2_UNITS_MEASUREMENT

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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 Created by C. Mani, Education Officer, KVS RO Silchar Next

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

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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: m etre, 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/m 3 , kg m/s 2 , kg m 2 /s 2 , etc. Home Next Previous

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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

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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: Quantity Dimension System of units fps cgs mks Length L f oot c enti metre m etre Mass M p ound g ramme k ilogramme Time T s econd s econd s econd S ysteme I nternationale 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

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Fundamental Units in SI system Quantity Symbol SI unit Symbol Main units Length L metre m Mass M kilogramme kg Time T second s Electric Current A ampere A Thermodynamic Temperature K kelvin K Light Intensity Cd candela cd Amount of substance mole mole mol Supplementary units Plane angle d θ radian rad Solid angle d Ω steradian sr Home Next Previous

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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) 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) Second 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) Definition of SI Units Home Next Previous

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Kelvin The kelvin, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (1967) Candela Mole The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×10 12 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. (1979) 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

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Plane angle Plane angle ‘d θ ’ is the ratio of arc ‘ds’ to the radius ‘r’. Its SI unit is ‘radian’. Solid angle Solid angle ‘d Ω ’ 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’. d θ ds r r O r d Ω r dA O d θ = ds r d Ω = dA r 2 Home Next Previous

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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. IMPORTANT Home Next Previous

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Name Symbol Value in SI Unit minute min 60 s hour h 60 min = 3600 s day d 24 h = 86400 s year y 365.25 d = 3.156 x 10 7 s degree 0 10 = (π / 180) rad litre l 1 dm 3 = 10 -3 m 3 tonne t 10 3 kg carat c 200 mg bar bar 0.1 MPa = 10 5 Pa curie ci 3.7 x 10 10 s -1 roentgen r 2.58 x 10 -4 C/kg quintal q 100 kg barn b 100 fm 2 = 10 -28 m 2 are a 1 dam 2 = 10 2 m 2 hectare ha 1 hm 2 = 10 4 m 2 standard atmosphere pressure atm 101325 Pa = 1.013 x 10 5 Pa Some Units are retained for general use (Though outside SI) Home Next Previous

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MEASUREMENT OF LENGTH The order of distances varies from 10 -14 m (radius of nucleus) to 10 25 m (radius of the Universe)   The distances ranging from 10 -5 m to 10 2 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

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P 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 D D R L b θ θ = LR D b D = D = b θ Home Next Previous

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d   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 Measurement of the size or angular diameter of an astronomical object α D D d = α D E Home Next Previous

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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 Gun fire Echo received Sound wave S S S = v x T 2 Home Next Previous

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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 10 8 m s -1 S = c x t 2 Home Next Previous

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

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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 (ml). Then we take 1 cm 3 of this solution and dilute it to 20 cm 3 , using alcohol. 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. 2. Avogadro ’ s Method: 20 x 20 1 acid per cm 3 of solution. cm 3 of oleic So, the concentration of the solution is Home Next Previous

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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 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 or 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. 20 x 20 1 nV cm 3 Amount of oleic acid in this solution = t = Volume of the film Area of the film t = 20 x 20 x A nV cm Home Next Previous

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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 10 26 m of the extent of the observable Universe. We also use certain special length units for short and large lengths which are given below: Range of Lengths Unit Symbol Value Definition 1 fermi 1 f 10 –15 m 1 angstrom 1 Å 10 –10 m 1 Astronomical Unit 1 AU 1.496 × 10 11 m Average distance of the Sun from the Earth 1 light year 1 ly 9.46 × 10 15 m The distance that light travels with speed of 3 × 10 8 m s –1 in 1 year 1 parsec 3.08 × 10 16 m (3.26 ly ) The distance at which average radius of Earth’s orbit subtends an angle of 1 arc second Home Next Previous

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S.No Size of the object or distance Length ( m) 1 Size of proton 10 -15 2 Size of atomic nucleus 10 -14 3 Size of the Hydrogen atom 10 -10 4 Length of a typical virus 10 -8 5 Wavelength of a light 10 -7 6 Size of the red blood corpuscle 10 -5 7 Thickness of a paper 10 -4 8 Height of the Mount Everest from sea level 10 4 9 Radius of the Earth 10 7 10 Distance of the moon from the earth 10 8 11 Distance of the Sun from the earth 10 11 12 Distance of the Pluto from the Sun 10 13 13 Size of our Galaxy 10 21 14 Distance of the Andromeda galaxy 10 22 15 Distance of the boundary of observable universe 10 26 Range and Order of Lengths (L longest : L shortest = 10 41 : 1) Home Next Previous

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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 ( 12 C 6 ) including the mass of electrons.   MEASUREMENT OF MASS 1 u = 1.66 × 10 –27 kg Home Next Previous

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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.  Methods of measuring mass 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 10 55 kg of the known Universe. Range of Masses Home Next Previous

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S.No . Object Mass (kg) 1 Electron 10 -30 2 Proton 10 -27 3 Uranium atom 10 -25 4 Red blood cell 10 -13 5 Dust particle 10 -9 6 Rain drop 10 -6 7 Mosquito 10 -5 8 Grape 10 -3 9 Human 10 2 10 Automobile 10 3 11 Boeing 747 10 8 12 Moon 10 23 13 Earth 10 25 14 Sun 10 30 15 Milky way galaxy 10 41 16 Observable Universe 10 55 Range and Order of Masses (M largest : M smallest = 10 85 : 1 ≈ (10 41 ) 2 ) Home Next Previous

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

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S.No . Event Time Intervals ( s) 1 Life span of most unstable particle 10 -24 2 Time required for light to cross a nuclear distance 10 -22 3 Period of X-rays 10 -19 4 Period of atomic vibrations 10 -15 5 Period of light wave 10 -15 6 Life time of an excited atom 10 -8 7 Period of radio wave 10 -6 8 Period of sound wave 10 -3 9 Wink of eye 10 -1 10 Time between successive human heart beats 10 0 11 Travel time for light from the Moon to the Earth 10 0 12 Travel time for light from the Sun to the Earth 10 2 13 Time period of a satellite 10 4 14 Rotation period of the Earth 10 5 15 Rotation and revolution periods of the Moon 10 6 16 Revolution period of the Earth 10 7 17 Travel time for light from the nearest star 10 8 18 Average human life span 10 9 19 Age of Egyptian pyramids 10 11 20 Time since dinosaurs became extinct 10 15 21 Age of the Universe 10 17 Range and Order of Time Intervals (T longest : T shortest = 10 41 : 1) Home Next Previous

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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. ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN MEASUREMENT Home Next Previous

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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 vernier callipers 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

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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.   (b) Imperfection in experimental technique or procedure: Home Next Previous

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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.   II. Random errors 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

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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

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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 |Δ a |. Note: In absence of any other method of knowing true value, we consider arithmetic mean as the true value. Absolute Error, Relative Error and Percentage Error The errors in the individual measurement values from the true value are: Δ a 1 = a 1 - a mean Δ a 2 = a 2 - a mean Δ a n = a n - a mean The Δ a calculated above may be positive or negative. But absolute error |Δ a | will always be positive. ---------------- ---------------- Home Next Previous

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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 Δ a mean . Thus, Δ a mean = (|Δ a 1 |+|Δ a 2 |+|Δ a 3 |+...+ |Δ a n |)/ n = ∑ |Δ a i |/ n i=1 n If we do a single measurement, the value we get may be in the range a mean ± Δ a mean This implies that any measurement of the physical quantity a is likely to lie between ( a mean + Δ a mean ) and ( a mean - Δ a mean ) Home Next Previous

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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 . When the relative error is expressed in per cent, it is called the percentage error ( δ a ). Relative error Relative error = Δ a mean a mean Relative error = Mean absolute error True value or Arithmetic Mean Percentage error Percentage error = Mean absolute error True value or Arithmetic Mean x 100% Percentage error δ a = Δ a mean a mean x 100% Home Next Previous

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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 ± Δ A , B ± Δ B respectively, where Δ A and Δ B are their absolute errors. Combination of Errors Let Z = A + B Z ± Δ Z = ( A ± Δ A ) + ( B ± Δ B ) = ( A + B) ± ( Δ A + Δ B ) = Z ± ( Δ A + Δ B ) ± Δ Z = ± ( Δ A + Δ B ) When two quantities are added, the absolute error in the final result is the sum of the individual errors. or Δ Z = ( Δ A + Δ B ) Home Next Previous

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(b) Error of a Difference: 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. Let Z = A - B Z ± Δ Z = ( A ± Δ A ) - ( B ± Δ B ) When two quantities are subtracted, the absolute error in the final result is the sum of the individual errors. Δ Z = ( Δ A + Δ B ) ± = ( A - B) ± Δ A Δ B = Z ± ( Δ A + Δ B ) (since ± and are the same) ± 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. or ± Δ Z = ± ( Δ A + Δ B ) Home Next Previous

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(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, 1 ± ΔZ Z = ΔB B 1 ± ΔA A ± ± ΔA ΔB A B ΔA ΔB A B is very small and hence negligible ± ΔZ Z = ΔB B ± ΔA A ± or = ΔZ Z ΔA A ΔB B + When two quantities are multiplied, the relative error in the final result is the sum of the relative errors of the individual quantities. Home Next Previous

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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 Z ΔA A ΔB B + Home Next Previous

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(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. Let Z = A B (A ± ΔA) (B ± ΔB) Z ± ΔZ = (A ± ΔA) Z ± ΔZ = B ΔB B 1 ± Z ± ΔZ = (A ± ΔA) B ΔB B 1 ± -1 (by Binomial Approximation) = A B ΔA B ± Z ± ΔZ ΔB B 1 ± ΔB B x A B ΔA B ± Z ± ΔZ = A B ± ± ΔB B x ΔA B Dividing LHS by Z and RHS by A / B and simplifying we have, ± ΔZ Z = ΔB B ± ΔA A ± ΔA ΔB B 2 is negligible or = ΔZ Z ΔA A ΔB B + When two quantities are divided, the relative error in the final result is the sum of the relative errors of the individual quantities. Home Next Previous

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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 Z = A B Applying log on both the sides, we have log Z = log A - log B Differentiating , we have = ΔZ Z ΔA A ΔB B - Logically an error can not be nullified by making another error. Therefore errors are not subtracted but only added up. = ΔZ Z ΔA A ΔB B + Math has to be bent to satisfy Physics in many situations! Think of more such situations!! 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. Home Next Previous

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

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

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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.   SIGNIFICANT FIGURES 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 μ m all have four SF. Home Next Previous

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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. (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. Rules for determining the number of significant figures Home Next Previous

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Any given number can be written in the form of a ×10 b in many ways; for example 350 can be written as 3.5×10 2 or 35×10 1 or 350×10 0 . a ×10 b means " a times ten raised to the power of b ", where the exponent b is an integer , and the coefficient a 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 3.5×10 2 . 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 . Scientific Notation Home Next Previous

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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. Rules for Arithmetic Operations with Significant Figures Home Next Previous

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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. Rounding off the Uncertain Digits Home Next Previous

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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 [L 3 ] The dimensions of force are [MLT -2 ] The dimensions of energy are [ML 2 T -2 ] DIMENSIONS OF PHYSICAL QUANTITIES Home Next Previous

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Dimensional Quantity Dimensional quantity is a physical quantity which has dimensions. For example: Speed, acceleration, momentum, torque, etc. Dimensionless quantity is a physical quantity which has no dimensions. For example: Relative density, refractive index, strain, etc. Dimensionless Quantity 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 is a constant which has no dimensions. For example: 5, -.0.38, e, π , etc. Dimensionless Constant Home Next Previous

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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° L 3 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.   DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS Example: (i) [V] = [M° L 3 T°] (ii) [v] = [M° L T -1 ] (iii) [a] = [M° L T –2 ] Home Next Previous

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Impulse and momentum Work, energy, torque, moment of force Angular momentum, Planck’s constant, rotational impulse Stress, pressure, modulus of elasticity, energy density Force constant, surface tension, surface energy Angular velocity, frequency, velocity gradient Gravitational potential, latent heat Thermal capacity, entropy, universal gas constant and Boltzmann’s const. Force, thrust Power, luminous flux Quantities having the same dimensional formulae  Dimensional formulae for physical quantities often used in Physics are given at the end. ( From Slide 63 ) Home Next Previous

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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. Similarly, physical quantities represented by symbols on both sides of a mathematical equation must have the same dimensions. DIMENSIONAL ANALYSIS AND ITS APPLICATIONS Dimensional Analysis can be used- To check the dimensional consistency of equations (Principle of homogeneity of dimensions). 2. To convert units in one system into another system. 3. To derive the relation between physical quantities based on certain reasonable assumptions. Dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. Home Next Previous

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The magnitudes of physical quantities may be added together or subtracted from one another only if they have the same dimensions. For example, initial velocity can be added to or subtracted from final velocity because they have same dimensional formula [M 0 LT -1 ] . But, force and momentum can not be added because their dimensional formulae are different and are [MLT -2 ] and [MLT -1 ] respectively. I. Checking the Dimensional Consistency of Equations The principle of homogeneity of dimensions: Home Next Previous

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Example: 1. To check the dimensional consistency of v 2 = u 2 + 2as The dimensions of the quantities involved in the equation are: [u] = [M 0 LT -1 ] [v] = [M 0 LT -1 ] [a] = [M 0 LT -2 ] [s] = [M 0 LT 0 ] Substituting the dimensions in the given equation, [M 0 LT -1 ] 2 = [M 0 LT -1 ] 2 + [M 0 LT -2 ] [M 0 LT 0 ] [M 0 L 2 T -2 ] = [M 0 L 2 T -2 ] + [M 0 L 2 T -2 ] Each term of the above equation is having same dimensions. Therefore, the given equation is dimensionally correct or dimensionally consistent. (Note that the constant 2 in the term ‘2as’ does not have dimensions) Home Next Previous

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Note: If an equation fails the 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. Example: Equations v 2 = u 2 - 2as or v 2 = u 2 + ½as are dimensionally consistent but are incorrect equations in mechanics. Albert Einstein tried his famous mass-energy equation as E = m / c 2 , E = m 2 / c, E = m 2 c, etc. Finally he settled with E = m c 2 using dimensions and then proved it with the help of Calculus. Home Next Previous

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2. To check the dimensional consistency of ½ mv 2 = mgh The dimensions of the quantities involved in the equation are: [m] = [ML 0 T 0 ] [v] = [M 0 LT -1 ] [g] = [M 0 LT -2 ] [h] = [M 0 LT 0 ] Substituting the dimensions in the given equation, [ML 0 T 0 ] [M 0 LT -1 ] 2 = [ML 0 T 0 ] [M 0 LT -2 ] [M 0 LT 0 ] [ML 2 T -2 ] = [ML 2 T -2 ] Each term of the above equation is having same dimensions. Therefore, the given equation is dimensionally correct or dimensionally consistent. (Note that the constant ½ in the term ‘½ mv 2 ’ does not have dimensions) Home Next Previous

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II. Conversion of units in one system into another system Units are derived from the dimensions and the dimensions are derived from the actual formulae of physical quantities. If the dimensions are known for a physical quantity, then it is easy to express it in fps, cgs , mks , SI systems or any other arbitrary chosen system. n 1 [M 1 a L 1 b T 1 c ] = n 2 [M 2 a L 2 b T 2 c ] M 1 M 2 T 1 T 2 L 1 L 2 a b c n 2 = n 1 n 1 and n 2 are the magnitudes in the respective systems of units. Smaller the unit bigger the magnitude of a physical quantity and vice versa. For example, 1 m = 100 cm (m is the bigger unit and cm is the smaller one) 1 N = 10 5 dynes (Newton is bigger and dyne is smaller) Home Next Previous

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Example: 1. To convert 1 joule in erg. ‘joule’ is unit of energy or work in SI system and ‘erg’ is the unit in cgs system. The dimensional formula of energy or work is [ML 2 T -2 ]. The units from dimensions in SI and cgs systems are kg m 2 s -2 and g cm 2 s -2 respectively. SI System cgs System Magnitude n 1 = 1 n 2 = ? Mass (M) 1 kg (=1000 g) 1 g Length (L) 1 m (= 100 cm) 1 cm Time (T) 1 s 1 s [ M a L b T c ] = [ML 2 T -2 ] Therefore, a=1, b=2, c=-2 M 1 M 2 T 1 T 2 L 1 L 2 a b c n 2 = n 1 1000 g 1 g 1 2 -2 n 2 = 1 100 cm 1 cm 1 s 1 s n 2 = 1 (1000) 1 (100) 2 (1) -2 n 2 = 10 7 1 joule = 10 7 erg Let n 1 joule = n 2 erg Home Next Previous

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2. To convert 1 newton into a system where mass is measured in mg, length in km and time in minute ‘newton or kg m s -2 ’ is unit of force in SI system and ‘mg km min -2 ’ is the unit in the new system. The dimensional formula of force is [MLT -2 ]. SI System New System Magnitude n 1 = 1 n 2 = ? Mass (M) 1 kg (=10 6 mg) 1 mg Length (L) 1 m (= 1/1000 km) 1 km Time (T) 1 s (= 1/60) 1 s [ M a L b T c ] = [MLT -2 ] Therefore, a=1, b=1, c=-2 M 1 M 2 T 1 T 2 L 1 L 2 a b c n 2 = n 1 10 6 mg 1 mg 1 1 -2 n 2 = 1 1/1000 km 1 km 1/60 s 1 s n 2 = 1 (10 6 ) 1 (10 -3 ) 1 (60) 2 n 2 = 3.6 x 10 6 1 newton = 3.6x10 6 mg km min -2 Let n 1 newton = n 2 mg km min -2 Home Next Previous

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III. Deducing Relation among the Physical Quantities 1. Consider a simple pendulum, having a bob attached to a string that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length ( l ), mass of the bob ( m ) and acceleration due to gravity ( g ). Derive the expression for its time period using method of dimensions. The dependence of time period T on the quantities l, g and m as a product may be written as: T = k l x m y g z where k is dimensionless constant and x, y and z are the exponents. The method of dimensions can sometimes be used to deduce relation among the physical quantities. For this we should know the dependence of the physical quantity on other quantities (upto three physical quantities or linearly independent variables) and consider it as a product type of the dependence. Example: Home Next Previous

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By substituting dimensions on both sides of T = k l x m y g z , we have [M 0 L 0 T] = [M 0 LT 0 ] x [ML 0 T 0 ] y [M 0 LT -2 ] z [M 0 L 0 T] = [M] y [L] x+z [T] -2z On equating the dimensions on both sides, we have y = 0 x + z = 0 –2z = 1 So that x = ½ , y = 0, z = -½ Then, T = k l ½ g –½ Or The dimensions of the quantities involved in the equation are: [m] = [ML 0 T 0 ] [l] = [M 0 LT 0 ] [g] = [M 0 LT -2 ] [T] = [M 0 L 0 T] T = k l g The value of k is 2 π and determined from other methods. T = 2 π l g Home Next Previous

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Demerits of Dimensional Analysis The dimensional analysis can not be used in the following cases: The value of constants in an equation can not be determined as the constants do not have dimensions. Only dimensional consistency and not the physical consistency can be tested. Dimensions can be found from the physical quantity, but physical quantity can not be always guessed from dimensions because two or more quantities may have same dimensions. The equation containing the dependency on more than 3 quantities can not be determined using only M, L and T. (Note that if 4 independent quantities are involved, then 4 variables and hence 4 simultaneous equations are required; hence there must be 4 fundamental dimensions) 5. The equation containing exponential, trigonometric, logarithmic functions, etc. can not be derived as they do not have dimensions. The equations having the relations other than products / quotients can not be derived. Home Next Previous

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Dimensional formulae for some physical quantities   Physical quantity Symbol Formula Dimensional formula Unit General   Length L Fundamental [M 0 LT 0 ] m Mass M Fundamental [ML 0 T 0 ] kg Time T Fundamental [M 0 L 0 T] s Area A Length x breadth [M 0 L 2 T 0 ] m 2 Volume V Length x breadth x height [M 0 L 3 T 0 ] m 3 Linear density   Mass / length [ML –1 T 0 ] kg m –1 Mass Density ρ Mass / volume [ML –3 T 0 ] Kg m –3 Specific gravity or relative density RD Density of the substance / density of water [M o L o T o ] –– Specific volume   1 / density [M –1 L 3 T 0 ] kg –1 m 3 Time period T Time taken for 1 oscillation [M 0 L 0 T] s Frequency ν 1 / time period [M 0 L 0 T –1 ] Hz or s -1 Home Next Previous

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Mechanics Distance, displacement, wavelength, focal length s, λ, f   [M 0 LT 0 ] m Speed v Distance / time [M 0 LT –1 ] m s –1 Velocity v Displacement / time [M 0 LT –1 ] m s –1 Velocity gradient   dv / dx [M 0 L 0 T –1 ] s –1 Acceleration or acceleration due to gravity a or g Velocity / time [M 0 LT –2 ] m s –2 Momentum p Mass x velocity [MLT –1 ] kg m s –1 Force F Mass x acceleration [MLT –2 ] kg m s –2 or newton ( N) Force constant or spring constant K Force / extension [ML 0 T –2 ] N m –1 Impulse j Force x time [MLT –1 ] N s or kg m s –1 Home Next Previous

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Gravitation Gravitational potential   G P E / mass [M 0 L 2 T –2 ] J kg –1 Universal gravitational constant G G = F d 2 /(m 1 m 2 ) [M –1 L 3 T –2 ] N m 2 kg –2 Intensity of gravitational field   F / m [M 0 L 1 T –2 ] N kg –1 Mechanics Work W Force x Displacement [ML 2 T –2 ] kg m 2 s –2 or joule (J) Energy E Capacity to do work [ML 2 T –2 ] joule (J) Energy density   Energy / volume [ML –1 T –2 ] J m –3 Power P (Work or energy) / time [ML 2 T –3 ] J s –1 or Watt (W ) Pressure P Force / area [ML –1 T –2 ] N m –2 or Pascal (Pa) Pressure head     [ M o LT o ] m Home Next Previous

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Rotational Motion Angle θ Arc/radius [M o L o T o ] rad Angular displacement θ   [M o L o T o ] rad Angular impulse   Torque x time [ML 2 T –1 ] N m s Angular velocity ω Angular displacement / time [M 0 L 0 T –1 ] rad s –1 Angular frequency ω Angular displacement / time [M 0 L 0 T –1 ] rad s –1 Angular momentum L Moment arm x linear momentum [ML 2 T –1 ] kg m 2 s –1 Torque or moment of force τ Force x moment arm [ML 2 T –2 ] N m Moment of inertia I Mass x radius 2 [ML 2 T 0 ] kg m 2 Home Next Previous

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Properties of matter Stress   Restoring force / area [ML –1 T –2 ] N m –2 or Pa Strain   Change in dimension / original dimension [M o L o T o ] –– Modulus of elasticity E Stress / strain [ML –1 T –2 ] N m –2 or Pa Bulk modulus K [ML –1 T –2 ] N m –2 or Pa Compressibility   1 / bulk modulus [M –1 LT 2 ] Pa –1 or N –2 m 2 Poisson’s ratio   Lateral strain / longitudinal strain [ M o L o T o ] –– Surface tension σ Force / length [ML 0 T –2 ] N m –1 or J m –2 Surface energy   Energy / area [ML 0 T –2 ] J m –2 Coefficient of viscosity η F = [ML –1 T –1 ] Poise Home Next Previous

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Heat Heat Q or H Energy [ML 2 T –2 ] J Temperature θ Fundamental [M o L o T o θ] kelvin ( K ) Temperature gradient   [M o L –1 T o θ] m –1 K Thermal capacity   Mass x specific heat or heat energy / temp. [ML 2 T –2 θ –1 ] J K -1 Water equivalent w   [ML o T o ] kg Specific heat Capacity s or c c = Q / (m θ) [M o L 2 T –2 θ –1 ] J kg –1 K –1 Ratio of specific Heats γ c p / c v [M o L o T o ] –– Latent heat L L = Q/m [M o L 2 T –2 ] J kg –1 Entropy   [ML 2 T –2 θ –1 ] J –1 Specific entropy   1/entropy [M –1 L –2 T 2 θ] K J –1 Joule’s constant or mechanical equivalent of heat J J = W / H [M o L o T o ] J cal –1 Home Next Previous

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Heat Calorific value     [M 0 L 2 T –2 ] J kg –1 Coefficient of linear or areal or volume expansion α, β, γ Change in dimension /(original dimension x temp.) [ M 0 L 0 T 0 θ -1 ] K –1 Coefficient of thermal conductivity K K = ( H.E. x thickness)/ ( area x temp . x time) [MLT –3 θ –1 ] W m –1 K –1 Pressure coefficient or volume coefficient     [M o L o T o θ –1 ] K –1 Boltzmann’s Constant K   [ML 2 T –2 θ –1 ] J K –1 Stefan’s constant σ [ML o T –3 θ –4 ] W m –2 K –4 Universal gas constant R Work / temperature [ML 2 T –2 θ –1 ] J mol –1 K -1 Home Next Previous

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Electricity Electric current I Fundamental [M 0 L 0 T 0 I] Ampere ( A) Electric charge or quantity of electric charge Q Current x time [M 0 L 0 TI] Coulomb or A s Electric dipole moment p Charge x dipole Length [M 0 LTI] C m Electric field strength or Intensity of electric field E Force / charge [MLT –3 I –1 ] N C –1 , or V m –1 Emf (or) electric potential Difference E Work / charge [ML 2 T –3 I –1 ] volt (V) Electric resistance R [ML 2 T –3 I –2 ] Ohm(Ω) Electric conductance   1 / resistance [M –1 L –2 T 3 I 2 ] Ohm –1 or mho or siemen Resistivity or specific Resistance ρ ρ = RA/l [ML 3 T –3 I –2 ] m Specific conductance or Conductivity   1 / specific Resistance [M –1 L –3 T 3 I 2 ] siemen / metre or S m –1 Electric capacitance C Charge / potential [M –1 L –2 T 4 I 2 ] C V –1 or farad Permittivity of free space ε 0 [M –1 L –3 T 4 I 2 ] C 2 N –1 m –2 Home Next Previous

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Magnetism Magnetic pole strength m m = M / l [M 0 LT 0 I] A m Magnetic dipole moment M Pole strength x dipole length [M 0 L 2 T 0 I] A m 2 Permeability of free space μ 0 [MLT –2 I –2 ] H m –1 or N A –2 Intensity of magnetization , magnetic field strength, magnetic intensity , magnetic moment density I Magnetic moment / Volume M 0 L –1 T 0 I] A m –1 Magnetic flux Ф b Magnetic induction x area [ML 2 T –2 I –1 ] weber ( Wb ) Magnetic induction, magnetic flux density, magnetic field B Force/(current x length ) [ML 0 T –2 I –1 ] N A –1 m –1 or tesla (T) Inductance or coefficient of self induction L L = 2E/(LI 2 ) [ML 2 T –2 I –2 ] henry (H) Home Next Previous

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Optics Luminous flux     [ML 2 T –3 ] lumen or J s –1 Refractive index μ c air / c med [M o L o T o ] –– Illumination ( Illuminance )     [ML 0 T –3 ] lux or lumen/metre 2   Decay constant   λ λ = 0.693 / half life [M o L o T –1 ] disintegrations per second   Planck’s constant h Energy / frequency [ML 2 T –1 ] J s   Wien’s constant b Wavelength x temp. [M 0 LT 0 θ] m K Home Next Previous

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Acknowledgement 1. Physics Part I for Class XI by NCERT ] Home End Previous

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