Page 1 CHAPTER TWO UNITS AND MEASUREMENT 2.1 INTRODUCTION Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. The result of a measurement of a physical quantity is expressed by a number (or numerical measure) accompanied by a unit. Although the number of physical quantities appears to be very large, we need only a limited number of units for expressing all the physical quantities, since they are inter- related with one another. The units for the fundamental or base quantities are called fundamental or base units. The units of all other physical quantities can be expressed as combinations of the base units. Such units obtained for the derived quantities are called derived units. A complete set of these units, both the base units and derived units, is known as the system of units. 2.2 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 2.1 Introduction 2.2 The international system of units 2.3 Measurement of length 2.4 Measurement of mass 2.5 Measurement of time 2.6 Accuracy, precision of instruments and errors in measurement 2.7 Significant figures 2.8 Dimensions of physical quantities 2.9 Dimensional formulae and dimensional equations 2.10 Dimensional analysis and its applications Summary Exercises Additional exercises 2020-21 Page 2 CHAPTER TWO UNITS AND MEASUREMENT 2.1 INTRODUCTION Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. The result of a measurement of a physical quantity is expressed by a number (or numerical measure) accompanied by a unit. Although the number of physical quantities appears to be very large, we need only a limited number of units for expressing all the physical quantities, since they are inter- related with one another. The units for the fundamental or base quantities are called fundamental or base units. The units of all other physical quantities can be expressed as combinations of the base units. Such units obtained for the derived quantities are called derived units. A complete set of these units, both the base units and derived units, is known as the system of units. 2.2 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 2.1 Introduction 2.2 The international system of units 2.3 Measurement of length 2.4 Measurement of mass 2.5 Measurement of time 2.6 Accuracy, precision of instruments and errors in measurement 2.7 Significant figures 2.8 Dimensions of physical quantities 2.9 Dimensional formulae and dimensional equations 2.10 Dimensional analysis and its applications Summary Exercises Additional exercises 2020-21 Table 2.1 SI Base Quantities and Units* 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 Table 2.1. Besides the seven base units, there are two more units that are defined for (a) plane angle d? as the ratio of length of arc ds to the radius r and (b) solid angle dO 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. 2.1(a) and (b) respectively. The unit for plane angle is radian with the symbol rad and the unit for the solid angle is steradian with the symbol sr. Both these are dimensionless quantities. UNITS AND MEASUREMENT 17 (a) (b) Fig. 2.1 Description of (a) plane angle d? and (b) solid angle dO . 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 ??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 m 2 s –1 , where the metre and the second are defined in terms of c and ??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 ??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 ??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 m 2 s –2 k –1 , where the kilogram, metre and second are defined in terms of h, c and ??cs. Amount of mole mol The mole, symbol mol, is the SI unit of amount of substance. One mole substance contains exactly 6.02214076×10 23 elementary entities. This number is the fixed numerical value of the Avogadro constant, N A , 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 540×10 12 Hz, K cd , to be 683 when expressed in the unit lm W –1 , which is equal to cd sr W –1 , or cd sr kg –1 m –2 s 3 , where the kilogram, metre and second are defined in terms of h, c and ??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. 2020-21 Page 3 CHAPTER TWO UNITS AND MEASUREMENT 2.1 INTRODUCTION Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. The result of a measurement of a physical quantity is expressed by a number (or numerical measure) accompanied by a unit. Although the number of physical quantities appears to be very large, we need only a limited number of units for expressing all the physical quantities, since they are inter- related with one another. The units for the fundamental or base quantities are called fundamental or base units. The units of all other physical quantities can be expressed as combinations of the base units. Such units obtained for the derived quantities are called derived units. A complete set of these units, both the base units and derived units, is known as the system of units. 2.2 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 2.1 Introduction 2.2 The international system of units 2.3 Measurement of length 2.4 Measurement of mass 2.5 Measurement of time 2.6 Accuracy, precision of instruments and errors in measurement 2.7 Significant figures 2.8 Dimensions of physical quantities 2.9 Dimensional formulae and dimensional equations 2.10 Dimensional analysis and its applications Summary Exercises Additional exercises 2020-21 Table 2.1 SI Base Quantities and Units* 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 Table 2.1. Besides the seven base units, there are two more units that are defined for (a) plane angle d? as the ratio of length of arc ds to the radius r and (b) solid angle dO 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. 2.1(a) and (b) respectively. The unit for plane angle is radian with the symbol rad and the unit for the solid angle is steradian with the symbol sr. Both these are dimensionless quantities. UNITS AND MEASUREMENT 17 (a) (b) Fig. 2.1 Description of (a) plane angle d? and (b) solid angle dO . 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 ??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 m 2 s –1 , where the metre and the second are defined in terms of c and ??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 ??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 ??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 m 2 s –2 k –1 , where the kilogram, metre and second are defined in terms of h, c and ??cs. Amount of mole mol The mole, symbol mol, is the SI unit of amount of substance. One mole substance contains exactly 6.02214076×10 23 elementary entities. This number is the fixed numerical value of the Avogadro constant, N A , 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 540×10 12 Hz, K cd , to be 683 when expressed in the unit lm W –1 , which is equal to cd sr W –1 , or cd sr kg –1 m –2 s 3 , where the kilogram, metre and second are defined in terms of h, c and ??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. 2020-21 PHYSICS 18 Table 2.2 Some units retained for general use (Though outside SI) Note that when mole is used, the elementary entities must be specified. These entities may be atoms, molecules, ions, electrons, other particles or specified groups of such particles. We employ units for some physical quantities that can be derived from the seven base units (Appendix A 6). Some derived units in terms of the SI base units are given in (Appendix A 6.1). Some SI derived units are given special names (Appendix A 6.2 ) and some derived SI units make use of these units with special names and the seven base units (Appendix A 6.3). These are given in Appendix A 6.2 and A 6.3 for your ready reference. Other units retained for general use are given in Table 2.2. Common SI prefixes and symbols for multiples and sub-multiples are given in Appendix A2. General guidelines for using symbols for physical quantities, chemical elements and nuclides are given in Appendix A7 and those for SI units and some other units are given in Appendix A8 for your guidance and ready reference. 2.3 MEASUREMENT OF LENGTH You are already familiar with some direct methods for the measurement of length. For example, a metre scale is used for lengths from 10 –3 m to 10 2 m. A vernier callipers is used for lengths to an accuracy of 10 –4 m. A screw gauge and a spherometer can be used to measure lengths as less as to 10 –5 m. To measure lengths beyond these ranges, we make use of some special indirect methods. 2.3.1 Measurement of Large Distances Large distances such as the distance of a planet or a star from the earth cannot be measured directly with a metre scale. An important method in such cases is the parallax method. When you hold a pencil in front of you against some specific point on the background (a wall) and look at the pencil first through your left eye A (closing the right eye) and then look at the pencil through your right eye B (closing the left eye), you would notice that the position of the pencil seems to change with respect to the point on the wall. This is called parallax. The distance between the two points of observation is called the basis. In this example, the basis is the distance between the eyes. To measure the distance D of a far away planet S by the parallax method, we observe it from two different positions (observatories) A and B on the Earth, separated by distance AB = b at the same time as shown in Fig. 2.2. We measure the angle between the two directions along which the planet is viewed at these two points. The ?ASB in Fig. 2.2 represented by symbol ? is called the parallax angle or parallactic angle. As the planet is very far away, 1, b D << and therefore, ? is very small. Then we approximately take AB as an arc of length b of a circle with centre at S and the distance D as 2020-21 Page 4 CHAPTER TWO UNITS AND MEASUREMENT 2.1 INTRODUCTION Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. The result of a measurement of a physical quantity is expressed by a number (or numerical measure) accompanied by a unit. Although the number of physical quantities appears to be very large, we need only a limited number of units for expressing all the physical quantities, since they are inter- related with one another. The units for the fundamental or base quantities are called fundamental or base units. The units of all other physical quantities can be expressed as combinations of the base units. Such units obtained for the derived quantities are called derived units. A complete set of these units, both the base units and derived units, is known as the system of units. 2.2 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 2.1 Introduction 2.2 The international system of units 2.3 Measurement of length 2.4 Measurement of mass 2.5 Measurement of time 2.6 Accuracy, precision of instruments and errors in measurement 2.7 Significant figures 2.8 Dimensions of physical quantities 2.9 Dimensional formulae and dimensional equations 2.10 Dimensional analysis and its applications Summary Exercises Additional exercises 2020-21 Table 2.1 SI Base Quantities and Units* 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 Table 2.1. Besides the seven base units, there are two more units that are defined for (a) plane angle d? as the ratio of length of arc ds to the radius r and (b) solid angle dO 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. 2.1(a) and (b) respectively. The unit for plane angle is radian with the symbol rad and the unit for the solid angle is steradian with the symbol sr. Both these are dimensionless quantities. UNITS AND MEASUREMENT 17 (a) (b) Fig. 2.1 Description of (a) plane angle d? and (b) solid angle dO . 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 ??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 m 2 s –1 , where the metre and the second are defined in terms of c and ??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 ??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 ??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 m 2 s –2 k –1 , where the kilogram, metre and second are defined in terms of h, c and ??cs. Amount of mole mol The mole, symbol mol, is the SI unit of amount of substance. One mole substance contains exactly 6.02214076×10 23 elementary entities. This number is the fixed numerical value of the Avogadro constant, N A , 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 540×10 12 Hz, K cd , to be 683 when expressed in the unit lm W –1 , which is equal to cd sr W –1 , or cd sr kg –1 m –2 s 3 , where the kilogram, metre and second are defined in terms of h, c and ??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. 2020-21 PHYSICS 18 Table 2.2 Some units retained for general use (Though outside SI) Note that when mole is used, the elementary entities must be specified. These entities may be atoms, molecules, ions, electrons, other particles or specified groups of such particles. We employ units for some physical quantities that can be derived from the seven base units (Appendix A 6). Some derived units in terms of the SI base units are given in (Appendix A 6.1). Some SI derived units are given special names (Appendix A 6.2 ) and some derived SI units make use of these units with special names and the seven base units (Appendix A 6.3). These are given in Appendix A 6.2 and A 6.3 for your ready reference. Other units retained for general use are given in Table 2.2. Common SI prefixes and symbols for multiples and sub-multiples are given in Appendix A2. General guidelines for using symbols for physical quantities, chemical elements and nuclides are given in Appendix A7 and those for SI units and some other units are given in Appendix A8 for your guidance and ready reference. 2.3 MEASUREMENT OF LENGTH You are already familiar with some direct methods for the measurement of length. For example, a metre scale is used for lengths from 10 –3 m to 10 2 m. A vernier callipers is used for lengths to an accuracy of 10 –4 m. A screw gauge and a spherometer can be used to measure lengths as less as to 10 –5 m. To measure lengths beyond these ranges, we make use of some special indirect methods. 2.3.1 Measurement of Large Distances Large distances such as the distance of a planet or a star from the earth cannot be measured directly with a metre scale. An important method in such cases is the parallax method. When you hold a pencil in front of you against some specific point on the background (a wall) and look at the pencil first through your left eye A (closing the right eye) and then look at the pencil through your right eye B (closing the left eye), you would notice that the position of the pencil seems to change with respect to the point on the wall. This is called parallax. The distance between the two points of observation is called the basis. In this example, the basis is the distance between the eyes. To measure the distance D of a far away planet S by the parallax method, we observe it from two different positions (observatories) A and B on the Earth, separated by distance AB = b at the same time as shown in Fig. 2.2. We measure the angle between the two directions along which the planet is viewed at these two points. The ?ASB in Fig. 2.2 represented by symbol ? is called the parallax angle or parallactic angle. As the planet is very far away, 1, b D << and therefore, ? is very small. Then we approximately take AB as an arc of length b of a circle with centre at S and the distance D as 2020-21 UNITS AND MEASUREMENT 19 t t t t the radius AS = BS so that AB = b = D ? where ? is in radians. D = b ? (2.1) Having determined D, we can employ a similar method to determine the size or angular diameter of the planet. If d is the diameter of the planet and a the angular size of the planet (the angle subtended by d at the earth), we have a = d/D (2.2) The angle a can be measured from the same location on the earth. It is the angle between the two directions when two diametrically opposite points of the planet are viewed through the telescope. Since D is known, the diameter d of the planet can be determined using Eq. (2.2). Example 2.1 Calculate the angle of (a) 1 0 (degree) (b) 1' (minute of arc or arcmin) and (c) 1?(second of arc or arc second) in radians. Use 360 0 =2p rad, 1 0 =60' and 1' = 60 ? Answer (a) We have 360 0 = 2p rad 1 0 = (p /180) rad = 1.745×10 –2 rad (b) 1 0 = 60' = 1.745×10 –2 rad 1' = 2.908×10 –4 rad 2.91×10 –4 rad (c) 1' = 60? = 2.908×10 –4 rad 1? = 4.847×10 –4 rad 4.85×10 –6 rad t Example 2.2 A man wishes to estimate the distance of a nearby tower from him. He stands at a point A in front of the tower C and spots a very distant object O in line with AC. He then walks perpendicular to AC up to B, a distance of 100 m, and looks at O and C again. Since O is very distant, the direction BO is practically the same as AO; but he finds the line of sight of C shifted from the original line of sight by an angle ? = 40 0 (? is known as ‘parallax’) estimate the distance of the tower C from his original position A. Fig. 2.3 Answer We have, parallax angle ? = 40 0 From Fig. 2.3, AB = AC tan ? AC = AB/tan? = 100 m/tan 40 0 = 100 m/0.8391 = 119 m t Example 2.3 The moon is observed from two diametrically opposite points A and B on Earth. The angle ? subtended at the moon by the two directions of observation is 1 o 54'. Given the diameter of the Earth to be about 1.276 × × × × × 10 7 m, compute the distance of the moon from the Earth. Answer We have ? = 1° 54 ' = 114 ' ( ) ( ) -6 114 60 4.85 10 = ' ' × × × rad = 3.32 × - 10 2 rad, since 6 1 4.85 10 . = " rad - × Also b = AB =1.276 m ×10 7 Hence from Eq. (2.1), we have the earth-moon distance, D b = /? = 1.276 10 3.32 10 7 -2 × × 8 3.84 10 m = × t Example 2.4 The Sun’s angular diameter is measured to be 1920''. The distance D of the Sun from the Earth is 1.496 × 10 11 m. What is the diameter of the Sun ? Fig. 2.2 Parallax method. 2020-21 Page 5 CHAPTER TWO UNITS AND MEASUREMENT 2.1 INTRODUCTION Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. The result of a measurement of a physical quantity is expressed by a number (or numerical measure) accompanied by a unit. Although the number of physical quantities appears to be very large, we need only a limited number of units for expressing all the physical quantities, since they are inter- related with one another. The units for the fundamental or base quantities are called fundamental or base units. The units of all other physical quantities can be expressed as combinations of the base units. Such units obtained for the derived quantities are called derived units. A complete set of these units, both the base units and derived units, is known as the system of units. 2.2 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 2.1 Introduction 2.2 The international system of units 2.3 Measurement of length 2.4 Measurement of mass 2.5 Measurement of time 2.6 Accuracy, precision of instruments and errors in measurement 2.7 Significant figures 2.8 Dimensions of physical quantities 2.9 Dimensional formulae and dimensional equations 2.10 Dimensional analysis and its applications Summary Exercises Additional exercises 2020-21 Table 2.1 SI Base Quantities and Units* 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 Table 2.1. Besides the seven base units, there are two more units that are defined for (a) plane angle d? as the ratio of length of arc ds to the radius r and (b) solid angle dO 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. 2.1(a) and (b) respectively. The unit for plane angle is radian with the symbol rad and the unit for the solid angle is steradian with the symbol sr. Both these are dimensionless quantities. UNITS AND MEASUREMENT 17 (a) (b) Fig. 2.1 Description of (a) plane angle d? and (b) solid angle dO . 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 ??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 m 2 s –1 , where the metre and the second are defined in terms of c and ??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 ??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 ??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 m 2 s –2 k –1 , where the kilogram, metre and second are defined in terms of h, c and ??cs. Amount of mole mol The mole, symbol mol, is the SI unit of amount of substance. One mole substance contains exactly 6.02214076×10 23 elementary entities. This number is the fixed numerical value of the Avogadro constant, N A , 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 540×10 12 Hz, K cd , to be 683 when expressed in the unit lm W –1 , which is equal to cd sr W –1 , or cd sr kg –1 m –2 s 3 , where the kilogram, metre and second are defined in terms of h, c and ??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. 2020-21 PHYSICS 18 Table 2.2 Some units retained for general use (Though outside SI) Note that when mole is used, the elementary entities must be specified. These entities may be atoms, molecules, ions, electrons, other particles or specified groups of such particles. We employ units for some physical quantities that can be derived from the seven base units (Appendix A 6). Some derived units in terms of the SI base units are given in (Appendix A 6.1). Some SI derived units are given special names (Appendix A 6.2 ) and some derived SI units make use of these units with special names and the seven base units (Appendix A 6.3). These are given in Appendix A 6.2 and A 6.3 for your ready reference. Other units retained for general use are given in Table 2.2. Common SI prefixes and symbols for multiples and sub-multiples are given in Appendix A2. General guidelines for using symbols for physical quantities, chemical elements and nuclides are given in Appendix A7 and those for SI units and some other units are given in Appendix A8 for your guidance and ready reference. 2.3 MEASUREMENT OF LENGTH You are already familiar with some direct methods for the measurement of length. For example, a metre scale is used for lengths from 10 –3 m to 10 2 m. A vernier callipers is used for lengths to an accuracy of 10 –4 m. A screw gauge and a spherometer can be used to measure lengths as less as to 10 –5 m. To measure lengths beyond these ranges, we make use of some special indirect methods. 2.3.1 Measurement of Large Distances Large distances such as the distance of a planet or a star from the earth cannot be measured directly with a metre scale. An important method in such cases is the parallax method. When you hold a pencil in front of you against some specific point on the background (a wall) and look at the pencil first through your left eye A (closing the right eye) and then look at the pencil through your right eye B (closing the left eye), you would notice that the position of the pencil seems to change with respect to the point on the wall. This is called parallax. The distance between the two points of observation is called the basis. In this example, the basis is the distance between the eyes. To measure the distance D of a far away planet S by the parallax method, we observe it from two different positions (observatories) A and B on the Earth, separated by distance AB = b at the same time as shown in Fig. 2.2. We measure the angle between the two directions along which the planet is viewed at these two points. The ?ASB in Fig. 2.2 represented by symbol ? is called the parallax angle or parallactic angle. As the planet is very far away, 1, b D << and therefore, ? is very small. Then we approximately take AB as an arc of length b of a circle with centre at S and the distance D as 2020-21 UNITS AND MEASUREMENT 19 t t t t the radius AS = BS so that AB = b = D ? where ? is in radians. D = b ? (2.1) Having determined D, we can employ a similar method to determine the size or angular diameter of the planet. If d is the diameter of the planet and a the angular size of the planet (the angle subtended by d at the earth), we have a = d/D (2.2) The angle a can be measured from the same location on the earth. It is the angle between the two directions when two diametrically opposite points of the planet are viewed through the telescope. Since D is known, the diameter d of the planet can be determined using Eq. (2.2). Example 2.1 Calculate the angle of (a) 1 0 (degree) (b) 1' (minute of arc or arcmin) and (c) 1?(second of arc or arc second) in radians. Use 360 0 =2p rad, 1 0 =60' and 1' = 60 ? Answer (a) We have 360 0 = 2p rad 1 0 = (p /180) rad = 1.745×10 –2 rad (b) 1 0 = 60' = 1.745×10 –2 rad 1' = 2.908×10 –4 rad 2.91×10 –4 rad (c) 1' = 60? = 2.908×10 –4 rad 1? = 4.847×10 –4 rad 4.85×10 –6 rad t Example 2.2 A man wishes to estimate the distance of a nearby tower from him. He stands at a point A in front of the tower C and spots a very distant object O in line with AC. He then walks perpendicular to AC up to B, a distance of 100 m, and looks at O and C again. Since O is very distant, the direction BO is practically the same as AO; but he finds the line of sight of C shifted from the original line of sight by an angle ? = 40 0 (? is known as ‘parallax’) estimate the distance of the tower C from his original position A. Fig. 2.3 Answer We have, parallax angle ? = 40 0 From Fig. 2.3, AB = AC tan ? AC = AB/tan? = 100 m/tan 40 0 = 100 m/0.8391 = 119 m t Example 2.3 The moon is observed from two diametrically opposite points A and B on Earth. The angle ? subtended at the moon by the two directions of observation is 1 o 54'. Given the diameter of the Earth to be about 1.276 × × × × × 10 7 m, compute the distance of the moon from the Earth. Answer We have ? = 1° 54 ' = 114 ' ( ) ( ) -6 114 60 4.85 10 = ' ' × × × rad = 3.32 × - 10 2 rad, since 6 1 4.85 10 . = " rad - × Also b = AB =1.276 m ×10 7 Hence from Eq. (2.1), we have the earth-moon distance, D b = /? = 1.276 10 3.32 10 7 -2 × × 8 3.84 10 m = × t Example 2.4 The Sun’s angular diameter is measured to be 1920''. The distance D of the Sun from the Earth is 1.496 × 10 11 m. What is the diameter of the Sun ? Fig. 2.2 Parallax method. 2020-21 PHYSICS 20 t Answer Sun’s angular diameter a = 1920" = × × - 1920 4.85 10 rad 6 = × - 9.31 10 rad 3 Sun’s diameter d D = a 3 11 9.31 10 1.496 10 m ? ? ? ? ? ? ? ? ? ? ? ? - = × × × 9 =1.39 10 m × t 2.3.2 Estimation of Very Small Distances: Size of a Molecule To measure a very small size, like that of a molecule (10 –8 m to 10 –10 m), we have to adopt special methods. We cannot use a screw gauge or similar instruments. Even a microscope has certain limitations. An optical microscope uses visible light to ‘look’ at the system under investigation. As light has wave like features, the resolution to which an optical microscope can be used is the wavelength of light (A detailed explanation can be found in the Class XII Physics textbook). For visible light the range of wavelengths is from about 4000 Å to 7000 Å (1 angstrom = 1 Å = 10 -10 m). Hence an optical microscope cannot resolve particles with sizes smaller than this. Instead of visible light, we can use an electron beam. Electron beams can be focussed by properly designed electric and magnetic fields. The resolution of such an electron microscope is limited finally by the fact that electrons can also behave as waves ! (You will learn more about this in class XII). The wavelength of an electron can be as small as a fraction of an angstrom. Such electron microscopes with a resolution of 0.6 Å have been built. They can almost resolve atoms and molecules in a material. In recent times, tunnelling microscopy has been developed in which again the limit of resolution is better than an angstrom. It is possible to estimate the sizes of molecules. A simple method for estimating the molecular size of oleic acid is given below. Oleic acid is a soapy liquid with large molecular size of the order of 10 –9 m. The idea is to first form mono-molecular layer of oleic acid on water surface. We dissolve 1 cm 3 of oleic acid in alcohol to make a solution of 20 cm 3 . Then we take 1 cm 3 of this solution and dilute it to 20 cm 3 , using alcohol. So, the concentration of the solution is equal to 1 20 20 3 × ? ? ? ? ? ? cm of oleic acid/cm 3 of solution. Next we lightly sprinkle some lycopodium powder on the surface of water in a large trough and we put one drop of this solution in the water. The oleic acid drop spreads into a thin, large and roughly circular film of molecular thickness on water surface. Then, we quickly measure the diameter of the thin film to get its area A. Suppose we have dropped n drops in the water. Initially, we determine the approximate volume of each drop (V cm 3 ). Volume of n drops of solution = nV cm 3 Amount of oleic acid in this solution = nV 1 20 20 × ? ? ? ? ? ? cm 3 This solution of oleic acid spreads very fast on the surface of water and forms a very thin layer of thickness t. If this spreads to form a film of area A cm 2 , then the thickness of the film t = Volume of the film Area of the film or, cm 20 20 nV t A = × (2.3) If we assume that the film has mono-molecular thickness, then this becomes the size or diameter of a molecule of oleic acid. The value of this thickness comes out to be of the order of 10 –9 m. Example 2.5 If the size of a nucleus (in the range of 10 –15 to 10 –14 m) is scaled up to the tip of a sharp pin, what roughly is the size of an atom ? Assume tip of the pin to be in the range 10 –5 m to 10 –4 m. Answer The size of a nucleus is in the range of 10 –15 m and 10 –14 m. The tip of a sharp pin is taken to be in the range of 10 –5 m and 10 –4 m. Thus we are scaling up by a factor of 10 10 . An atom roughly of size 10 –10 m will be scaled up to a size of 1 m. Thus a nucleus in an atom is as small in size as the tip of a sharp pin placed at the centre of a sphere of radius about a metre long. t 2020-21Read More

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