CHAPTER TWO UNITS AND MEASUREMENT 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 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 interrelated 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, was developed and recommended by General Conference on Weights and Measures in 1971 for 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 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. 2.1(a) and (b) respectively. The unit for plane angle is radian with the symbol rad and the unit for the (b) solid angle is steradian with the symbol sr. Both Fig. 2.1 Description of (a) plane angle dθ and these are dimensionless quantities. (b) solid angle d Ω . Table 2.1 SI Base Quantities and Units* Base quantity Name Symbol SI Units Definition Length Mass Time Electric current Thermo dynamic Temperature Amount of substance Luminous intensity metre kilogram second ampere kelvin mole candela m kg s A K mol cd The metre is the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second. (1983) The kilogram is equal to the mass of the international prototype of the kilogram (a platinum-iridium alloy cylinder) kept at international Bureau of Weights and Measures, at Sevres, near Paris, France. (1889) The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom. (1967) The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 210√7 newton per metre of length. (1948) The kelvin, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (1967) The mole is the amount of substance of a system, which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon - 12. (1971) The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 5401012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. (1979) * 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. 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√3m to 102 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√5m. 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 θ is called the parallax angle or parallactic angle. b As the planet is very far away, <<1, and D therefore, θ 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 the radius AS = BS so that AB = b = D θ where AO; but he finds the line of sight of C shifted θ is in radians. from the original line of sight by an angle θ = 400 (θ is known as «parallax») estimateb the distance of the tower C from his originalD= (2.1)θ position A. • • • Having determinedD, we can employ a similar Answer We have, parallax angle θ = 400 method to determine the size or angular diameter From Fig. 2.3, AB = AC tan θ of the planet. If d is the diameter of the planet AC = AB/tanθ = 100 m/tan 400 and α the angular size of the planet (the angle = 100 m/0.8391 = 119 m • subtended by d at the earth), we have α = d/D (2.2) Example 2.3 The moon is observed from The angle α can be measured from the same two diametrically opposite points A and B • location on the earth. It is the angle between on Earth. The angle θ subtended at the the two directions when two diametrically moon by the two directions of observation opposite points of the planet are viewed through is 1o 54′. Given the diameter of the Earth to the telescope. Since D is known, the diameter d be about 1.276 × 107 m, compute the of the planet can be determined using Eq. (2.2). distance of the moon from the Earth. Example 2.1 Calculate the angle of Answer We have θ = 1°54 ′ = 114 ′ (a) 10 (degree) (b) 1′ (minute of arc or arcmin) ′′ -6 )= 114 × 60 × 4.85 ×10 radand (c) 1″(second of arc or arc second) in ( )( radians. Use 3600=2π rad, 10=60′ and − = 3.32 × 10 2 rad,1′ = 60 ″ since 1"=4.85 10 6 rad.Answer (a) We have 3600 = 2π rad 10 = (π /180) rad = 1.74510√2 rad Also b=AB =1.276 × 107m (b) 10 = 60′ = 1.74510√2 rad Hence from Eq. (2.1), we have the earth-moon1′ = 2.90810√4 rad . 2.9110√4 rad distance,(c) 1′ = 60″ = 2.90810√4 rad Db=/θ 1″ = 4.84710√4 rad . 4.8510√6 rad • 71.276 × 10 =Example 2.2 A man wishes to estimate 3.32 × 10-2 the distance of a nearby tower from him. He stands at a point A in front of the tower =3.84 108m • C and spots a very distant object O in line with AC. He then walks perpendicular to Example 2.4 The Sun»s angular diameter AC up to B, a distance of 100 m, and looks is measured to be 1920′′. The distanceD of at O and C again. Since O is very distant, the Sun from the Earth is 1.496 1011 m. the direction BO is practically the same as What is the diameter of the Sun ? Answer Sun»s angular diameter α = 1920" = 1920 × 4.85 × 10 −6 rad = ×3rad 9.31 10−Sun»s diameter d = α D ⎛−3 ⎞⎛ 11 ⎞9.31 10 ⎟× 1.496 ×10 ⎟ m=⎜× ⎜⎝⎠⎝ ⎠ =1.39 109m • 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 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 cm3 of oleic acid in alcohol to make a solution of 20 cm3. Then we take 1 cm3 of this solution and dilute it to 20 cm3, using alcohol. So, the concentration of the solution is equal to 1 cm3 of oleic acid/cm3 of 20 20 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 cm3). Volume ofn drops of solution = nV cm3 Amount of oleic acid in this solution⎛ 1 ⎞ 3nV ⎜⎟ cm= ⎝ 20 × 20 ⎠ 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 cm2, then the thickness of the film Volume of the film t = Area of the film nV or, t cm (2.3)20 20 A 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. tExample 2.5 If the size of a nucleus (inmolecules in a material. In recent times, the range of 10√15 to 10√14 m) is scaled uptunnelling microscopy has been developed in to the tip of a sharp pin, what roughly iswhich again the limit of resolution is better than the size of an atom ? Assume tip of the pinan angstrom. It is possible to estimate the sizes to be in the range 10√5m to 10√4m. 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 1010. 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. • 2.3.3 Range of Lengths The sizes of the objects we come across in the universe vary over a very wide range. These may vary from the size of the order of 10√14 m of the tiny nucleus of an atom to the size of the order of 1026 m of the extent of the observable universe. Table 2.6 gives the range and order of lengths and sizes of some of these objects. We also use certain special length units for short and large lengths. These are 1 fermi = 1 f = 10√15 m 1 angstrom = 1 Å = 10√10 m 1 astronomical unit = 1 AU (average distanceof the Sun from the Earth) = 1.496 × 1011 m 1 light year = 1 ly= 9.46 × 1015 m (distance that light travels with velocity of3 × 108 m s√1 in 1 year) 1 parsec = 3.08 × 1016 m (Parsec is the distance at which average radius of earth»s orbit subtends an angle of 1 arc second) 2.4 MEASUREMENT OF MASS Mass is a basic property of matter. It does not depend on the temperature, pressure or location of the object in space. The SI unit of mass is kilogram (kg). The prototypes of the International standard kilogram supplied by the International Bureau of Weights and Measures (BIPM) are available in many other laboratories of different countries. In India, this is available at the National Physical Laboratory (NPL), New Delhi. While dealing with atoms and molecules, the kilogram is an inconvenient unit. In this case, there is an important standard unit of mass, called the unified atomic mass unit(u), which has been established for expressing the mass of atoms as 1 unified atomic mass unit = 1u = (1/12) of the mass of an atom of carbon-12 isotope ( 12 C) including the mass of electrons6 10√27= 1.66 kg Mass of commonly available objects can be determined by a common balance like the one used in a grocery shop. Large masses in the universe like planets, stars, etc., based on Newton»s law of gravitation can be measured by using gravitational method (See Chapter 8). For measurement of small masses of atomic/subatomic particles etc., we make use of mass spectrograph in which radius of the trajectory is proportional to the mass of a charged particle moving in uniform electric and magnetic field. 2.4.1 Range of Masses The masses of the objects, we come across in the universe, vary over a very wide range. These may vary from tiny mass of the order of 10-30 kg of an electron to the huge mass of about 1055 kg of the known universe. Table 2.4 gives the range and order of the typical masses of various objects. 2.5 MEASUREMENT OF TIME To measure any time interval we need a clock. We now use an atomic standard of time, which is based on the periodic vibrations produced in a cesium atom. This is the basis of the cesium clock, sometimes called atomic clock, used in the national standards. Such standards are available in many laboratories. In the cesium atomic clock, the second is taken as the time needed for 9,192,631,770 vibrations of the radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium-133 atom. The vibrations of the cesium atom regulate the rate of this cesium atomic clock just as the vibrations of a balance wheel regulate an ordinary wristwatch or the vibrations of a small quartz crystal regulate a quartz wristwatch. The cesium atomic clocks are very accurate. In principle they provide portable standard. The national standard of time interval «second» as well as the frequency is maintained through four cesium atomic clocks. A cesium atomic clock is used at the National Physical Laboratory (NPL), New Delhi to maintain the Indian standard of time. In our country, the NPL has the responsibility of maintenance and improvement of physical standards, including that of time, frequency, etc. Note that the Indian Standard Time (IST) is linked to this set of atomic clocks. The efficient cesium atomic clocks are so accurate that they impart the uncertainty in time realisation as Ø 1 × 10√13, i.e. 1 part in 1013. This implies that the uncertainty gained over time by such a device is less than 1 part in 1013; they lose or gain no more than 3 μs in one year. In view of the tremendous accuracy in time measurement, the SI unit of length has been expressed in terms the path length light travels in certain interval of time (1/299, 792, 458 of a second) (Table 2.1). The time interval of events that we come across in the universe vary over a very wide range. Table 2.5 gives the range and order of some typical time intervals. You may notice that there is an interesting coincidence between the numbers appearing in Tables 2.3and 2.5. Note that the ratio of the longest and shortest lengths of objects in our universe is about 1041. Interestingly enough, the ratio of the longest and shortest time intervals associated with the events and objects in our universe is also about 1041. This number, 1041 comes up again in Table 2.4, which lists typical masses of objects. The ratio of the largest and smallest masses of the objects in our universe is about (1041)2. Is this a curious coincidence between these large numbers purely accidental ? 2.6 ACCURACY, PRECISION OFINSTRUMENTS AND ERRORS IN MEASUREMENT Measurement is the foundation of all experimental science and technology. The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error. Every calculated quantity which is based on measured values, also has an error. We shall distinguish between two terms: accuracy and precision. The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured. The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument. For example, suppose the true value of a certain length is near 3.678 cm. In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 3.5 cm, while in another experiment using a measuring device of greater resolution, say 0.01 cm, the length is determined to be 3.38 cm. The first measurement has more accuracy (because it is closer to the true value) but less precision (its resolution is only 0.1 cm), while the second measurement is less accurate but more precise. Thus every measurement is approximate due to errors in measurement. In general, the errors in measurement can be broadly classified as (a) systematic errors and (b) random errors. Systematic errors The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematic errors are : (a) Instrumental errors that arise from the errors due to imperfect design or calibration of the measuring instrument, zero error in the instrument, etc. For example, the temperature graduations of a thermometer may be inadequately calibrated (it may read 104 °C at the boiling point of water at STP whereas it should read 100 °C); in a vernier callipers the zero mark of vernier scale may not coincide with the zero mark of the main scale, or simply an ordinary metre scale may be worn off at one end. (b) Imperfection in experimental technique or procedure To determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature. Other external conditions (such as changes in temperature, humidity, wind velocity, etc.) during the experiment may systematically affect the measurement. (c) Personal errors that arise due to an individual»s bias, lack of proper setting of the apparatus or individual»s carelessness in taking observations without observing proper precautions, etc. For example, if you, by habit, always hold your head a bit too far to the right while reading the position of a needle on the scale, you will introduce an error due to parallax. Systematic errors can be minimised by improving experimental techniques, selecting better instruments and removing personal bias as far as possible. For a given set-up, these errors may be estimated to a certain extent and the necessary corrections may be applied to the readings. Random errors Therandom errors are those errors, which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking readings, etc. For example, when the same person repeats the same observation, it is very likely that he may get different readings everytime. Least count error The smallest value that can be measured by the measuring instrument is called its least count. All the readings or measured values are good only up to this value. The least count error is the error associated with the resolution of the instrument. For example, a vernier callipers has the least count as 0.01 cm; a spherometer may have a least count of 0.001 cm. Least count error belongs to the category of random errors but within a limited size; it occurs with both systematic and random errors. If we use a metre scale for measurement of length, it may have graduations at 1 mm division scale spacing or interval. Using instruments of higher precision, improving experimental techniques, etc., we can reduce the least count error. Repeating the observations several times and taking the arithmetic mean of all the observations, the mean value would be very close to the true value of the measured quantity. 2.6.1 Absolute Error, Relative Error and Percentage Error (a) Suppose the values obtained in several measurements are a1, a2, a3...., an . The arithmetic mean of these values is taken as the best possible value of the quantity under the given conditions of measurement as : a = (a+a+a+...+a ) / n (2.4)mean123nor, n as to underestimate the true value of the quantity. The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement. This is denoted by |Δa |. In absence of any other method of knowing true value, we considered arithmatic mean as the true value. Then the errors in the individual measurement values from the true value, are Δa = a1 √ a ,1meanΔa = a √ a ,22mean.... .... .... .... .... .... Δa = a √ a nnmean The Δa calculated above may be positive in certain cases and negative in some other cases. But absolute error |Δa| will always be positive. (b) The arithmetic mean of all theabsolute errors is taken as the final or mean absolute error of the value of the physical quantity a. It is represented by Δamean . Thus, Δa = (|Δa |+|Δa |+|Δa |+...+ |Δa |)/n mean123n(2.6) n ∑=i1 If we do a single measurement, the value we get may be in the range a Ø Δa meanmean i.e. a = a Ø Δa meanmean or,a √ Δa ≤ a ≤ a + Δa meanmean meanmean (2.8) This implies that any measurement of the physical quantity a is likely to lie between (a+ Δa ) and (a− Δa ).mean mean mean mean=|Δai|/n (2.7) ∑=i1 (δa). The relative error is the ratio of theThis is because, as explained earlier, it is mean absolute error Δa to the meanreasonable to suppose that individual mean measurements are as likely to overestimate value amean of the quantity measured. a/n (c) Instead of the absolute error, we often usei (2.5) =amean the relative error or the percentage error Relative error = Δa /a (2.9) = 2.624 s meanmean = 2.62 s When the relative error is expressed in per As the periods are measured to a resolutioncent, it is called the percentage error (δa). of 0.01 s, all times are to the second decimal; it is proper to put this mean period also to theThus, Percentage error second decimal. δa = (Δa /a ) 100% (2.10)meanmeanThe errors in the measurements are Let us now consider an example. 2.63 s √ 2.62 s = 0.01 s 2.56 s √ 2.62 s = √ 0.06 s Example 2.6 Two clocks are being tested 2.42 s √ 2.62 s = √ 0.20 s against a standard clock located in a 2.71 s √ 2.62 s = 0.09 s national laboratory. At 12:00:00 noon by 2.80 s √ 2.62 s = 0.18 s the standard clock, the readings of the two• Note that the errors have the same units as the clocks are : quantity to be measured. Clock 1 Clock 2 The arithmetic mean of all the absolute errors (for arithmetic mean, we take only theMonday 12:00:05 10:15:06 magnitudes) isTuesday 12:01:15 10:14:59 Wednesday 11:59:08 10:15:18 ΔΤ = [(0.01+ 0.06+0.20+0.09+0.18)s]/5Thursday 12:01:50 10:15:07 mean= 0.54 s/5Friday 11:59:15 10:14:53 = 0.11 s Saturday 12:01:30 10:15:24 Sunday 12:01:19 10:15:11 That means, the period of oscillation of the simple pendulum is (2.62 Ø 0.11) s i.e. it liesIf you are doing an experiment that requires between (2.62 + 0.11) s and (2.62 √ 0.11) s orprecision time interval measurements, which between 2.73 s and 2.51 s. As the arithmeticof the two clocks will you prefer ? mean of all the absolute errors is 0.11 s, there is already an error in the tenth of a second.Answer The range of variation over the seven Hence there is no point in giving the period to adays of observations is 162 s for clock 1, and hundredth. A more correct way will be to write31 s for clock 2. The average reading of clock 1 is much closer to the standard time than the T = 2.6 Ø 0.1 s average reading of clock 2. The important point Note that the last numeral 6 is unreliable, sinceis that a clock»s zero error is not as significant it may be anything between 5 and 7. We indicatefor precision work as its variation, because a this by saying that the measurement has two«zero-error» can always be easily corrected. significant figures. In this case, the twoHence clock 2 is to be preferred to clock 1. • significant figures are 2, which is reliable and 6, which has an error associated with it. YouExample 2.7 We measure the period of will learn more about the significant figures inoscillation of a simple pendulum. In section 2.7. successive measurements, the readings For this example, the relative error or theturn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s • percentage error isand 2.80 s. Calculate the absolute errors, relative error or percentage error. 01. δa =× 100 = 4 % • Answer The mean period of oscillation of the 2.6 pendulum( 2.63 + 2.56 + 2.42 + 2.71 +)2.80 s 2.6.2 Combination of ErrorsT = 5 If we do an experiment involving several 13.12 measurements, we must know how the errorss= in all the measurements combine. For example, How will you measure the length of a line? What a naïve question, at this stage, you might say! But what if it is not a straight line? Draw a zigzag line in your copy, or on the blackboard. Well, not too difficult again. You might take a thread, place it along the line, open up the thread, and measure its length. Now imagine that you want to measure the length of a national highway, a river, the railway track between two stations, or the boundary between two states or two nations. If you take a string of length 1 metre or 100 metre, keep it along the line, shift its position every time, the arithmetic of man-hours of labour and expenses on the project is not commensurate with the outcome. Moreover, errors are bound to occur in this enormous task. There is an interesting fact about this. France and Belgium share a common international boundary, whose length mentioned in the official documents of the two countries differs substantially! Go one step beyond and imagine the coastline where land meets sea. Roads and rivers or the common boundary between two states, etc. Railway tickets come with the distance between stations printed on them. We have «milestones» all along the roads indicating the distances to various towns. So, how is it done? One has to decide how much error one can tolerate and optimise cost-effectiveness. If you want smaller errors, it will involve high technology and high costs. Suffice it to say that it requires fairly advanced level of physics, mathematics, engineering and technology. It belongs to the areas of fractals, which has lately become popular in theoretical physics. Even then one doesn»t know how much to rely on the figure that props up, as is clear from the story of France and Belgium. Incidentally, this story of the France-Belgium discrepancy appears on the first page of an advanced Physics book on the subject of fractals and chaos! density is the ratio of the mass to the volume of the substance. If we have errors in the measurement of mass and of the sizes or dimensions, we must know what the error will be in the density of the substance. To make such estimates, we should learn how errors combine in various mathematical operations. For this, we use the following procedure. (a) Error of a sum or a difference Suppose two physical quantities A and B have measured values A Ø ΔA, B Ø ΔB respectively where ΔA and ΔB are their absolute errors. We wish to find the error ΔZ in the sum Z = A + B. We have by addition, Z ± ΔZ = (A Ø ΔA) + (B Ø ΔB). The maximum possible error in Z ΔZ = ΔA + ΔB For the difference Z = A √ B, we have Z Ø Δ Z = (A Ø ΔA) œ (B Ø ΔB) = (A √ B) ± ΔA Ø ΔB or, Ø ΔZ = Ø ΔA Ø ΔB The maximum value of the error ΔZ is again ΔA + ΔB. Hence the rule : When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities. have fairly mild bends as compared to a coastline. Even so, all documents, including our school books, contain information on the length of the coastline of Gujarat or Andhra Pradesh, tExample 2.8 The temperatures of two bodies measured by a thermometer are t1 = 20 0C Ø 0.5 0C and t2 = 50 0C Ø 0.5 0C. Calculate the temperature difference and the error theirin. Answer t′ = t2√t1 = (50 0CØ0.5 0C)√ (200CØ0.5 0C) t′ = 30 0C Ø 1 0C • (b) Error of a product or a quotient Suppose Z = AB and the measured values of A and B are A Ø ΔA and B Ø ΔB. Then Z Ø ΔZ = (A Ø ΔA) (B Ø ΔB) = AB Ø B ΔA Ø A ΔB Ø ΔA ΔB. Dividing LHS by Z and RHS by AB we have, 1Ø(ΔZ/Z) = 1 Ø (ΔA/A) Ø (ΔB/B) Ø (ΔA/A)(ΔB/B). Since ΔA and ΔB are small, we shall ignore their product. Hence the maximum relative error ΔZ/ Z = (ΔA/A) + (ΔB/B). You can easily verify that this is true for division also. Hence the rule : When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers. Then,Example 2.9 The resistanceR = V/I where ΔZ/Z = (ΔA/A) + (ΔA/A) = 2 (ΔA/A).V = (100 Ø 5)V and I = (10 Ø 0.2)A. Find the Hence, the relative error in A2 is two times thepercentage error in R. error in A. Answer The percentage error in V is 5% and in In general, if Z = Ap Bq/Cr I it is 2%. The total error in R would therefore Then, be 5% + 2% = 7%. • ΔZ/Z = p (ΔA/A) + q (ΔB/B) + r (ΔC/C). • Hence the rule : The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity. tExample 2.11 Find the relative error in Z, if Z = A4B1/3/CD3/2. Answer The relative error in Z is ΔZ/Z = 4(ΔA/A) +(1/3) (ΔB/B) + (ΔC/C) + (3/2) (ΔD/D). Answer (a) The equivalent resistance of series • combination R =R1 + R2 = (100 Ø 3) ohm + (200 Ø 4) ohm = 300 Ø 7 ohm. (b) The equivalent resistance of parallel combination RR 200R′= 12 = = 66.7 ohmR1 +R23 11 1 =+Then, from R′ R1 R2 we get, ΔR′ΔR1 +ΔR2= 2 22R′ R1 R2 2 ΔR12 ΔR2Δ= R′ R( ) ′ R12 +( ) R′ R22 ⎛66.7 ⎞2 ⎛66.7 ⎞2 = 3 + 4⎜⎟⎜⎟⎝100 ⎠⎝200 ⎠= 1.8 Then, R′=66.7 ±1.8 ohm (Here, ΔR is expresed as 1.8 instead of 2 to keep in confirmity with the rules of significant figures.) • (c) Error in case of a measured quantity raised to a power Suppose Z = A2, Answer g = 4π2L/T2 t Δt ΔT Δt =Here, T = and Δ=T . Therefore, . nn Tt The errors in both L and t are the least count errors. Therefore, (Δg/g) = (ΔL/L) + 2(ΔT/T )0.1 12 0.027 = 20.0 90 Thus, the percentage error in g is 100 (Δg/g) = 100(ΔL/L) + 2 × 100 (ΔT/T ) = 3% • 2.7 SIGNIFICANT FIGURES As discussed above, every measurement involves errors. Thus, the result of measurement should be reported in a way that indicates the precision of measurement. Normally, the reported result of measurement is a number that includes all digits in the number that are known reliably plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures. If we say the period of oscillation of a simple pendulum is 1.62 s, the digits 1 and 6 are reliable and certain, while the digit 2 is uncertain. Thus, the measured value has three significant figures. The length of an object reported after measurement to be 287.5 cm has four significant figures, the digits 2, 8, 7 are certain while the digit 5 is uncertain. Clearly, reporting the result of measurement that includes more digits than the significant digits is superfluous and also misleading since it would give a wrong idea about the precision of measurement. The rules for determining the number of significant figures can be understood from the following examples. Significant figures indicate, as already mentioned, the precision of measurement which depends on the least count of the measuring instrument. A choice of change of different units does not change the number of significant digits or figures in a measurement. This important remark makes most of the following observations clear: (1) For example, the length 2.308 cm has four significant figures. But in different units, the same value can be written as 0.02308 m or 23.08 mm or 23080 μm. All these numbers have the same number of significant figures (digits 2, 3, 0, 8), namely four. This shows that the location of decimal point is of no consequence in determining the number of significant figures. The example gives the following rules : • All the non-zero digits are significant. • All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all. • If the number is less than 1, the zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant. [In 0.00 2308, the underlined zeroes are not significant]. • The terminal or trailing zero(s) in a number without a decimal point are not significant. [Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s) being not significant.] However, you can also see the next observation. • The trailing zero(s) in a number with a decimal point are significant. [The numbers 3.500 or 0.06900 have four significant figures each.] (2) There can be some confusion regarding the trailing zero(s). Suppose a length is reported to be 4.700 m. It is evident that the zeroes here are meant to convey the precision of measurement and are, therefore, significant. [If these were not, it would be superfluous to write them explicitly, the reported measurement would have been simply 4.7 m]. Now suppose we change units, then 4.700 m = 470.0 cm = 4700 mm = 0.004700 km Since the last number has trailing zero(s) in a number with no decimal, we would conclude erroneously from observation (1) above that the number has two significant figures, while in fact, it has four significant figures and a mere change of units cannot change the number of significant figures. (3) To remove such ambiguities in determining the number of significant figures, the best way is to report every measurement in scientific notation (in the power of 10).In this notation, every number is expressed as a 10b, where a is a number between 1 and 10, and b is any positive or negative exponent (or power) of 10. In order to get an approximate idea of the number, we may round off the number a to 1 (for a ≤ 5) and to 10 (for 5