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(b) Measuring the meter
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Physics is an experimental science. Experiments require measurements, and we generally use numbers to describe the results of measurements. Any number that is used to describe a physical phenomenon quantitatively is called a physical quantity. For example, two physical quantities that describe you are your weight and your height. Some physical quantities are so fundamental that we can define them only by describing how to measure them. Such a definition is called an operational definition. Two examples are measuring a distance by using a ruler and measuring a time interval by using a stopwatch. In other cases we define a physical quantity by describing how to calculate it from other quantities that we can measure. Thus we might define the average speed of a moving object as the distance traveled (measured with a ruler) divided by the time of travel (measured with a stopwatch). When we measure a quantity, we always compare it with some reference standard. When we say that a Ferrari 458 Italia is 4.53 meters long, we mean that it is 4.53 times as long as a meter stick, which we define to be 1 meter long. Such a standard defines a unit of the quantity. The meter is a unit of distance, and the second is a unit of time. When we use a number to describe a physical quantity, we must always specify the unit that we are using; to describe a distance as simply “4.53” wouldn’t mean anything. To make accurate, reliable measurements, we need units of measurement that do not change and that can be duplicated by observers in various locations. The system of units used by scientists and engineers around the world is commonly called “the metric system”.
The laws of physics are expressed in terms of basic quantities that require a clear definition. In mechanics, the three basic quantities are length (L), mass (M), and time (T). All other quantities in mechanics can be expressed in terms of these three.
Table of Units for Three SI
Base Quantities ;
Quantity Unit Name Unit Symbol
Length meter m
Time second s
Mass kilogram kg
If we are to report the results of a measurement to someone who wishes to reproduce this measurement, a standard must be defined. It would be meaningless if a visitor from another planet were to talk to us about a length of 8 “glitches” if we do not know the meaning of the unit glitch. On the other hand, if someone familiar with our system of measurement reports that a wall is 2 meters high and our unit of length is defined to be 1 meter, we know that the height of the wall is twice our basic length unit. Likewise, if we are told that a person has a mass of 75 kilograms and our unit of mass is defined to be 1 kilogram, then that person is 75 times as massive as our basic unit.1 Whatever is chosen as a standard must be readily accessible and possess some property that can be measured reliably—measurements taken by different people in different places must yield the same results.
The International System of Units:
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In 1971, the 14th General Conference on Weights and Measures picked seven quantities as base quantities, thereby forming the basis of the International System of Units, abbreviated SI from its French name and popularly known as the metric system. Table shows the units for the three base quantities—length, mass, and time—that we use in the early chapters of this book. These units were defined to be on a “human scale.”
Many SI derived units are defined in terms of these base units. For example, the SI unit for power, called the watt (W), is defined in terms of the base units for mass, length, and time.
1 watt = 1 W = 1 kg . m² / s³, (1-1)
, where the last collection of unit symbols is read as kilogram-me
Known universe 1 ' 1053 Our galaxy 2 ' 1041 Sun 2 ' 1030 Moon 7 ' 1022 Asteroid Eros 5 ' 1015 Small mountain 1 ' 1012 Ocean liner 7 ' 107 Elephant 5 ' 103 Grape 3 ' 10$3 Speck of dust 7 ' 10$10 Penicillin molecule 5 ' 10$17 Uranium atom 4 ' 10$25 Proton 2 ' 10$27 Electron 9 ' 10$3 squared per second cubed.
To express the very large and very small quantities we often run into in physics, we use scientific notation, which employs powers of 10. In this notation,
3 560 000 000 m = 3.56 × 10⁹ m (1-2)
and
0.000 000 492 s = 4.92 × 10-⁷ s. (1-3)
Scientific notation on computers sometimes takes on an even briefer look, as in 3.56 E9 and 4.92 E–7, where E stands for “exponent of ten.” It is briefer still on some calculators, where E is replaced with an empty space.
As a further convenience when dealing with very large or very small measurements, we use the prefixes listed in Table . As you can see, each prefix represents a certain power of 10, to be used as a multiplication factor. Attaching a prefix to an SI unit has the effect of multiplying by the associated fact. Thus, we can express a particular electric power as
1.27 × 10⁹ watts = 1.27 gigawatts = 1.27 GW (1-4)
or a particular time interval as
2.35 × 10-⁹ s = 2.35 nanoseconds = 2.35 n s. (1-5)
Some prefixes, as used in milliliter, centimeter, kilogram, and megabyte, are probably familiar to you.
Changing Units:
Changing Units We often need to change the units in which a physical quantity is expressed. We do so by a method called chain-link conversion. In this method, we multiply the original measurement by a conversion factor (a ratio of units that is equal to unity). For example, because 1 min and 60 s are identical time intervals, we have
1 min / 60 s = 1 and 60 s / 1 min = 1.
Thus, the ratios (1 min)/(60 s) and (60 s)/(1 min) can be used as conversion factors. This is not the same as writing or 60 " 1; each number and its unit must be treated together.
Because multiplying any quantity by unity leaves the quantity unchanged, we can introduce conversion factors wherever we find them useful. In chain-link conversion, we use the factors to cancel unwanted units. For example, to convert 2 min to seconds, we have
2 min = (2 min)(1) = (2 min) (60 s / 1 min) = 120 s (1-6)
If you introduce a conversion factor in such a way that unwanted units do not cancel, invert the factor and try again. In conversions, the units obey the same algebraic rules as variables and numbers.
Appendix D gives conversion factors between SI and other systems of units, including non-SI units still used in the United States. However, the conversion factors are written in the style of “1 min " 60 s” rather than as a ratio. So, you need to decide on the numerator and denominator in any needed ratio.
Length
In 1792, the newborn Republic of France established a new system of weights and measures. Its cornerstone was the meter, defined to be one ten-millionth of the distance from the north pole to the equator. Later, for practical reasons, this Earth standard was abandoned and the meter came to be defined as the distance between two fine lines engraved near the ends of a platinum–iridium bar, the standard meter bar, which was kept at the International Bureau of Weights and Measures near Paris. Accurate copies of the bar were sent to standardizing laboratories throughout the world. These secondary standards were used to produce other, still more accessible standards, so that ultimately every measuring device derived its authority from the standard meter bar through a complicated chain of comparisons. Eventually, a standard more precise than the distance between two fine scratches on a metal bar was required. In 1960, a new standard for the meter, based on the wavelength of light, was adopted. Specifically, the standard for the meter was redefined to be 1 650 763.73 wavelengths of a particular orange-red light emitted by atoms of krypton-86 (a particular isotope, or type, of krypton) in a gas discharge tube that can be set up anywhere in the world. This awkward number of wavelengths was chosen so that the new standard would be close to the old meter-bar standard.
By 1983, however, the demand for higher precision had reached such a point that even the krypton-86 standard could not meet it, and in that year a bold step was taken. The meter was redefined as the distance traveled by light in a specified time interval. In the words of the 17th General Conference on Weights and Measures;
👉 The meter is the length of the path traveled by light in a vacuum during a time interval of 1/299 792 458 of a second.
This time interval was chosen so that the speed of light c is exactly
c = 299 792 458 m/s.
Measurements of the speed of light had become extremely precise, so it made sense to adopt the speed of light as a defined quantity and to use it to redefine the meter. Table in below shows a wide range of lengths, from that of the universe (top line) to those of some very small objects.
Table of Some Approximate Length
Measurement Length in Meters
Distance to the first galaxies formed 2 × 10²⁶
Distance to the Andromeda galaxy 2 × 10²²
Distance to the nearby star Proxima Centauri 4 × 10¹⁶
Distance to Pluto 6 × 10¹²
Radius of Earth 6 × 10⁶
Height of Mt. Everest 9 × 10³
Thickness of this page 1 × 10-⁴
Length of a typical virus 1 × 10-⁸
Radius of a hydrogen atom 5 × 10-¹¹
Radius of a proton 1 × 10-¹⁵
Time:
Time has two aspects. For civil and some scientific purposes, we want to know the time of day so that we can order events in sequence. In much scientific work, we want to know how long an event lasts. Thus, any time standard must be able to answer two questions: “When did it happen?” and “What is its duration?” Table 1-4 shows some time intervals. Any phenomenon that repeats itself is a possible time standard. Earth’s rotation, which determines the length of the day, has been used in this way for centuries; Fig. 1-1 shows one novel example of a watch based on that rotation. A quartz clock, in which a quartz ring is made to vibrate continuously, can be calibrated against Earth’s rotation via astronomical observations and used to measure time intervals in the laboratory. However, the calibration cannot be carried out with the accuracy called for by modern scientific and engineering technology.
Table of Some Approximate Time Intervals
Measurement Time Interval in Seconds
Lifetime of the proton (predicted) 3 × 10⁴⁰
Age of the universe 5 × 10¹⁷
Age of the pyramid of Cheops 1 × 10¹¹
Human life expectancy 2 × 10⁹
Length of a day 9 × 10⁴
Time between human heartbeats 8 × 10-¹
Lifetime of the muon 2 × 10-⁶
Shortest lab light pulse 1 × 10-¹⁶
Lifetime of the most unstable particle 1 × 10-²³
The Planck time a 1 × 10-⁴³
To meet the need for a better time standard, atomic clocks have been developed. An atomic clock at the National Institute of Standards and Technology (NIST) in Boulder, Colorado, is the standard for Coordinated Universal Time (UTC) in the United States. Its time signals are available by shortwave radio (stations WWV and WWVH) and by telephone (303-499-7111). Time signals (and related information) are also available from the United States Naval Observatory at website http://tycho.usno.navy.mil/time.html. (To set a clock extremely accurately at your particular location, you would have to account for the travel time required for these signals to reach you.) Figure 1-2 shows variations in the length of one day on Earth over a 4-year period, as determined by comparison with a cesium (atomic) clock. Because the variation displayed by Fig. 1-2 is seasonal and repetitious, we suspect the rotating Earth when there is a difference between Earth and atom as timekeepers. The variation is due to tidal effects caused by the Moon and to large-scale winds. The 13th General Conference on Weights and Measures in 1967 adopted a standard second based on the cesium clock;
👉 One second is the time taken by 9 192 631 770 oscillations of the light (of a specified wavelength) emitted by a cesium-133 atom.
Atomic clocks are so consistent that, in principle, two cesium clocks would have to run for 6000 years before their readings would differ by more than 1 s. Even such accuracy pales in comparison with that of clocks currently being developed; their precision may be 1 part in 10¹⁸—that is, 1 s in 1 × 10¹⁸ s (which is about 3 × 10¹⁰ y)
Mass :
The Standard Kilogram
The SI standard of mass is a cylinder of platinum and iridium that is kept at the International Bureau of Weights and Measures near Paris and assigned, by international agreement, a mass of 1 kilogram. Accurate copies have been sent to standardizing laboratories in other countries, and the masses of other bodies can be determined by balancing them against a copy. Table 1-5 shows some masses expressed in kilograms, ranging over about 83 orders of magnitude. The U.S. copy of the standard kilogram is housed in a vault at NIST. It is removed, no more than once a year, for the purpose of checking duplicate copies that are used elsewhere. Since 1889, it has been taken to France twice for recomparison with the primary standard.
Table of Some Approximate Masses
Object Mass in Kilograms
Known universe 1 × 10⁵³
Our galaxy 2 × 10⁴¹
Sun 2 × 10³⁰
Moon 7 × 10²²
Asteroid Eros 5 × 10¹⁵
Small mountain 1 × 10¹²
Ocean liner 7 × 10⁷
Elephant 5 × 10³
Grape 3 × 10-³
Speck of dust 7 × 10-¹⁰
Penicillin molecule 5 × 10-¹⁷
Uranium atom 4 × 10-²⁵
Proton 2 × 10-²⁷
Electron 9 × 10-³¹
A Second Mass Standard The masses of atoms can be compared with one another more precisely than they can be compared with the standard kilogram. For this reason, we have a second mass standard. It is the carbon-12 atom, which, by international agreement, has been assigned a mass of 12 atomic mass units (u). The relation between the two units is
1 u = 1.660 538 86 × 10-²⁷ kg, (1-7)
with an uncertainty of %10 in the last two decimal places. Scientists can, with reasonable precision, experimentally determine the masses of other atoms relative to the mass of carbon-12. What we presently lack is a reliable means of extending that precision to more common units of mass, such as a kilogram.
Density:
As we shall discuss further in Chapter 14, density r (lowercase Greek letter rho) is the mass per unit volume:
p = m/v (1-8)
Densities are typically listed in kilograms per cubic meter or grams per cubic centimeter. The density of water (1.00 gram per cubic centimeter) is often used as a comparison. Fresh snow has about 10% of that density; platinum has a density that is about 21 times that of water.
Review & Summary
Measurement in Physics :
Physics is based on measurement of physical quantities. Certain physical quantities have been chosen as base quantities (such as length, time, and mass); each has been defined in terms of a standard and given a unit of measure (such as meter, second, and kilogram). Other physical quantities are defined in terms of the base quantities and their standards and units.
SI Units :
The unit system emphasized in this book is the International System of Units (SI). The three physical quantities displayed in Table 1-1 are used in the early chapters. Standards, which must be both accessible and invariable, have been established for these base quantities by international agreement. These standards are used in all physical measurement, for both the base quantities and the quantities derived from them. Scientific notation and the prefixes of Table 1-2 are used to simplify measurement notation.
Changing Units :
Conversion of units may be performed by using chain-link conversions in which the original data are multiplied successively by conversion factors written as unity and the units are manipulated like algebraic quantities until only the desired units remain.
Length :
The meter is defined as the distance traveled by light during a precisely specified time interval. Time The second is defined in terms of the oscillations of light emitted by an atomic (cesium-133) source. Accurate time signals are sent worldwide by radio signals keyed to atomic clocks in standardizing laboratories.
Mass :
The kilogram is defined in terms of a platinum– iridium standard mass kept near Paris. For measurements on an atomic scale, the atomic mass unit, defined in terms of the atom carbon-12, is usually used.
Density :
The density r of a material is the mass per unit volume:
p = m/v
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