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CHAPTER ONE UNITS AND MEASUREMENT 1.1 INTRODUCTION Measurement of any physical quantity involves comparison with a certain ba...

CHAPTER ONE UNITS AND MEASUREMENT 1.1 INTRODUCTION Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. The result of a 1.1 Introduction measurement of a physical quantity is expressed by a 1.2 The international system of number (or numerical measure) accompanied by a unit. units Although the number of physical quantities appears to be 1.3 Significant figures very large, we need only a limited number of units for expressing all the physical quantities, since they are inter- 1.4 Dimensions of physical related with one another. The units for the fundamental or quantities base quantities are called fundamental or base units. The 1.5 Dimensional formulae and units of all other physical quantities can be expressed as dimensional equations combinations of the base units. Such units obtained for the 1.6 Dimensional analysis and its derived quantities are called derived units. A complete set applications of these units, both the base units and derived units, is known as the system of units. Summary Exercises 1.2 THE INTERNATIONAL SYSTEM OF UNITS In earlier time scientists of different countries were using different systems of units for measurement. Three such systems, the CGS, the FPS (or British) system and the MKS system were in use extensively till recently. The base units for length, mass and time in these systems were as follows : In CGS system they were centimetre, gram and second respectively. In FPS system they were foot, pound and second respectively. In MKS system they were metre, kilogram and second respectively. The system of units which is at present internationally accepted for measurement is the Système Internationale d’ Unites (French for International System of Units), abbreviated as SI. The SI, with standard scheme of symbols, units and abbreviations, developed by the Bureau International des Poids et measures (The International Bureau of Weights and Measures, BIPM) in 1971 were recently revised by the General Conference on Weights and Measures in November 2018. The scheme is now for 2024-25 2 PHYSICS international usage in scientific, technical, industrial and commercial work. Because SI units used decimal system, conversions within the system are quite simple and convenient. We shall follow the SI units in this book. In SI, there are seven base units as given in (a) Table 1.1. Besides the seven base units, there are two more units that are defined for (a) plane angle dθ as the ratio of length of arc ds to the radius r and (b) solid angle dΩ as the ratio of the intercepted area dA of the spherical surface, described about the apex O as the centre, to the square of its radius r, as shown in Fig. 1.1(a) and (b) respectively. The unit for plane angle (b) is radian with the symbol rad and the unit for the solid angle is steradian with the symbol sr. Both these are Fig. 1.1 Description of (a) plane angle dθ and dimensionless quantities. (b) solid angle dΩ. Table 1.1 SI Base Quantities and Units* Base SI Units quantity Name Symbol Definition Length metre m The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299792458 when expressed in the unit m s–1 , where the second is defined in terms of the caesium frequency ∆ν cs. Mass kilogram kg The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015×10–34 when expressed in the unit J s, which is equal to kg m2 s–1, where the metre and the second are defined in terms of c and ∆ν cs. Time second s The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency ∆ν cs, the unperturbed ground- state hyperfine transition frequency of the caesium-133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to s–1. Electric ampere A The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602176634×10–19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of ∆ν cs. Thermo kelvin K The kelvin, symbol K, is the SI unit of thermodynamic temperature. dynamic It is defined by taking the fixed numerical value of the Boltzmann constant Temperature k to be 1.380649×10–23 when expressed in the unit J K–1, which is equal to kg m2 s–2 k–1, where the kilogram, metre and second are defined in terms of h, c and ∆ν cs. Amount of mole mol The mole, symbol mol, is the SI unit of amount of substance. One mole substance contains exactly 6.02214076×1023 elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit mol–1 and is called the Avogadro number. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles. Luminous candela cd The candela, symbol cd, is the SI unit of luminous intensity in given direction. intensity It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540×1012 Hz, Kcd, to be 683 when expressed in the unit lm W–1, which is equal to cd sr W–1, or cd sr kg–1m–2s3, where the kilogram, metre and second are defined in terms of h, c and ∆ν cs. * The values mentioned here need not be remembered or asked in a test. They are given here only to indicate the extent of accuracy to which they are measured. With progress in technology, the measuring techniques get improved leading to measurements with greater precision. The definitions of base units are revised to keep up with this progress. 2024-25 UNITS AND MEASUREMENT 3 Table 1.2 Some units retained for general use (Though outside SI) Note that when mole is used, the elementary the first uncertain digit are known as entities must be specified. These entities significant digits or significant figures. If we may be atoms, molecules, ions, electrons, say the period of oscillation of a simple other particles or specified groups of such pendulum is 1.62 s, the digits 1 and 6 are particles. reliable and certain, while the digit 2 is We employ units for some physical quantities uncertain. Thus, the measured value has three that can be derived from the seven base units significant figures. The length of an object (Appendix A 6). Some derived units in terms of reported after measurement to be 287.5 cm has the SI base units are given in (Appendix A 6.1). four significant figures, the digits 2, 8, 7 are Some SI derived units are given special names certain while the digit 5 is uncertain. Clearly, (Appendix A 6.2 ) and some derived SI units make reporting the result of measurement that use of these units with special names and the includes more digits than the significant digits seven base units (Appendix A 6.3). These are is superfluous and also misleading since it given in Appendix A 6.2 and A 6.3 for your ready would give a wrong idea about the precision of reference. Other units retained for general use measurement. are given in Table 1.2. The rules for determining the number of Common SI prefixes and symbols for multiples significant figures can be understood from the and sub-multiples are given in Appendix A2. following examples. Significant figures General guidelines for using symbols for physical indicate, as already mentioned, the precision quantities, chemical elements and nuclides are given in Appendix A7 and those for SI units and of measurement which depends on the least some other units are given in Appendix A8 for count of the measuring instrument. A choice your guidance and ready reference. of change of different units does not change the number of significant digits or 1.3 SIGNIFICANT FIGURES figures in a measurement. This important As discussed above, every measurement remark makes most of the following involves errors. Thus, the result of observations clear: measurement should be reported in a way that (1) For example, the length 2.308 cm has four indicates the precision of measurement. significant figures. But in different units, the Normally, the reported result of measurement same value can be written as 0.02308 m or 23.08 is a number that includes all digits in the mm or 23080 µm. number that are known reliably plus the first All these numbers have the same number of digit that is uncertain. The reliable digits plus significant figures (digits 2, 3, 0, 8), namely four. 2024-25 4 PHYSICS This shows that the location of decimal point is negative exponent (or power) of 10. In order to of no consequence in determining the number get an approximate idea of the number, we may of significant figures. round off the number a to 1 (for a ≤ 5) and to 10 The example gives the following rules : (for 5