UNIT AND MEASUREMENT

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UNIT MEASUREMENT AND

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THE INTERNATIONAL SYSTEM OF UNITS • In CGS system they were centimeter, 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 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. 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 Seven Base SI Units:

The Seven Base SI Units Quantity Unit Symbol Length meter m Mass kilogram kg Temperature kelvin K Time second s Amount of Substance mole mol Luminous Intensity candela cd Electric Current ampere a

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Measurement of lemgth

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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 . 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 . We measure the angle between the two directions along which the planet is viewed at these two points. The ∠ASB represented by symbol θ is called the parallax angle or parallactic angle . As the planet is very far away, B << 1. 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 θ is in radians. D =b θ 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 α the angular size of the planet (the angle subtended by d at the earth), α = d/D Measurement of Large Distances

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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. 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 focused 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 ! 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, tunneling 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 10–9m Estimation of Very Small Distances: Size of a Molecule

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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. We 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 distance from earth to sun) = 1.496 × 1011 m 1 light year = 1 ly = 9.46 × 1015 m (distance that travels with velocity of 3 × 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)

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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). 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 including the mass of electrons = 1.66 × 10–27 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. MEASUREMENT OF MASS

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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. 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.

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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. 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.

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Systematic errors The systematic errors are those errors that tend to be in one direction, either positive or negative. Systematic errors can be minimized 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. Some of the sources of systematic errors are Instrumental errors that arise from the errors due to imperfect design or calibration of the measuring instrument, zero error in the instrument, etc. The temperature graduations of a thermometer may be inadequately calibrated 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 maybe worn off at one end.

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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 during the experiment may systematically affect the measurement. 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 ex, 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.

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Random errors The random 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.

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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. Least count error belongs to the category of random errors but within a limited size; it occurs with both systematic and random errors. 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.

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Absolute Error, Relative Error and Percentage Error 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 mean = (a1+a2+a3+...+an ) / n This is because, as explained earlier, it is reasonable to suppose that individual measurements are as likely to overestimate as to underestimate the true value of the quantity. The magnitude of the difference between the true value of the quantity and the individual measurement value is called the absolute error of the measurement. This is denoted by | Δ a | . Instead of the absolute error, we often use the relative error or the percentage error (δ a ). T he relative error is the ratio of the mean absolute error Δ a mean to the mean value a mean of the quantity measured . Relative error = Δ a mean / a mean When the relative error is expressed in per cent, it is called the percentage error (δ a ).Thus, Percentage error δ a = (Δ a mean / a mean ) × 100%

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Combination of Errors

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Error of a sum or a difference 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. Suppose two physical quantities A and B have measured values A ± Δ A , B ± Δ B respectively where Δ A and Δ B are their absolute errors. We wish to find the error Δ Z in the sum Z = A + B . We have by addition, Z ± Δ Z The maximum possible error in Z Δ Z = Δ A + Δ B For the difference Z = A – B , we have Z ± Δ Z = (A ± Δ A ) – ( B ± Δ B ) = ( A – B ) ± Δ A ± Δ B or, ± Δ Z = ± Δ A ± Δ B The maximum value of the error Δ Z is again Δ A + Δ B .

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Error of a product or a quotient When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers. Suppose Z = AB and the measured values of A and B are A ± Δ A and B ± Δ B . Then Z ± Δ Z = ( A ± Δ A ) ( B ± Δ B ) = AB ± B Δ A ± A Δ B ± Δ A Δ B . Dividing LHS by Z and RHS by AB we have, 1±(Δ Z/Z ) = 1 ± (Δ A / A ) ± (Δ B / B ) ± (Δ A / A )(Δ B / B ). Since Δ A and Δ B are small, we shall ignore their product. Hence the maximum relative error Δ Z / Z = (ΔA/A) + (ΔB/B). You can easily verify that this is true for division also.

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Error in case of a measured quantity raised to a power The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity .Suppose Z = A2 ,Then, Δ Z / Z = (Δ A / A ) + (Δ A / A ) = 2 (Δ A / A ). Hence, the relative error in A2 is two times the error in A . In general, if Z = Ap Bq/Cr Then, Δ Z / Z = p (Δ A / A ) + q (Δ B/B ) + r (Δ C / C ).

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SIGNIFICANT FIGURES 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

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• 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. • The terminal or trailing zero(s) in a number without a decimal point are not significant. • The trailing zero(s) in a number with a decimal point are significant. • For a number greater than 1, without any decimal, the trailing zero(s) are not significant. • For a number with a decimal, the trailing zero(s) are significant . • The multiplying or dividing factors which are neither rounded numbers nor numbers representing measured values are exact and have infinite number of significant digits.

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Rules for Arithmetic Operations with Significant Figures The result of a calculation involving approximate measured values of quantities (i.e. values with limited number of significant figures) must reflect the uncertainties in the original measured values. It cannot be more accurate than the original measured values themselves on which the result is based. In general, the final result should not have more significant figures than the original data from which it was obtained. It would be clearly absurd and irrelevant to record the calculated value of density to such a precision when the measurements on which the value is based, have much less precision. (1) In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures. (2) In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

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Rounding off the Uncertain Digits The result of computation with approximate numbers, which contain more than one uncertain digit, should be rounded off. The rules for rounding off numbers to the appropriate significant figures are obvious in most cases. The rule by convention is that the preceding digit is raised by 1 if the insignificant digit to be dropped is more than 5, and is left unchanged if the latter is less than 5. if the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by 1. In any involved or complex multi-step calculation, you should retain, in intermediate steps, one digit more than the significant digits and round off to proper significant figures at the end of the calculation.

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Rules for Determining the Uncertainty in the Results of Arithmatic Calculations If the length and breadth of a thin rectangular sheet are measured, using a metre scale as 16.2 cm and, 10.1 cm respectively, there are three significant figures in each measurement. If a set of experimental data is specified to n significant figures, a result obtained by combining the data will also be valid to n significant figures. However, if data are subtracted, the number of significant figures can be reduced. The relative error of a value of number specified to significant figures depends not only on n but also on the number itself. intermediate results in a multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement.

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DIMENSIONS OF PHYSICAL QUANTITIES The nature of a physical quantity is described by its dimensions. All the physical quantities represented by derived units can be expressed in terms of some combination of seven fundamental or base quantities. The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity. Note that using the square brackets [ ] round a quantity means that we are dealing with ‘the dimensions of ’ the quantity. In mechanics, all the physical quantities can be written in terms of the dimensions [L], [M] and [T].

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DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity. An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity. Thus, the dimensional equations are the equations, which represent the dimensions of a physical quantity in terms of the base quantities. The dimensional equation can be obtained from the equation representing the relations between the physical quantities. The dimensional formulae of a large number and wide variety of physical quantities, derived from the equations representing the relationships among other physical quantities and expressed in terms of base quantities

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DIMENSIONAL ANALYSIS AND ITS APPLICATIONS The recognition of concepts of dimensions, which guide the description of physical behaviour is of basic importance as only those physical quantities can be added or subtracted which have the same dimensions. A thorough understanding of dimensional analysis helps us in deducing certain relations among different physical quantities and checking the derivation, accuracy and dimensional consistency or homogeneity of various mathematical expressions. When magnitudes of two or more physical quantities are multiplied, their units should be treated in the same manner as ordinary algebraic symbols. We can cancel identical units in the numerator and denominator. The same is true for dimensions of a physical quantity. Similarly, physical quantities represented by symbols on both sides of a mathematical equation must have the same dimensions.

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Checking the Dimensional Consistency of Equations The magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions. In other words, we can add or subtract similar physical quantities. Thus, velocity cannot be added to force, or an electric current cannot be subtracted from the thermodynamic temperature. This simple principle called the principle of homogeneity of dimensions in an equation is extremely useful in checking the correctness of an equation. If the dimensions of all the terms are not same, the equation is wrong. Dimensions are customarily used as a preliminary test of the consistency of an equation, when there is some doubt about the correctness of the equation. It is uncertain to the extent of dimensionless quantities or functions. The arguments of special functions, such as the trigonometric, logarithmic and exponential functions must be dimensionless. if an equation fails this consistency test, it is proved wrong, but if it passes, it is not proved right. Thus, a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be wrong.

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Dimensional analysis is very useful in deducing relations among the interdependent physical quantities. However, dimensionless constants cannot be obtained by this method. The method of dimensions can only test the dimensional validity, but not the exact relationship between physical quantities in any equation. It does not distinguish between the physical quantities having same dimensions. 