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UNITS AND MEASUREMENT

UNITS AND MEASUREMENT. Physical Quantity – Fundamental & Derived Quantities Unit – Fundamental & Derived Units Characteristics of Standard Unit fps, cgs, mks & SI System of Units Definition of Fundamental SI units Measurement of Length – Large Distances and Small Distances

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UNITS AND MEASUREMENT

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  1. UNITS AND MEASUREMENT • Physical Quantity – Fundamental & Derived Quantities • Unit – Fundamental & Derived Units • Characteristics of Standard Unit • fps, cgs, mks & SI System of Units • Definition of Fundamental SI units • Measurement of Length – Large Distances and Small Distances • Measurement of Mass and Measurement of Time • Accuracy, Precision of Instruments and Errors in Measurements • Systematic Errors and Random Errors • Absolute Error, Relative Error and Percentage Error • Combination of Errors in Addition, Subtraction, Multiplication, Division and Exponent. • Significant Figures, Scientific Notation and Rounding off Uncertain Digits • Dimensions, Dimensional Formulae and Dimensional Equations • Dimensional Analysis – Applications- I, II & III and Demerits Next Created by C. Mani, Education Officer, KVS RO Silchar

  2. Physical Quantity A quantity which is measurable is called ‘physical quantity’. Fundamental Quantity A physical quantity which is the base and can not be derived from any other quantity is called ‘fundamental quantity’. Examples:Length, Mass, Time, etc. Derived Quantity A physical quantity which can be derived or expressed from base or fundamental quantity / quantities is called ‘derived quantity’. Examples:Speed, velocity, acceleration, force, momentum, torque, energy, pressure, density, thermal conductivity, resistance, magnetic moment, etc. Home Next Previous

  3. Unit Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. Fundamental Units The units of the fundamental or base quantities are called fundamental or base units. Examples:metre, kilogramme, second, etc. Derived Units The units of the derived quantities which can be expressed from the base or fundamental quantities are called derived units. Examples:metre/sec, kg/m3, kg m/s2, kg m2/s2, etc. Home Next Previous

  4. System of Units A complete set of both fundamental and derived units is known as the system of units. Characteristics of Standard Units A unit selected for measuring a physical quantity must fulfill the following requirements: • It should be well defined. • It should be of suitable size i.e. it should neither be too large nor too small in comparison to the quantity to be measured. • It should be reproducible at all places. • It should not change from place to place or time to time. • It should not change with the physical conditions such as temperature, pressure, etc. • It should be easily accessible. Home Next Previous

  5. Various System of Units In earlier time, various systems like ‘fps’, ‘cgs’ and ‘mks’ system of units were used for measurement. They were named so from the fundamental units in their respective systems as given below: Systeme Internationale d’ unites (SI Units) The SI system 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. This is the system of units which is at present accepted internationally. SI system uses decimal system and therefore conversions within the system are quite simple and convenient. Home Next Previous

  6. Fundamental Units in SI system Home Next Previous

  7. Definition of SI Units Metre 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) Kilogramme 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) Second 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) Ampere 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 2×10–7 newton per metre of length. (1948) Home Next Previous

  8. Kelvin The kelvin, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (1967) Candela The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. (1979) Mole The mole is the amount of substance of a system, which contains as many elementary entities as there are atoms in 0.012 kilogramme of carbon-12. (1971) Home Next Previous

  9. dθ dΩ Plane angle Plane angle ‘dθ’ is the ratio of arc ‘ds’ to the radius ‘r’. Its SI unit is ‘radian’. ds ds r dθ = r r O Solid angle Solid angle ‘dΩ’ is the ratio of the intercepted area ‘dA’ of the spherical surface described at the apex ‘O’ as the centre, to the square of its radius ‘r’. Its SI unit is ‘steradian’. dA r r dA dΩ = r2 O Home Next Previous

  10. IMPORTANT The following conventions are adopted while writing a unit: (1) Even if a unit is named after a person the unit is not written in capital letters. i.e. we write joules not Joules. (2) For a unit named after a person the symbol is a capital letter e.g. for joules we write ‘J’ and the rest of them are in lowercase letters e.g. second is written as ‘s’. (3) The symbols of units do not have plural form i.e. 70 m not 70 ms or 10 N not 10 Ns. (4) Punctuation marks are not written after the unit       e.g. 1 litre = 1000 cc not 1000 c.c. Home Next Previous

  11. Some Units are retained for general use (Though outside SI) Home Next Previous

  12. MEASUREMENT OF LENGTH The order of distances varies from 10-14 m (radius of nucleus) to 1025 m (radius of the Universe) The distances ranging from 10-5 m to 102 m can be measured by direct methods which involves comparison of the distance or length to be measured with the chosen standard length. Example: i) A metre rod can be used to measure distance as small as 10-3 m. ii) A vernier callipers can be used to measure as small as 10-4 m. iii) A screw gauge is used to measure as small as 10-5 m. For very small distances or very large distances indirect methods are used.  Home Next Previous

  13. Measurement of Large Distances • The following indirect methods may be used to measure very large distances: • Parallax method • Let us consider a far away planet ‘P’ at a distance ‘D’ from our two eyes. • Suppose that the lines joining the planet to the left eye (L) and the right eye (R) subtend an angle θ (in radians). • The angle θ is called ‘parallax angle’ or ‘parallactic angle’ and the distance LR = b is called ‘basis’. • As the planet is far away, b/D << 1, and therefore θ is very small. • Then, taking the distance LR = b as a circular arc of radius D, we have P θ D D R L b LR b θ = = D D b D = θ Home Next Previous

  14. Measurement of the size or angular diameter of an astronomical object If ‘d’ is the diameter of the planet and ‘α’ is the angular size of the planet (the angle subtended by d at the Earth E), then α = d/D The angle α 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 from d D D α E d = α D Home Next Previous

  15. Echo method or Reflection method • This method is used to measure the distance of a hill. • A gun is fired towards the hill and the time interval between the instant of firing the gun and the instant of hearing the echo of the gun shot is noted. • This is the time taken by the sound to travel from the observer to the hill and back to the observer. • If v= velocity of sound; • S = the distance of hill from the observer • and • T = total time taken, then S S Sound wave v x T S = Echo received Gun fire 2 Home Next Previous

  16. In place of sound waves, LASER can be used to measure the distance of the Moon from the Earth. LASER is a monochromatic, intense and unidirectional beam. • If ‘t’ is the time taken for the LASER beam in going to and returning from the Moon, then the distance can be calculated from the formula where c = 3 x 108 m s-1 c x t S = 2 Home Next Previous

  17. Estimation of Very Small Distances 1. Using Electron Microscope: 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. 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.  Home Next Previous

  18. 2. Avogadro’s Method: 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 (ml). Then we take 1 cm3 of this solution and dilute it to 20 cm3, using alcohol. So, the concentration of the solution is cm3of oleic acid per cm3 of solution. 1 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. 20 x 20 Home Next Previous

  19. 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 of n drops of solution = nVcm3 Amount of oleic acid in this solution = 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 1 nV cm3 Volume of the film 20 x 20 t = t = 20 x 20 x A Area of the film or nV cm 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 10–9m. Home Next Previous

  20. Range of Lengths The size of the objects we come across in the Universe varies 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 also use certain special length units for short and large lengths which are given below: Home Next Previous

  21. (Llongest : Lshortest = 1041 : 1) Range and Order of Lengths Home Next Previous

  22. MEASUREMENT OF MASS The SI unit of mass is kilogram (kg). The prototypes of the International standard kilogramme 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 kilogramme 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 = 1 u One unified mass unit is equal to (1/12) of the mass of an atom of Carbon-12 isotope (12C6 ) including the mass of electrons.   1 u = 1.66 × 10–27 kg Home Next Previous

  23. Methods of measuring mass • By using a common balance. • Large masses in the Universe like planets, stars, etc., • based on Newton’s law of gravitation can be measured by • using gravitational method. • 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.  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. Home Next Previous

  24. (Mlargest : Msmallest = 1085 : 1 ≈ (1041)2) Range and Order of Masses Home Next Previous

  25. MEASUREMENT OF TIME  We 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. 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. A cesium atomic clock is used at the National Physical Laboratory (NPL), New Delhi to maintain the Indian standard of time. Home Next Previous

  26. Range and Order of Time Intervals (Tlongest : Tshortest = 1041 : 1) Home Next Previous

  27. ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN MEASUREMENT Error: The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error. Accuracy: The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision: Precision tells us to what resolution or limit the quantity is measured. Example: Suppose the true value of a certain length is near 2.874 cm. In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 2.7 cm, while in another experiment using a measuring device of greater resolution, say 0.01 cm, the length is determined to be 2.69 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. Home Next Previous

  28. In general, the errors in measurement can be broadly classified as • (I) Systematic errors and (II) Random errors • I. 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: • The instrumental errors that arise from the errors due to imperfect design • or calibration of the measuring instrument, zero error in the instrument, • etc. • 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 verniercallipers the zero mark of vernier scale may not coincide with • the zero mark of the main scale; • (iii) An ordinary metre scale may be worn off at one end. Home Next Previous

  29. (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. • (c) Personal errors: • The personal errors 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. • Example: • If you 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 minimized by • improving experimental techniques, • selecting better instruments and • removing personal bias as far as possible.   Home Next Previous

  30. II. 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, personal errors by the observer taking readings, etc. Example: When the same person repeats the same observation, it is very likely that he may get different readings every time. Least count error Least count: The smallest value that can be measured by the measuring instrument is called its least count. The least count error is the error associated with the resolution of the instrument. Home Next Previous

  31. Example: (i) A Vernier callipers has the least count as 0.01 cm; (ii) A spherometer may have a least count of 0.001 cm. 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. Note: Least count error belongs to Random errors category but within a limited size; it occurs with both systematic and random errors. Home Next Previous

  32. Absolute Error, Relative Error and Percentage Error Absolute error The magnitude of the difference between the individual measurement value and the true value of the quantity is called the absolute error of the measurement. This is denoted by |Δa|. Note: In absence of any other method of knowing true value, we consider arithmetic mean as the true value. The errors in the individual measurement values from the true value are: Δa1= a1 - amean Δa2= a2 - amean ---------------- ---------------- Δan= an - amean The Δa calculated above may be positive or negative. But absolute error |Δa| will always be positive. Home Next Previous

  33. The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity a. It is represented by Δamean. Thus, Δamean= (|Δa1|+|Δa2|+|Δa3|+...+ |Δan|)/n n = ∑ |Δai|/n i=1 If we do a single measurement, the value we get may be in the range amean ± Δamean This implies that any measurement of the physical quantity a is likely to lie between (amean + Δamean) and (amean - Δamean) Home Next Previous

  34. Relative error The relative error is the ratio of the mean absolute error Δameanto the mean value ameanof the quantity measured. Mean absolute error Mean absolute error Percentage error = Relative error = Relative error = True value or Arithmetic Mean True value or Arithmetic Mean Percentage error When the relative error is expressed in per cent, it is called the percentage error (δa). Δamean Δamean amean amean x 100% Percentage error δa = x 100% Home Next Previous

  35. Combination of Errors In an experiment involving several measurements, the errors in all the measurements get combined. Example: Density is the ratio of the mass to the volume of the substance. If there are errors in the measurement of mass and of the sizes or dimensions, then there will be error in the density of the substance.  (a) Error of a Sum: Suppose two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors. Let Z = A + B Z ± ΔZ = (A ± ΔA) + (B ± ΔB) = (A + B) ± (ΔA + ΔB) = Z ± (ΔA + ΔB) ± ΔZ = ± (ΔA + ΔB) When two quantities are added, the absolute error in the final result is the sum of the individual errors. ΔZ = (ΔA + ΔB) or Home Next Previous

  36. (b) Error of a Difference: Suppose two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors. Let Z = A - B Z ± ΔZ = (A ± ΔA) - (B ± ΔB) = (A - B) ± ΔA ΔB ± = Z ± (ΔA + ΔB) (since ± and are the same) ± ± ΔZ = ± (ΔA + ΔB) When two quantities are subtracted, the absolute error in the final result is the sum of the individual errors. ΔZ = (ΔA + ΔB) or 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. Home Next Previous

  37. (c) Error of a Product: Suppose two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors. Let Z = A x B Z ± ΔZ = (A ± ΔA) x (B ± ΔB) Z ± ΔZ = AB ± A ΔB ± B ΔA ± ΔA ΔB Dividing LHS by Z and RHS by AB we have, ΔA ΔB ΔA ΔB = 1 ± ± ± A B A B 1 ± ± ΔA ΔB ΔZ ΔA ΔB ΔZ is very small and hence negligible A A B B Z Z When two quantities are multiplied, the relative error in the final result is the sumof the relative errors of the individual quantities. ΔZ ΔB ΔA = ± ± = + or Z B A Home Next Previous

  38. Error of a Product:ALITER Suppose two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors. Let Z = A x B Applying log on both the sides, we have log Z = log A + log B Differentiating, we have ΔZ ΔB ΔA = + Z B A Home Next Previous

  39. (d) Error of a Quotient: Suppose two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors. 1 1 ± 1 ± Let ΔA ± (A ± ΔA) Dividing LHS by Z and RHS by A / B and simplifying we have, B Z ± ΔZ = (B ± ΔB) (A ± ΔA) Z ± ΔZ = ΔA ΔB B B2 ± -1 (A ± ΔA) ΔB ΔB ΔB ΔB ΔZ ΔA A A A A ΔB Z ± ΔZ = Z = = is negligible Z B B A B B B B B B B B ΔA When two quantities are divided, the relative error in the final result is the sum of the relative errors of the individual quantities. B ΔA ΔB ΔZ ΔB ΔA ± x x ± ± Z ± ΔZ = = ± ± ± Z ± ΔZ = + B B or Z B A (by Binomial Approximation) Home Next Previous

  40. Error of a Quotient: ALITER Suppose two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively, where ΔA and ΔB are their absolute errors. Let Applying log on both the sides, we have log Z = log A - log B Differentiating, we have Logically an error can not be nullified by making another error. Therefore errors are not subtracted but only added up. A Math has to be bent to satisfy Physics in many situations! Think of more such situations!! Z = B Rule: When two quantities are multiplied or divided, the relative error in the final result is the sum of the relative errors in the individual quantities. ΔZ ΔZ ΔB ΔB ΔA ΔA = = + - Z Z B B A A Home Next Previous

  41. (e) Error of an Exponent (Power): Suppose a physical quantity A has measured values A ± ΔA where ΔA is its absolute error. Let Z = Ap where p is a constant. Z = A x A x A x ………x A (p times) Z ± ΔZ = (A ± ΔA) x (A ± ΔA) x (A ± ΔA) x ……. x (A ± ΔA) (p times) ΔA ΔA ΔZ ΔZ ΔA ΔA ΔA + + ……… (p times as per the product rule for errors) Note: If p is negative, |p| is taken because errors due to multiple quantities get added up. A A = = p + + or Z Z A A A Rule: The relative error in a physical quantity raised to the power p is the p times the relative error in the individual quantity. Home Next Previous

  42. (f) Error of an Exponent (Power): ALITER Suppose a physical quantity A has measured values A ± ΔA where ΔA is its absolute error. Let Z = Ap where p is a constant. Applying log on both the sides, we have (Whether p is positive or negative errors due to multiple quantities get added up only) log Z = |p| log A Differentiating, we have Z = ΔA |p| A ΔZ ΔZ ΔA ΔC ΔB = = q r p Z Z B C A Ap x Bq In general, if , then Cr Note: Cr is in Denominator, but the relative error is added up. + + Home Next Previous

  43. SIGNIFICANT FIGURES 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. Example: (i) The period of oscillation of a simple pendulum is 2.36 s; the digits 2 and 3 are reliable and certain, while the digit 6 is uncertain. Thus, the measured value has three significant figures. (ii) 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.   Note: A choice of change of different units does not change the number of significant digits or figures in a measurement. Eg. The length 1.205 cm, 0.01205, 12.05 mm and 12050 μm all have four SF. Home Next Previous

  44. Rules for determining the number of significant figures • 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. • (iv) The terminal or trailing zero(s) in a number without a decimal point are • not significant. • (v) The trailing zero(s) in a number with a decimal point are significant. Home Next Previous

  45. Scientific Notation Any given number can be written in the form of a×10b in many ways; for example 350 can be written as 3.5×102 or 35×101 or 350×100. a×10bmeans"a times ten raised to the power of b", where the exponentb is an integer, and the coefficienta is any real number called the significand or mantissa (the term "mantissa" is different from "mantissa" in common logarithm). If the number is negative then a minus sign precedes a (as in ordinary decimal notation). In normalized scientific notation, the exponent b is chosen such that the absolute value of a remains at least one but less than ten (1 ≤ |a| < 10). For example, 350 is written as 3.5×102. This form allows easy comparison of two numbers of the same sign in a, as the exponent b gives the number's order of magnitude. Home Next Previous

  46. Rules for Arithmetic Operations with Significant Figures • In arithmetic operations the final result should not have more significant figures than the original data from which it was obtained. • 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) Addition or subtraction: • The final result should retain as many decimal places as are there in the number with the least decimal places. Home Next Previous

  47. Rounding off the Uncertain Digits • Rounding off a number means dropping of digits which are not significant. The following rules are followed for rounding off the number: • If the digits to be dropped are greater than five, then add one to the • preceding significant figure. • 2. If the digit to be dropped is less than five then it is dropped without • bringing any change in the preceding significant figure. • If the digit to be dropped is five, then the preceding digit will be left • unchanged if the preceding digit is even and it will be increased by • one if it is odd. • In any involved or complex multi-step calculation, one 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. Home Next Previous

  48. DIMENSIONS OF PHYSICAL QUANTITIES • The nature of a physical quantity is described by its dimensions. • All the physical quantities can be expressed in terms of the seven base or fundamental quantities viz. mass, length, time, electric current, thermodynamic temperature, intensity of light and amount of substance, raised to some power. • The dimensions of a physical quantity are the powers (or exponents) to which the fundamental or base quantities are raised to represent that quantity. • Note: • Using the square brackets [ ] around a quantity means that we are dealing with ‘the dimensions of’ the quantity.  • Example: • The dimensions of volume of an object are [L3] • The dimensions of force are [MLT-2] • The dimensions of energy are [ML2T-2] Home Next Previous

  49. Dimensional Quantity Dimensional quantity is a physical quantity which has dimensions. For example: Speed, acceleration, momentum, torque, etc. Dimensionless Quantity Dimensionless quantity is a physical quantity which has no dimensions. For example: Relative density, refractive index, strain, etc. Dimensional Constant Dimensional constant is a constant which has dimensions. For example: Universal Gravitational constant, Planck’s constant, Hubble constant, Stefan constant, Wien constant, Boltzmann constant, Universal Gas constant, Faraday constant, etc. Dimensionless Constant Dimensionless constant is a constant which has no dimensions. For example: 5, -.0.38, e, π, etc. Home Next Previous

  50. 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. • Example: • The dimensional formula of the volume is [M° L3 T°], • The dimensional formula of speed or velocity is [M° L T-1] • (iii) The dimensional formula of acceleration is [M° L T–2] • An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity.   Example: (i) [V] = [M° L3 T°] (ii) [v] = [M° L T-1] (iii) [a] = [M° L T–2] Home Next Previous

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