1 UNIT & DIMENSION, MEASUREMENT Physical quantity: A quantity that can be measured by instrument, clearly defined and has proper units is called physical quantity. Physical quantities are classified as fundamental and derived quantities. Unit For Measurement The ‘Standared’ of measurement of a physical quantity is called the unit of that physical quantity N = Numerical value of the measure of the quantity and U = Unit of the quantity. Fundamental units: The physical quantity which does not depend on any other physical quantity is called a fundamental physical quantity such as length; mass and time are called fundamental units. Derived units: The units that can be obtained from fundamental units are called derived units. System of units: There are three systems of units. Name of system Fundamental unit of Length Mass Time F.P.S. Foot Pound Second C.G.S. Centimetre Gram Second M.K.S. (S.I.) Meter Kilogram Second In physics SI system is based on seven fundamental and two supplementary units. (i) Fundamental units: S.No. Basic Physics Fundamental Unit Symbol 1. Mass kilogram kg 2. Length meter m 3. Time second s 4. Electric current ampere A 5. Temperature Kelvin or Celsius K or C 6. Luminous intensity candela Cd 7. Quantity of matter mole Mol
2 (ii) Supplementary units: S.No. Supplementary Physical Quantities Supplementary unit Symbol 1. Plane angle Radian rad 2. Solid Angle Steradian sr Practical Units (1) Length (i) 1 fermi = 1 fm = 10–15 m (ii) 1 X-ray unit = 1XU = 10–13 m (iii) 1 angstrom = 1Å = 10–10 m = 10–8 cm = 10–7 mm = 0.1 mm (iv) 1 micron = m = 10–6 m (v) 1 astronomical unit= 1 A.U. = 1. 49 1011 m 1.5 1011 m 108 km (vi) 1 Light year = 1 ly = 9.46 1015 m (vii) 1 Parsec = 1pc = 3.26 light year (2) Mass (i) Chandra Shekhar unit : 1 CSU = 1.4 times the mass of sun = 2.8 1030 kg (ii) Metric tonne : 1 Metric tonne = 1000 kg (iii) Quintal : 1 Quintal = 100 kg (iv) Atomic mass unit (amu): amu = 1.67 10– 27 kg Mass of proton or neutron is of the order of 1 amu (3) Time (i) Year : It is the time taken by the Earth to complete 1 revolution around the Sun in its orbit. (ii) Lunar month : It is the time taken by the Moon to complete 1 revolution around the Earth in its orbit. 1 L.M. = 27.3 days (iii) Solar day : It is the time taken by Earth to complete one rotation about its axis with respect to Sun. Since this time varies from day to day, average solar day is calculated by taking average of the duration of all the days in a year and this is called Average Solar day. 1 Solar year = 365.25 average solar day or average solar day 365.25 1 the part of solar year (iv) Sedrial day : It is the time taken by earth to complete one rotation about its axis with respect to a distant star. 1 Solar year = 366.25 Sedrial day = 365.25 average solar day Thus 1 Sedrial day is less than 1 solar day. (v) Shake : It is an obsolete and practical unit of time. 1 Shake = 10– 8 sec
3 DEFINITIONS OF BASE UNITS: (i) Meter: The currently accepted definition of meter is the length of path travelled by light in vacuum in 1/299,792,458th second. (ii) Kilogram: Kilogram is the fundamental unit of mass. It is defined as the mass of a specific cylinder of platinum - iridium kept at the International Bureau of Weights and Measures in Paris. .(iii) Second: Second is the fundamental unit of time. It is defined as 86,400th part of a mean solar day. Second is accurately measured by an atomic clock. (iv) Coulomb: Coulomb is the fundamental unit of charge. It is defined as the charge required to obtain 9109 Newton of force between two equal charges separated at a distance of one meter in vacuum. (v) Candle: Candle is the fundamental unit of luminous intensity. It is defined as luminous intensity observed from a source of monochromatic light of frequency 5401012 Hz, that has an intensity of 1/683 watt per steradian. (vi) Kelvin: Kelvin is the fundamental unit of temperature. It has value of zero where the molecular activity of gases cease. (vii) Mole: Mole is the fundamental unit of quantity of matter. It is defined as amount of substance of a system that contains as many elementary particle as there are in 0.012 kg of carbon-12 (C-12). SI Prefixes and Symbols Used to Denote Powers of 10 Prefix Multiple Symbol yotta 1024 Y zetta 1021 Z exa 1018 E peta 1015 P tera 1012 T giga 109 G mega 106 M kilo 103 k hecto 102 h deca 10 da deci 10-1 d centi 10-2 c milli 10-3 m micro 10-6 mu nano 10-9 n pico 10-12 p femto 10-15 f atto 10-18 a zepto 10-21 Z yotto 10-24 Y
4 Dimensions Dimensions of a physical quantity are the powers to which the fundamental quantities must be raised to represent the given physical quantity. In mechanics all physical quantities can be expressed in terms of mass (M), length (L) and time (T). Example : Force = mass x acceleration = Or, So, the dimensions of force are 1 in mass, 1 in length and – 2 in time. Dimensionless quantity In the equation then the quantity is called dimensionless. Examples : Strain, specific gravity, angle. They are ratio of two similar quantities. A dimensionless quantity has same numeric value in all system of units. Definition of a Dimensional Formula The formulae or expressions that indicate how and which fundamental quantities are there in a physical quantity are called as the Dimensional Formula of the Physical Quantity. Dimensional formulae are helpful in finding units from one system to another. They have numerous real-life applications and form the basic aspect of units and measurements. Let’s assume that there is a physical quantity X that depends on base dimensions M (Mass), L (Length) and T (Time) with respective powers a, b and c, then its dimensional formula can be written as: [M aL bT c ] Note that a dimensional formula will always be closed in a square bracket [ ]. Also, when it comes to dimensional formulae of trigonometric, solid angle and plane angle, they cannot be defined as these are the types of quantities that are dimensionless in nature. When velocity is defined with the help of fundamental units of mass, length and time, we get when there is no mass, M0 = 1 (algebraic theory of indices). This is basically the dimensional formula for velocity and we can conclude that: Unit of velocity varies as per the unit of length and time but is independent of mass. When it comes to the unit of velocity, the power of L and T are 1 and -1 respectively. E.g. Formula for density is ML-3 Formula for force is MLT-2
5 BASIC PHYSICAL QUANTITIES PHYSICAL QUANTITY SYMBO L DIMENSIO N MEASUREMENT UNIT UNIT Length s L Meter m Mass M M Kilogram Kg Time t T Second Sec Electric charge q Q Coulomb C luminous intensity I C Candela Cd Temperature T K Kelvin oK Angle none Radian None Mechanical Physical Quantities (derived) PHYSICAL QUANTITY EQUATION SYMBOL Dimensional Formula MEASURMENT (in SI) UNIT Area A [M 0L 2 T0 ] square meter m2 Volume V [M 0L 3 T0 ] cubic meter m3 velocity v [M 0L T-1 ] meter per second m/sec angular velocity [M 0L 0T -1 ] radians per second 1/sec acceleration a [M 0L T-2 ] meter per square second m/sec2 angular acceleration [M 0L 0T -2 ] radians per square second 1/sec2 Force F [ M 0L T-2 ] Newton Kg m/sec2 Energy E [ M 0L 2 T-2 ] Joule Kg m2 /sec2 Work W [ M 0L 2 T-2 ] Joule Kg m2 /sec2 Heat Q [ M 0L 2 T-2 ] Joule Kg m2 /sec2 Torque [ M 0L 2 T-2 ] Newton meter Kg m2 /sec2 Power P [ M 0L 2 T-3 ] watt or joule/sec Kg m2 /sec3 Density D or [ M L3 T0 ] kilogram per cubic meter Kg/m3 pressure P [ M 0L -1 T-2 ] Newton per square meter Kg m-1 /sec2
6 impulse p [ M L T-1 ] Newton second Kg m/sec Inertia I [ M L2 T0 ] Kilogram square meter Kg m2 Force Constant k [ M L0 T-2 ] Newton per meter Nm-1 Latent Heat L [ M0 L2 T-2 ] Joule Per kg J/kg entropy S [ M L2 T-2K -1 ] joule per degree Kg m 2 /sec2K Volume rate of flow Q [ M0 L3 T-1 ] cubic meter per second m 3 /sec kinematic viscosity [ M0 L2 T-1 ] square meter m 2 /sec per second dynamic viscosity [ M L-1 T-1 ] Newton second per square meter Kg/m sec specific weight [ M L-2 T-2 ] Newton per cubic meter Kg m-2 /sec2 Application of Dimensional Analysis (1) To find the unit of a physical quantity in a given system of units : To write the definition or formula for the physical quantity we find its dimensions. Now in the dimensional formula replacing M, L and T by the fundamental units of the required system we get the unit of physical quantity. However, sometimes to this unit we further assign a specific name, e.g., Work = Force Displacement So [W] = [MLT–2] [L] = [ML2T –2] So its unit in C.G.S. system will be g cm 2 /s2 which is called erg while in M.K.S. system will be kg-m2 /s 2 which is called joule. (2) To find dimensions of physical constant or coefficients : As dimensions of a physical quantity are unique, we write any formula or equation incorporating the given constant and then by substituting the dimensional formulae of all other quantities, we can find the dimensions of the required constant or coefficient. (i) Gravitational constant : According to Newton’s law of gravitation SIGNIFICANT FIGURES The measurement of any physical quantity by any instrument is not absolutey correct. The degree of accuracy is shown by the significant figures. Significant figures do not change if we measure a physical quantity in different units. Example: Now, both have three significant figures.
7 Rules for Significant Figures Example: (i) All non-zero digits are significant figures. (ii) All zeros occuring between non-zero digits are significant figures: (iii) All zero to the right of the last non zero digit are not significant. (iv) All zero to the right of a derived point and to the left of a non zero digit are not significant. (v) All zeros to the right of a decimal point and to the right of a no-digit are significant. ROUNDING OFF THE MEASUREMENTS The certain rules are applied in order to rounding off the measurements : a. If the digit to be dropped is less than 5, then the preceding digit remains unchanged. For Example : The number 9.64 in rounded off to 9.6. b. If the digit to dropped is greater than 5, then the preceding digit is raised by 1. For Example : The 9.77 is rounded off to 9.8. c. The number 9.65 is rounded off to 9.6, the number 9.650 is rounded off to 9.6. d. The number 9.75 is rounded off to 9.8 Topic : Units, Dimensions and Measurement Prepared By : Mukesh Kumar Lect. in Physics, GSSS Behal (Bilaspur) HP Mob. No. 97368-78563 Email: mksrattan197@gmail.com
8 CONVERSION FACTORS (i) 1 A.U = 1.4961011m (ii) 1X-ray unit = 10-13m (iii) 1foot = 30.48 cm (iv) 1Chandra Shekhar limit (CSL) = 1.4 times the mass of sun (v) 1 metric Ton = 1000kg (vi) 1pound = 0.4537kg (vii) 1 atomic mass unit (a.m.u) = 1.67 10- 27kg (viii) 1shake = 10-8kg (ix) 1 year = 365.25d = 3.156107 s (x) 1 carat = 200mg (xi) 1 bar = 0.1 M Pa = 105Pa (xii) 1curie = 3.71010s -1 (xiii) 1 roentgen = 2.58 10-4 C/kg (xiv) 1quintal = 100kg (xv) 1barn = 10-28m 2 (xvi) 1standard atmospheric pressure = 1.013105 Pa or N/m2 (xvii) 1mm of Hg = 133N/m2 (xviii) 1horse power = 746w (xix) Gas constant, R = 8.36j/mol k = 8.3610-7erg/mol k = 2cal/mol (xx) 1 Weber = 108 maxwell (xxi) 1 tesla = 1wb/m2 = 104 gauss (xxii) 1amp turn/meter = 410-3 oersted (xxiii) 1electron volt (eV) = 1.6 10-19J (xxiv) 1calorie = 4.19J (xxv) 1watt-hour = 3.6 103 J ELECTRICAL PHYSICAL QUANTITIES (DERIVED) Physical Quantity Symbo l Dimension Formula Measurement Unit Unit Electric current I [A] Ampere C/sec emf, voltage, potential E [ M L2 T-3A -1 ] Volt Kg m2 /sec2C resistance or impedance R [ M L2 T-3A -2 ] ohm Kgm2 /secC2 Electric conductivity [ M-1L -3T 3A 2 ] mho secC2 /Kg m3 capacitance C [ M-1L -2T 4A 2 ] Farad sec2C 2 /Kgm2 inductance L [ M L2 T-2A -2 ] Henry Kg m2 /C2 Current density J [ M0 L-2 T0A] ampere per square meter C/sec m2
9 Charge density [ M0 L-3 TA] coulomb per cubic meter C/m3 Magnetic induction B [ M L0 T-2A -1 ] weber per square meter Kg/sec C magnetic intensity H [ ML-1T 0A] ampere per meter C/m sec Gravitational Constant G [ M-1 L3 T-2 ] Newton meter square per mass square Nm2 /kg2 Electric field intensity E [ M L T-3A -1 ] volt/meter or newton/coulomb Kg m/sec2 C Planks Constant h [ M L2 T-1 ] Joule second Js Coefficient of viscosity [ M L-1 T-1 ] Decapoise daP permittivity, [ M-1L -3T 4A 2 ] farad per meter sec2C 2 /Kgm3 dielectric constant K [ M0L 0T 0 ] None None frequency f or [ M0L 0T -1 ] Hertz sec-1 angular frequency [ M0L 0T -1 ] radians per second sec-1 Wave length [ M0LT0 ] Meters M Quantities Having same Dimensions Dimension Quantity [M 0L 0T –1] Frequency, angular frequency, angular velocity, velocity gradient and decay constant [M 1L 2T –2 ] Work, internal energy, potential energy, kinetic energy, torque, moment of force [M 1L –1T –2] Pressure, stress, Young’s modulus, bulk modulus, modulus of rigidity, energy density [M 1L 1T –1 ] Momentum, impulse [M 0L 1T –2 ] Acceleration due to gravity, gravitational field intensity [M 1L 1T –2 ] Thrust, force, weight, energy gradient [M 1L 2T –1] Angular momentum and Planck’s constant [M 1L 0T –2 ] Surface tension, Surface energy (energy per unit area)
10 Heat Quantity Unit Dimension Temperature (T) Kelvin [M 0L 0T 0 1 ] Heat (Q) Joule [ML2T – 2 ] Specific Heat (c) Joule/kg-K [M 0L 2T – 2 –1] Thermal capacity Joule/K [M 1L 2T – 2 –1] Latent heat (L) Joule/kg [M 0L 2T – 2 ] Gas constant (R) Joule/mol-K [M 1L 2T – 2 – 1] Boltzmann constant (k) Joule/K [M 1L 2T – 2 – 1] Coefficient of thermal conductivity (K) Joule/m-s-K [M 1L 1T – 3 – 1] Stefan's constant () Watt/m2 -K 4 [M1L 0T – 3 – 4] Wien's constant (b) Metre-K [M0L 1T 0 1] Planck's constant (h) Joule-s [M 1L 2T –1 ] Coefficient of Linear Expansion () Kelvin–1 [M 0L 0T 0 –1] Mechanical equivalent of Heat (J) Joule/Calorie [M 0L 0T 0 ] Vander wall’s constant (a) Newton-m4 [ML5T – 2 ] Vander wall’s constant (b) m 3 [M 0L 3T 0 ] Electricity Quantity Unit Dimension Electric charge (q) Coulomb [M0L 0T 1A 1 ] Electric current (I) Ampere [M0L 0T 0A 1 ] Capacitance (C) Coulomb/volt or Farad [M–1L – 2T 4A 2 ] Electric potential (V) Joule/coulomb [M1L 2T –3A –1 ] Permittivity of free space (0) 2 2 Newton - metre Coulomb [M–1L –3T 4A 2 ] Dielectric constant (K) Unitless [M0L 0T 0 ] Resistance (R) Volt/Ampere or ohm [M1L 2T – 3A – 2 ] Resistivity or Specific resistance () Ohm-metre [M1L 3T – 3A – 2]
11 Quantity Unit Dimension Coefficient of Self-induction (L) ampere volt second or henry or ohmsecond [M1L 2T – 2A – 2] Magnetic flux () Volt-second or weber [M1L 2T –2A –1] Magnetic induction (B) ampere metre newton 2 ampere metre Joule 2 second metre volt or Tesla [M1L 0T – 2A – 1] Magnetic Intensity (H) Ampere/metre [M0L – 1T 0A 1 ] Magnetic Dipole Moment (M) Ampere-metre2 [M0L 2T 0A 1 ] Permeability of Free Space (0) 2 ampere Newton or ampere metre Joule 2 or ampere metre Volt second or metre Ohm second or metre henry [M1L 1T –2A –2] Surface charge density () 2 Coulomb metre [M0L –2T 1A 1 ] Electric dipole moment (p) Coulomb metre [M0L 1T 1A 1 ] Conductance (G) (1/R) 1 ohm [M–1L –2T 3A 2 ] Conductivity () (1/) 1 1 ohm metre [M–1L –3T 3A 2 ] Current density (J) Ampere/m2 M 0L –2T 0A 1 Intensity of electric field (E) Volt/metre, Newton/coulomb M 1L 1T –3A –1 Rydberg constant (R) m –1 M 0L –1T 0
12 Uses of Dimensional equation A. To change the value of a physical quantity from one system to another : Example : Convert 1 newton into dyne. Force = mass x acceleration In this way the conversion factor for any derived physical quantity can be calculated if the dimensional formula of the derived quantity is known. Principle of Homogeneity of Dimensional Equation According to this principle, the dimensions of fundamental quantities on left hand side of an equation must be equal to the dimensions of the fundamental quantities on the right hand side of that equation. Let us consider three quantities A, B and C such that C = A + B. Therefore, according to this principle, the dimensions of C are equal to the dimensions of A and B. For example: Dimensional equation of v = u + at is: [M0 L T-1] = [M0 L T-1] + [M0 L T-1] X [M0 L0 T] = [M0 L T-1] B. Homogenity of Dimensions in an equation : On the basis of this principle the accuracy of an equation can be checked. In a physical equation every term should have the same dimensions. Example : The given relation is v = u + at Dimensionally: As in the above equation dimensions of both sides are the same. Hence, this formula is correct dimensionally. Example : Consider the formula. As in the above equation dimensions of both sides are not same, this formula is not correct dimensionally.
13 Example : In van-der Wall’s equation = RT, obtain the dimension of the constant a and b, where P is pressure, V is volume, T is temperature and R is gas constant. Solution : The given equation is As pressure can be added only to pressure, therefore, represents pressure P. Again, from volume V, one can subtract only the volume. Therefore, ‘b’ must represent volume i.e., C. Deducing Relation Among the Physical Quantities : The relation between different physical quantities can be established with the help of dimensions. It will be possible if we know the quantities on which a particular physical quantity depends. Example : Einstein Mass-Energy Relation : When mass is converted into energy, let the energy produced depends on the mass (m) and speed of light (c) i.e., Solution: where K is a dimensionless quantity. Equating the exponents of similar quantities a = 1 and b = 2 So the required physical relation becomes, E = mc2 The value of dimensionless constant is found unity through experiment. E = mc2 Example : To find the expression for the time period of a simple pendulum which depends on the its length, mass of the bob and acceleration due to gravity. Solution : We know where K is a dimensionless constant and a, b and c are exponents which we have to evaluate. Taking the dimensions of both sides.
14 Putting these values in the above given equation : Example : Assuming that the critical velocity of flow of a liquid through a narrow tube depends on the radius of the tube, density of the liquid and viscosity of the liquid, find an expression for critical velocity. Solution : Suppose, where r = radius of the tube, ρ = density of the liquid, η = co-efficient of viscosity of the liquid and a, b and c are the unknown powers to be determined. Equating the powers of M, L and T, we have Limitations of Theory of Dimensions Although dimensional analysis is very useful but it is not universal, it has some limitatios as given below. a. The dimensional formula for many physical quantities is the same. For example the dimensional formula for work, energy and torque is b. Numerical constant (k), having no dimensions such 3/4, e, 2π etc, can not be deduced by the method of dimensions. c. The method of dimensions can not be used to derive relations other than produces of power functions. For example : d. The method of dimensions can not be applied to derive formula if in mechanics a physical quantity depends on more than three physical quantities.Example : can not be derived by theory of dimensions but its dimensional correctness can be checked. e. Even if a physical quantity depends on 3 physical quantities, out of which two have same dimensions, the formula can not be derived by theory of dimensions. Errors in Measurement The errors in the measurement of a physical quantity is the difference between the true value and the measurement value of the physical quantity.
15 i.e Error = True value - Measured value. Suppose, is the error in measurement of x then Fractional error and percentage error Example : Suppose a quantity x depends on three quantities A, B and C, as : If ΔA, ΔB, ΔC and ΔX are the errors in A, B, C and x respectively, then This given fractional error in x. The maximum fractional error is obtained by taking all signs as positive. ∴ Maximum percentage error in x is : Accuracy and Precision Accuracy and Inaccuracy The accuracy of a measurement is how close a result comes to the true value. Precision and Imprecision Precision refers to how well measurements agree with each other in multiple tests. Graphical representation Accurate and Precise Accurate but not Precise Precise but not Accurate Not Precise or Accurate
16 TYPES OF ERROR 1. Systematic Error (determinate error) The error is reproducible and can be discovered and corrected. 2. Random Error (indeterminate error) Caused by uncontrollable variables, which can not be defined/eliminated. a). Instrument errors - failure to calibrate, degradation of parts in the instrument, power fluctuations, variation in temperature, etc. Can be corrected by calibration or proper instrumentation maintenance. b). Method errors - errors due to no ideal physical or chemical behavior - completeness and speed of reaction, interfering side reactions, sampling problems Can be corrected with proper method development. c). Personal errors - occur where measurements require judgment, result from prejudice, color acuity problems. Can be minimized or eliminated with proper training and experience. Systematic (determinate) errors OBJECTIVE 1. Which of the following sets cannot enter into the list of fundamental quantities in any system of units? (A) length, time and mass (B) mass, time and velocity (C) length, time and velocity (D) gravitational constant Solution:(C) Since velocity is derivable from length and time therefore it cannot be grouped with length and time as fundamental quantity. 2. Sleman is S.I unit for (A) Specific-Conductance (B) Inductance (C) Capacitance (D) Pressure Solution 2: (A) 3. A science student takes 100 observations in an experiment. Second time he takes 500 observations in the same experiment. By doing so the possible error becomes (A) 5 times (B) 1/5 times (C) Unchanged (D) None of these Solution 3: (B) (1/5 times) 4. The unit of surface energy per unit area may be expressed (A) Nm–2 (B) Nm–1 (C) Nm (D) Nm2 Solution 4: (B) Surface energy per unit Surface energy per unit area = Energy Area Force displacement Area 1 2 N m Nm m
17 5. Density of a liquid is 13.6 gcm–3. Its value in SI units is (A) 136.0kgm–3 (B) 13600kgm–3 (C) 13.60kgm–3 (D) 1.360kgm–3 Solution 5: (B) Density = 13.6 g cm–3 = 3 2 3 13.6 10 kg (10 m) = 13600 kg m–3 [ 1 g = 10–3 kg, 1 cm = 10–2 m] 6. If the size of a unit be represented by k and is numerical value as n, then (A) n k (B) n 1 k (C) n k2 (D) n 1 k Solution:(B) Value = nk. Since value is fixed therefore nk = constant. 7. The SI unit of the universal gas constant R is (A) Erg K–1 MOL–1 (B) Watt K-1 MOL-1 (C) Newton-1 MOL-1 (D) Jule–1 MOL–1 Solution 7: (B) Joule (mol k) 8. The maximum error in the measurement of mass and density of the cube are 3% and 9% respectively. The maximum error in the measurement of length will be (A) 9% (B) 3% (C) 4% (D) 2% Solution 8: (C) Density = Mass(M) Volume(V) = M V V = M 3 = M Max. fractional error 3 M M Percentage error 3 % = 3% + 9% 12 % % 3 % = 4 % 9. The SI unit of electrochemical equivalent is (A) kg C (B) C kg-1 (C) kg C-1 (D) kg2C -1 Solution:(C) According to Faraday's first law of electrolysis, m = ZQ or m Z Q . So, SI unit of Z is kg C-1 10. Which of the following has a dimensional constant (A) Refractive index (B) Passion’s ratio (C) Relative velocity (D) Gravitation of constant Solution 10: (D) All of physical quantity has no dimension except gravitational force so correct.
18 11. The dimensions of surface tension length are (A) ML0T -2 (B) MLT-2 (C) ML-1T –2 (D) ML–2T –2 Solution 11: (B) ML0T –2 L= MLT-2 12 Pick the odd man out (A) Weight (B) Thrust (C) Electromotive force (D) Force Solution 12: (C) 13 Dimension formula for luminous flux is (A) ML2T -2 (B) ML2T -3 (C) ML2T –1 (D) None of these Solution 13: (D) 14. If w, x, y and z are mass, length, time and current respectively, then 2 3 x w y z has dimensional formula same as (A) electric potential (B) capacitance (C) electric field (D) permittivity Sol. : (A) 2 2 3 3 x w ML y z T A 3 2 2 3 x w ML T [Work] y z AT [Charge] 3 3 y w [Work] [Potential] y z [Charge] 15. MLT-1 T-1 are the dimension of (A) Power (B) Momentum (C) Force (D) Couple Solution 15: (C) 16. The unit of impulse is the same as that of (A) Moment of force (B) Linear momentum (C) Rate of change of linear momentum (D) Force Solution 16: (B) Impulse = Force time = MLT–2 T = MLT–1 i.e. Dimension of linear momentum 17. The dimensions of capacitance are (A) M-1L -2TI2 (B) M-1L -2T 2 I -2 (C) ML-2T -2I -1 (D) M-1L -2T 4 I 2 Solution 17: (D) C = Q V C = 2 2 2 2 2 2 Q Q T T I W / Q W T W = 2 2 2 2 T I ML T = M–1L –2T 4 I 2 18. The dimension of angular momentum length are (A) MLT-1 (B) ML3T -1 (C) ML-1T (D) ML0T -2 Solution 18: (B) ML3T –1 19. The SI unit of the universal gas constant R is (A) erg K-1mol-1 (B) watt K-1mol-1 (C) newton K-1mol-1(D) joule K-1mol-1 Solution 19: (D)
19 20. The dimension of plank’s are the same as those of (A) energy (B) power (C) angular frequency (D) angular momentum Solution 20: (D) E = h Plank’s constant h = E Dimension of (h) = 2 2 ML T 1/ T h = ML2T –1 Dimension of angular momentum = ML2T –1 21. The volume V of water passing any point of a uniform tube during t seconds is related to the cross-sectional area A of the tube and velocity u of water by the relation V Au t Which one of the following will be true? (A) = = (B) = (C) = (D) Solution 21: (B) V = k. u t L3 = k (L2 ) . (LT–1) . (T) L3 = k . L(2 + ) T– + 2 + = 3 – + = 0 = , 2 + = 3 so are can conclude that = 22. Which one of the following relations is dimensionally consistent where h is height to which a liquid of density rises in a capillary tube of radius, r, T is the surface ension of the liquid, the angle of contact and g the acceleration due to gravity (A) 2 cos T h r g (B) 2 cos Tr h g (C) 2 cos 2 g h Tr (D) 2 cos Tr g h Solution 22: (A) 23 The dimension of calories are (A) ML2T -2 (B) MLT-2 (C) ML2T –1 (D) ML2T -1 Solution 23: (A) Calories is unit of energy so dimension of calories is = ML2T –2 24. The dimension of potential difference length are (A) ML3T -3I -1 (B) MLT-2I -1 (C) ML2T -2I (D) MLT-2I Solution 24: (A) V = 2 2 work ML T length L charge Q T T = ML3T –3I –1 25. What is the power of a 100 W bulb in cgs units? (A) 106 erg/s (B) 107 erg/s (C) 109 erg/s (D) 1011 erg/s Solution:(C) 100 W = 100 J s-1= 100 107 ergs-1 .
20 26. A quantity X is given by 0l v t , where 0 is the permittivity of free space, l is the length, v is a potential difference and t is a interval. The dimensional formula for X is the same as that of (A) Resistance (B) Charge (C) Voltage (D) Current Solution: (D) x = 0 v l t = 2 2 C N M M C M N sec = C Thatisunitof current sec 27. Let (0) denote the dimensional formula for the permittivity of the vacuum, and (0) that of the permeability of the vacuum. If M = mass, L = length, T = time and A = electric current (A) [0] =[M–1L –3T 2A] (B) [0] = [M–L –3T 4A 2 ] (C) [0] = [MLT–2A –2] (D) [0] = [M– 1L –3T –2A] Solution: (B) 28. The dimensions of 2 2 ( ) 0 e hc are (A) (A2L 3T 4M –4) (B) (A–2T –4L 3M) (C) (A0M 0L 0T 0 ) (D) (AT2L –3M –1) Solution: (C) 2 2 2 1 C C (N m ) Joule sec meter sec 29. Density of liquid is 15.7 g cm-3. Its value in the International System of Units is (A) 15.7 kg m-3 (B) 157 kg m-3 (C) 1570 kg m-3 (D) 15700 kg m-3 Solution:(D) 15.7 g cm-3 = 15.7 10-3kg(10- 2m)-3= 15700 kg m-3 30. On the basis of dimensional equation, the maximum number of unknown that can be found is (A) One (B) Two (C) Three (D) Four Solution: (C) 31. If v stands for velocity of sound, E is elasticity and d the density, then find x in the equation v = x d E (A) 1 (B) ½ (C) 2 (D) –½ Solution: (D) V = x d E LT–1 = x 3 x 2 2 ML MLT /L LT–1 = L–3x + x T2x –1 = 2x x = 1 2 32. The dimension of ½0E 2 (0 is permittivity of free space and E is electric field) are (A) MLT-1 (B) ML2T -2 (C) ML-1T -2 (D) ML2T -1 Solution:(B) Dimension of energy = ML2T –2
21 33. A weber is equivalent to (A) A m-2 (B) A m-1 (C) A m2 (D) T m-2 Solution:(D) 1 T = 1 Wb m-2 34. With the usual notation, the equation 2 tan rg v said to give the angle of banking is (A) Numerical correct only (B) Dimensionally correct only (C) Both numerical & dimensionally correct (D) Neither numerical nor dimensionally correct Solution: (C) 35. When light travels through glass, the refractive index is found to vary with the wavelength as = A + B/ 2 , what is dimension of B ? (A) L (B) L2 (C) L-1 (D) L-2 Solution:(B) Dimension of wavelength = L2 Dimension of refractive index = M0L 0T 0 Dimension of B is L2 36. The dimension of ½0E 2 (0 is permittivity of free space and E is electric field) are (A) MLT-1 (B) ML2T -2 (C) ML-1T -2 (D) ML2T -1 Solution: (B) 37. A travelling wave in a stretched string is described by the equation y = A sin (kx-t) The dimension of k is (A) M0L -1T 0 (B) M0L 0T 0 (C) M0L 2T 0 (D) MLT-1 Solution: (A) k = 2 Dimension of k = L–1 38. Dimension formula of Stefan’s constant (A) ML2T -2 -4 (B) ML2T -3 -4 (C) ML0T -3 -4 (D) M0LT-1 Solution: (C) 39. Of the following quantities, which one has dimensions different from the remaining three (A) Energy per unit volume (B) Force per unit area (C) Product of voltage and charge per unit volume (D) Angular momentum Solution: (C) 40. The dimension equation for magnetic flux is (A) ML2T -2I -1 (b) ML2T -2I -2 (C) ML-2T -2I -1 (d) ML-2T -2I -2 Solution: (A) = B. A = F IL . A = 2 2 MLT L I L = ML2T –2I –1
22 41 The dimension of the Rydberg constant are (A) M0L -1T (B) MLT1 (C) M0L –1T 0 (D) ML0T 2 Solution: (C) M0L –1T 0 42. The pairs of physical quantities which have same dimension are (A) Reynolds number and coefficient of friction (B) Latent heat and gravitational potential (C) Curie and frequency of light wave (D) Plank’s constant and torque Solution: (B) 43. In the relation x = 3yz2, x and z represent the dimensions of capacitance and magnetic induction respectively. What will be the dimension of y (a) M-3L -2T 4Q 4 (B) M2L -2T 4Q 4 (c) M-2L -2T 4Q 4 (D) M-3L -2T 4Q Solution: (A) x = 3yz2 dimension of y = 2 x z = 2 Capicitance (Magnetic induction) = 1 2 2 2 1 1 2 M L T Q (MT Q ) = M-3 L -2 T +4 Q 4 44. A sextant is used to measure (A) area of hill (B) height of an object (C) breadth of a tower (D) volume of the building. Solution:(B) The height of a tree, building, tower, hill etc. can be determined with the help of a sextant. 45. What is the dimensional formula of coefficient of linear expansion? (A) [ML2T -2K -1] (B) [MLT-2K -1] (C) [M0L 0TK-1] (D) [M0L 0T 0K -1] Solution:(D) lt = l0(1 + t) or t 0 0 l l l t 46. A pressure of 106 dyne cm-2 is equivalent to (A) 105 N m-2 (B) 104 N m-2 (C) 106 N m-2 (D) 107 N m-2 Solution:(A) Remember the conversion factor of 10. 47. The Vander Waal’s equation for a gas is (P+a/v2 )(V-b) = nRT. The ratio b/a will have the following dimensional formula (A) M-1L -2T 2 (B) M-1L -1T -1 (C) ML2T 2 (D) MLT-2 Solution: (A) Dimension of (b) = L3 Dimension of (a) = ML5 T-2 Dimension of b a = 3 1 2 2 5 2 L M L T ML T 48. If the time period of a drop of liquid of density d, radius r, vibrating under surface tension s is given by the formula 1/ 2 ( ) a b c t d r s and if a = 1, c = -1, then b is (A) 1 (B) 2 (C) 3 (D) 4
23 Solution: (C) T = (M L-3) a/2 Lb/2 (ML0T -2) c/2 M0L 0 T = Ma/2 + c/2 L-3a/2+b/2 T-c 3a b 0 2 2 - 3 a + b =0 b = 3 a b = 3 1 b = 3 49. If P represents radiation pressure, C represents speed of light and Q represents radiation energy striking a unit area per second, then the non-zero integers x, y and z, such that P xQ yC z is dimensionless are (A) x=1, y=1, z=1 (B) x=1, y=-1, z=1 (C) x=-1,y=1,z=1 (D) x=1, y=1, z=-1 Solution: (B) M0L 0T 0 = Px Qy C z = (M L-1 T-2) x (ML2T -2) y (LT-1) z x + y = 0 x = -y -x + 2y +z = 0 -2x – 2y –z = 0 x = -y 50. In the relation Z P e kE ,P is pressure Z is distance k is Boltzman constant and is the temperature. The dimension formula of will be (A) M0L 2T 0 (B) M1L 2T -1 (C) ML0T -1(D) M0L 2T 1 Solution: (A) z k is dimension less quantity dimension of = k z = 2 2 1 ML T K K L = MLT–2 Dimension of is equal to dimension of pressure P P = ML–1T –2 = 2 MLT = 2 1 2 MLT ML T = M0L 2T 0 51. Velocity v, acceleration a and force f are taken as fundamental quantities, then angular momentum will have the dimension (A) fv2 a -2 (B) f2 v 2 a -2 (C) fv3 a -2 (D) None of these Solution: (D) Angular momentum (L) v x a y fz MLT–2 = x y z 1 2 2 LT LT MLT MLT–2 = Mz Lx +y + z T–x –2y –2z 1 = z 1 = x + y + z –2 = – x – 2y –2z z = 1 x + y = 0 x = –y –2 = –x + 2x –2 0 = x, y = 0 Angular momentum(L) = f
24 52. Fund the unit of acceleration time? (A) ms–1 (B) ms–3 (C) ms+1 (D) ms+2 Solution: (A) Acceleration = 1 2 m Velocity V m s C time t s ms time S s s 53. What is the unit of current Resistance. (A) amps (B) volt (C) coulomb (D) farad Solution: (B) 54. What will be equivalent energy of 5eV in joule? (A) 8.0 × 10–22J (B) 8.0 × 10–19J (C) 8.0 × 10–25J (D) 8.0 × 10–26J Solution: (B) 55. One joule is the equivalent of ? (A) 1 calorie 5.19 (B) 1 calorie 4.19 (C) 1 calorie 6.19 (D) 1 calorie 8.19 Solution: (B) 1 calorie 4.19 56. Least count of screw gauge depend on ? (A) Main scale division (B) circular scale (C) no. of circular scale division (D) Main scale division & no. of circular scale division Solution: (D) 57. Least count of vernier calipers depend on? (A) Main scale division (B) vernier scale (C) no. of vernier scale division (D) Main scale division & no. of vernier scale division Solution: (D) 58. Least count of spherometer depend on ? (A) Main scale division (B) circular scale (C) no. of circular scale division (D) Main scale division & no. of circular scale division Solution: (D) 59. What is the dimension of angular frequency time? (A) Dimension less (B) sec–2 (C) sec–3 (D) sec+1 Solution: (A) 60. What is the dimension of wave length Frequency? (A) M (B) LT–1 (C) T (D) MT–1 Solution: (B) Some Tips 1. The dimensions of many physical quantities, especially those in heat, thermodynamics, electricity and magnetism in terms of mass, length and time alone become irrational. Therefore, SI is adopted which uses 7 basic units.
25 2. The dimensions of a physical quantity are the powers to which basic units (not fundamental units alone) should be raised to represent the derived unit of that physical quantity. 3. The dimensional formula is very helpful in writing the unit of a physical quantity in terms of the basic units. 4. The dimensions of a physical quantity do not depend on the system of units. 5. A physical quantity that does not have any unit must be dimensionless. 6. The pure numbers are dimensionless. 7. Generally, the symbols of those basic units, whose dimension (power) in the dimensional formula is zero, are omitted from the dimensional formula. 8. It is wrong to say that the dimensions of force are MLT–2. On the other hand we should say that the dimensional formula for force is MLT–2 and that the dimensions of force are 1 in mass, 1 in length and –2 in time. 9. Physical quantities defined as the ratio of two similar quantities are dimensionless. 10. The physical relation involving logarithm, exponential, trigonometric ratios, numerical factors etc. cannot be derived by the method of dimensional analysis. 11. Physical relations involving addition or subtraction sign cannot be derived by the method of dimensional analysis. 12. If units or dimensions of two physical quantities are same, these need not represent the same physical characteristics. For example torque and work have the same units and dimensions but their physical characteristics are different. 13. The standard units must not change with space and time. That is why atomic standard of length and time have been defined. Attempts are being made to define the atomic standard for mass as well. 14. The unit of time, the second, was initially defined in terms of the rotation of the earth around the sun as well as that about its own axis. This time standard is subjected to variation with time. Therefore, the atomic standard of time has been defined. 15. Any repetitive phenomenon, such as an oscillating pendulum, spinning of earth about its axis, etc can be used to measure time. 16. The product of numerical value of the physical quantity (n) and its unit (U) remains constant. That is : nU = constant or n1U1 = n2U2. 17. The product of numerical value (n) and unit (U) of a physical quantity is called magnitude of the physical quantity. Thus : Magnitude = nU 18. Poiseuille (unit of viscosity) = pascal (unit of pressure) × second. That is : Pl : Pa- s.
26 19. The unit of power of lens (dioptre) gives the ability of the lens to converge or diverge the rays refracted through it. 20. The order of magnitude of a quantity means its value (in suitable power of 10) nearest to the actual value of the quantity. 21. Angle is exceptional physical quantity, which though is a ratio of two similar physical quantities (angle = arc / radius) but still requires a unit (degrees or radians) to specify it along with its numerical value. 22. Solid angle subtended at a point inside the closed surface is 4 steradian. 23. A measurement of a physical quantity is said to be accurate if the systematic error in its measurement is relatively very low. On the other hand, the measurement of a physical quantity is said to be precise if the random error is small. 24. A measurement is most accurate if its observed value is very close to the true value. 25. Errors are always additive in nature. 26. For greater accuracy, the quantity with higher power should have least error. 27. The absolute error in each measurement is equal to the least count of the measuring instrument. 28. Percentage error = relative error × 100. 29. The unit and dimensions of the absolute error are same as that of quantity itself. 30. Absolute error is not dimensionless quantity. 31. Relative error is dimensionless quantity. 32. Least Count = Number of parts on vernier scale (n) value of 1 part on main scale (s) 33. Least count of vernier callipers = vernie r scale ( ) value of 1 part of main scale ( ) value of 1 part of s v Least count of vernier calliper = 1 MSD – 1 VSD where MSD = Main Scale Division VSD = Vernier Scale Division 34. Least count of screw guaze = No. of parts on circular scale ( ) Pitch( ) n p 35. Smaller the least count, higher is the accuracy of measurement. 36. Larger the number of significant figures after the decimal in a measurement, higher is the accuracy of measurement. 37. Significant figures do not change if we measure a physical quantity in different units. 38. Significant figures retained after mathematical operation (like addition, subtraction, multiplication and division) should be equal to the minimum significant figures involved in any physical quantity in the given operation. 39. Significant figures are the number of digits
27 upto which we are sure about their accuracy. 40. If a number is without a decimal and ends in one or more zeros, then all the zeros at the end of the number may not be significant. To make the number of significant figures clear, it is suggested that the number may be written in exponential form. For example 20300 may be expressed as 203.00×102 , to suggest that all the zeros at the end of 20300 are significant. 41. 1 inch = 2.54 cm 1 foot = 12 inches = 30.48 cm = 0.3048 m 1 mile = 5280 ft = 1.609 km 42. 1 yard = 0.9144 m Topic : Units, Dimensions and Measurement Prepared By : Mukesh Kumar Lect. in Physics, GSSS Behal (Bilaspur) HP Mob. No. 97368-78563 Email: mksrattan197@gmail.com