, the unit of volume is a cube whose length of side is equal to the unit of length.
Less obvious derived units are speed and velocity, and acceleration which are terms used in describing the motion of a body. When a body moves its speed is the ratio of the distance it travels in a small interval of time to the time required. It is thus measurable in terms of a length divided by a time, and so requires no other units than those of length and time already defined. We may express a speed in meters per second, kilometers per hour, yards per minute, or any other convenient length and time units.
The velocity of a moving body at a given instant is its speed in a particular direction. For example, two bodies having the same speeds, but one moving eastward and the other northward are said to have different velocities. A point on the rim of a flywheel rotating uniformly describes a circular path at uniform speed, but since its direction of motion is changing continuously, its velocity is also changing continuously.
Quantities like velocity which have both magnitudes and directions are called vector quantities.
The acceleration of a body is its rate of change of velocity. When the body is moving in a straight line this becomes equal to its rate of change of speed. For example, when an automobile is moving along a straight road, if it increases its speed it is said to be positively accelerated; if it decreases its speed the acceleration is negative. We commonly speak of the foot pedal for the gasoline feed as the ‘accelerator’. The brake, however, is just as truly an accelerator. If an automobile is increasing its speed uniformly at the rate of a mile per hour each second, we say that the acceleration is 1 mile per hour per second. This is clearly equal to 1.47 feet per second for each second, or to 47.7 centimeters per second for each second. From this we see that an acceleration involves the measurement of a distance, and the division of this by two measured time intervals. If we make these two time intervals the same, then acceleration becomes: (distance/time)/time, or distance/ (time)^{2}. Thus an acceleration of one cm./sec^{2}. means that the body is changing its velocity by an amount of 1 centimeter per second during each second.
Acceleration, like velocity, is also a vector quantity. Its direction is that of the change of velocity. What we mean by this can be shown by representing the velocity by an arrow whose length is proportional to the speed, and whose direction is that of the motion. Suppose the motion is curvilinear with the speed continuously varying. The velocity vectors represented by arrows for successive times will have different directions and lengths. If we take two of these arrows representing the motion at two successive times only a short interval apart and place them with their feathered ends at the same point, their tips will not coincide. Now if we place a small arrow with its tail at the tip of the first arrow, and its tip at the tip of the second, this small arrow will represent, both in magnitude and direction, the change of velocity during the time interval considered. The average acceleration during that time is the ratio of the change of velocity to the time required to affect the change, and has the same direction as the change of the velocity.
If this type of construction is tried with respect to uniform circular motion, it will be seen immediately that the velocity is continuously changing in a direction toward the center of the circle. Consequently the acceleration is also toward the center of the circle. If the motion is not at constant speed this will not be true.
Force. We come now to the concept of force. Our primitive experience with force is by means of our muscular sense of pushing and pulling. We can render this measurable by means of the stretch of springs, or the pull of gravity on bodies of known mass. A dynamic method of measuring force is by means of the acceleration of a body of known mass. For example, suppose we construct a small car with as nearly as possible frictionless bearings, and run it on a straight horizontal track. Suppose that we pull the car by means of a stretched spring or rubber band kept at constant tension. The car will accelerate uniformly in the direction of the pull. Now, if we load the car with different masses and repeat the experiment, for the same tension of the spring the acceleration will be greater when the load is decreased, and less when it is increased. If we keep the load constant and employ different tensions on the spring, the acceleration will increase as the tension is increased.
Quantitatively, after correcting for any residual friction, what we learn in this manner is that the acceleration of the car is directly proportional to the tension of the spring, or to the applied force, and inversely proportional to the total mass of the car and its contents.
By experiments similar to this it has been shown quite generally and very exactly that this is true for any kind of a body undergoing any kind of an acceleration: The acceleration is proportional to the applied force (or resultant of the applied forces where several act simultaneously), and inversely proportional to the mass. The direction of the acceleration is the same as that of the applied force. Conversely, the applied force has the direction of the acceleration and its magnitude is proportional to the acceleration and to the mass of the body accelerated.
Since we already know how to measure acceleration in terms of length and time, and how to measure mass, this last fact enables us to measure forces in terms of masses and accelerations.
In this manner we define a unit of force to be that force which causes a unit of mass to move with a unit of acceleration.
In the Metric system, using the gram, the centimeter, and the second as our units of mass, length and time, respectively, the unit of force is that amount of force which will cause 1 gram of mass to move with an acceleration of 1 centimeter per second for each second the force is applied. This amount of force we call a dyne.
At the latitude of New York the pull of gravity on a mass is such that if it is free to move with no other forces acting upon it, starting from rest it will move in the direction of the force exerted by gravity with a uniform acceleration of 980 cm./sec., or 32.2 ft./sec.^{2}. Since this is true for a mass of any size, then for a 1-gram mass the force must be 980 dynes, since the acceleration in this case is 980 times as great as that produced by a force of 1 dyne. For a mass of m grams the total force would have to he m times as great as for one gram in order to have the same acceleration.
We can obtain an approximate idea of the size of a dyne if we consider that a nickel coin (5 cents) has a mass of 5 grams. The force exerted by gravity upon this is therefore 5 x 980, or 4,900 dynes. Thus, approximately, a dyne is one five-thousandths part of the force exerted by gravity upon a nickel.
Engineers frequently use another method of measuring force. They take as
their unit of force the pull of gravity on a unit of mass, or its weight. The difficulty with this is that gravity is not the same at different parts of the earth. It varies with elevation above sea level, with the latitude, and with certain other random disturbing factors. Hence, to be exact we must define what the value of gravity is to be. This is commonly taken to be 980.665 cm./sec.^{2} which is approximately the mean value of gravity at sea level and latitude 45°. The pull of this standard gravity on a 1-pound mass is a pound weight. The corresponding pull of gravity on a kilogram of mass is a kilogram weight. Since a pound is 453.592 grams, and the attraction of this standard gravity on a gram mass is 980.665 dynes, it follows that a pound weight is the product of these two figures, or 444,820 dynes.
Work. When a force acts upon a body and causes it to move, work is said to be done. A unit of work is defined to be that which is done when a unit of force causes its point of application to move a unit of distance in the direction in which the force acts. In the English system when the unit of length is the foot and the unit of force the pound, the unit of work is the foot-pound. Hence the total number of foot-pounds of work done by a given force is the product of the force in pounds by the distance its point of application is moved in the direction of action of the force, in feet. The simplest example is afforded by the lifting of a weight. It requires 1 foot-pound of work to lift a 1-pound mass a height of 1 foot.
In the Metric system when a force of 1 dyne causes its point of application to move in the direction of the force a distance of 1 centimeter, the work performed is defined to be 1 erg. Like the dyne, the erg is a very small quantity so that a larger unit of work is useful. We obtain such a larger unit if we arbitrarily define 10,000,000 ergs to be one joule.
The conversion factors between the English and the Metric units of work are easily obtained by computing in both systems of units the work done in lifting a pound mass a height of 1 foot against standard gravity. In the English units this is simply 1 foot-pound. In Metric units the force, as we have already noted, is 444,820 dynes, and the distance 30.4800 centimeters. The work is therefore the product of these quantities, or 13,558,200 ergs or 1.35582 joules. Inversely, a joule is 0.73756 foot-pounds, or the amount of work required to lift a pound mass a height of 8.84 inches, and an erg is one ten-millionth of this amount of work.
Power. Power is the time rate of doing work. In the Metric system, when work is performed at the rate of 1 joule per second, the power is defined to be 1 watt. Work at the rate of 1,000 joules per second is a thousand watts or a kilowatt. In the English system, the unit of work is the horsepower. This unit was defined by James Watt, who attempted to determine the rate at which a draft horse could do work so that he could use this for rating the power of his steam engines. The result he achieved was that 1 horsepower is a rate of doing work of 33,000 foot-pounds per minute, or 550 foot-pounds per second. Since a kilowatt is 1,000 joules per second, or 737.56 foot-pounds per second, it follows that this is equal to 1.3410 horsepower, or that a horsepower is equal to 745.70 watts, or 0.74570 kilowatts.
A kilowatt-hour is the amount of work done by a kilowatt of power in 1 hour; a horsepower-hour is the amount of work done by a horsepower in 1 hour. These are accordingly units of work, the kilowatt-hour being 1,000 joules per second for 3,600 seconds, or 3,600,000 joules, and the horsepower-hour 33,000 foot-pounds per minute for 60 minutes or 1,980,000 foot-pounds. Also 1 kilowatt-hour bears to a horsepower-hour the same ratio as the kilo watt to the horsepower.
Conversion Factors. While all of the conversions between the foregoing units of measurement are easily derived in the manner we have just seen, it is convenient to have at hand a table of conversion factors for ready reference. Such a table containing the factors that are most often used is given below. In this let us introduce for the first time here a system of notation for writing numbers that is widely used by scientists and engineers but may not be familiar to some of the readers. When dealing with very large numbers or very small decimal fractions it is bothersome and confusing to have to write out numbers like 2,684,500; which is the number of joules in a horsepower-hour, or 0.000,000,737,56 which is the number of foot-pounds in an erg. We may note that
2,684,500 = 2.6845×1,000,000 = 2.6845×10^{6},
and similarly, that
0.00000073756 = 7.3756×^{1}/_{10,000,000} = 7.3756×10^{−7}.
Any number, large or small, can be written in this manner which has many advantages over the longhand method. In the following table this system will be used for the very large and very small numbers.
In this table the factors are expressed to five or six significant figures. For all ordinary calculations only the first three or four figures are needed and all the rest can be dropped or set equal to zero. They are only needed when very exact measurements have been made and hence very exact calculations required. Most measurements are not more accurate than 1 part in 1,000, and calculation more exact than this is meaningless for such measurements.
Examples of Work and Power. Lest we lose sight of the fundamental simplicity of the concepts of work and power and become confused by the array of conversion factors, let us consider a few simple examples.Examples of Work and Power
- THE POWER IN CLIMBING STAIRS. How much power does a man generate in climbing stairs, for example? At an average rate of walking a man will climb a height of about 36 feet per minute. In so doing he is lifting his own weight. Suppose he weighs 150 pounds. Then his rate of doing work is 5,400 foot-pounds per minute, or 90 foot-pounds per second. Since a horsepower is 550 foot-pounds per second, and a watt is 0.73756 foot-pounds per second, it follows that he generates 0.164 horsepower, or 122 watts. This is in round numbers one-sixth of a horsepower. If he ran up the stairs six times as fast he would generate 1 horsepower. Running at such a rate, however, could only be maintained for a few seconds. Even walking at the above rate can be continued by few people for more than a few minutes. For example, few people can walk steadily, without stopping for rest, from the ground to the top of the Washington Monument which is over 500 feet high. Climbing for 8 hours would give an average rate much smaller than that of walking up a few flights of stairs, and so would reduce correspondingly the average power generated.
- LIFTING PACKAGES. Suppose a workman lifts packages from the ground to trucks 4 feet above the ground. In 6 hours he lifts 65 tons. How much work does he do, and what is the average power? The work done is 520,000 foot-pounds. This is 0.26 horsepower-hour, or 0.20 kilowatt-hour. The power averaged is 24 foot-pounds per second which is 3.3 watts, or 0.044 horsepower.
- PUMPING WATER. A man pumps water for 10 hours with a handpump. In that time he raises 14,000 gallons a height of 10 feet. What is his work and his average power? A gallon of water weighs 8.337 pounds. The work done is therefore 1,170,000 foot-pounds, or 0.44 kilowatt-hour. The average power is 44 watts.
- SHOVELING LOOSE DIRT. In 10 hours a man shovels 25 tons of loose dirt over a wall 5 feet 3 inches high. What is the work and average power? The work done is 262,500 foot-pounds, or 0.10 kilowatt-hour. The power is 7.28 foot-pounds per second, or 10 watts.
- CARRYING A HOD. In 6 hours a man carrying a hod raises 17 tons of plaster 12 feet. The work is 408,000 foot-pounds, or 0.154 kilowatthour. The average power is 18.8 foot-pounds per second, or 25 watts.
- PUSHING A WHEELBARROW. A man with a wheelbarrow raises 51 tons of concrete a height of 3 feet in 10 hours. The work done is 306,000 foot-pounds, or 0.115 kilowatt-hour. The average power is 8.5 foot-pounds per second, or 11.5 watts.
These examples give one a very good idea of how much useful work a man can do in a day. In work of these kinds we have counted only the useful work accomplished. In each case the work actually done was greater than that computed. In the wheelbarrow problem the total work performed should include the repeated lifting of both the wheelbarrow and the man himself. If the wheelbarrow load was 200 pounds and the man and empty wheelbarrow weighed another 200 pounds, then it is clear that the actual work performed would be twice that computed, not allowing for the friction of the wheelbarrow.
A kilowatt-hour of work will lift a ton weight a quarter of a mile high; a kilowatt of power will do this in one hour of time. Working under the most efficient conditions, it would take at least 13 men to do the same amount of work in the same time. Under less efficient conditions the number of men would be correspondingly greater.
This same kilowatt-hour is the unit for which we pay our monthly electric light bill at a domestic rate of 5—7 cents each. Commercial rates on electric power range from a few mills to a cent or so per kilowatt-hour. A workman whose pay is less than 25 cents per hour is working at practically starvation wages. The conjunction of these two facts is of rather obvious social significance.
References:
This Mechanical World, Mott-Smith.
A Textbook of Physics, Vol. I, Grimselil.