Units & Measurements

Units & Measurements Physical Quantity (II) M.K.S, In this system length is measured in meter, mass in kilogram and time in second. A quantity which can be measured is called physical quantity e.g length mass, time force. In order to (III) F.P.S. In this system length is measured in foot, mass measure a physical quantity a standard is required. in pound and time in second. This standard is called unit of that physical (IV) S.I. System, It is a International System of units. In quantity. this system seven fundamental quantities are taken, A physical quantity is represented completely by its which are given below : magnitude and unit Fundamental Quantity Unit Symbol Physical quantity = magnitude x unit 1. Length metre m 1 2. Mass Kilogram kg Q = nu n  u 3. Time Second S means larger the unit, smaller will be the magnitude. 4. Electrier Current ambere A Fundamental and Derived Quantities 5. Temperature Kelvin K The physical quantities which are independent of other 6. Amount of Substance mole mol quanties are called Fundamental Quantities. 7. Luminous inensity candela Cd. The quantities which are expressed in terms of fundamental quantities are called Derived Dimension Quantities. The powers to which fundamental quantities must be raised in order to express the given physical Fundamental and Derived unit quantity are called its dimension. Units of fundamental physical quantities are called e.g. force = mass x acceleration. fundamental units and units of derived physical quantities are called derived unit. velocity = mass × time System of Units mass  length/ time (i) CGS system - In this system length is measured in = time centimeter, mass is measured in gram and time is = mass × length × (time)–2 measured in second. Units & Measurements Thus dimension of force are 1 in mass, 1 in length, and -2 in time. For convenience mass, length and time are denoted by M,L and T. The electric current, the temperature, the amount of substance and Luiminous intensity are denoted by symbols I, K, mol and candela respectively. The physical quantity which is expressed in terms of base quantities are enclosed in square brackets. Thus equation for force in written as [Force] = MLT–2 Dimensional Formula The dimensional formula of a physical quantity is the symbolic representation of its dimensions. Thus the dimensional formula of force is MLT–2. Dimensional Equation The dimensional equation of a physical quantity is written as [Quantity] = Dimensional formula. Thus [Force] = [MLT–2] or [F] = [MLT–2] Important Dimensions in Physics S. No. Physical quantity Symbol (a) Mechanics Dimensional formulae 1. Speed c,u,v,w [M0LT–1] ms 1 2. Velocity c,u,v,w [M0LT–1] ms 1 3. Acceleration a [M0LT–2] ms 2 4. Velocity gradient [M0L0T–1] s–1 5. Area S.A. a [M0L2T0] m2 6. Angle  [M0L0T0] radian (rad) 7. Solid angle  [M0L0T0] steradian (sr) 8. Angular velocity  [M0L0T–1] rad s-1 9. Angular frequency  [M0L0T–1] s-1 10. Angular acceleration  [M0L0T–2] rad s-2 11. Momentum P [MLT–1] kg m s-1 12. Angular momentum L [ML2T–1] kg m2 s–1 13. Force F [MLT–2] newton (N) 14. Torque, couple  [ML2T–2] Nm or joule (J) 15. Impulse I [MLT–1] Ns or kg ms-1 16. Frequency f,  [M0L0T–1] hertz (Hz) 17. Time period T [M0L0T1] S 18. Volume V [M0L3T0] m3 19. Density d [ML–3T0] kg m-3 20. Pressure p, P [ML–1T–2] pascal (pa) or [Nm–2] 21. Radius of gyration K [M0LT0] m 22. Moment of intrtia I [ML2T0] kg m2 23. Strain  [M0L0T0]  24. Stress  [ML–1T–2] pascal (pa) or [Nm–2] dv dr Commonly used unit Units & Measurements 25. Young’s modulus Y [ML–1T–2] Pa or Nm-2 26. Bulk modulus B [M0L–1T–2] Pa or Nm-2 27. Modulus of rigidity  [ML–1T–2] Pa or Nm-2 28. Poisson’s ratio  [M0L0T0] - 29. Surface Tension T,  [ML0T–2] Nm–1 or Jm-2 30. Planck’s constant h [ML2T–1] Js (b) Electricity Quantity Unit Dimension Electric Charge (q) Coulomb [M0L0T1A1] Electric current (I) Ampere [M0L0T0A1] Electric potential (V) Joule/ coulomb [M1L2T–3A–1] Capacitance (C) Coulomb/ volt or Farad [M–1L–2T4A2] Permittivity of free space (0) Coulomb2 or ferad per meter Newtonmetre2 [M–1L–3T4A2] Dielectric constant (K) Unitless [M0L0T0] Resistance (R) Volt/Ampere or ohm [M1L3T–3A–2] Resisvity or Specific Ohm-metre [M1L3T–3A–2] volt  sec ond or henry or ohm-second ampere [M1L2T–2A–2] Magnetic flux () Volt-second or weber [M1L2T–2A–1] Magnetic induction (B) Joule newton volt  sec ond orTesla [M1L0T–2A–1] ampere  metre ampere  metre 2 metre 2 resistance (p) Coefficient of Selfinduction (c) Thermodynamics Quantity Unit Dimension Temperature (T) Kelvin [M0L0T01] Heat (Q) Joule [ML2T–2] Specific Heat (c) Joule/kg-K [M0L2T–2–1] Thermal capacity Joule/K [M1L2T–2–1] Latent heat (L) Joule/Kg [M0L2T–2] Units & Measurements Gas constant (R) Joule/mol-K [M1L2T–2–1] Boltzmann constant (k) Joule/K [M1L2T–2–1] conductivity (K) Joule/m-s-K [M1L1T–3–1] Stefan’s constant () Watt/m2-K4 [M1L0T–3–4] Wien’s constant (b) Metre-Kelvin [M0L1T01] Planck’s constant (h) Joule-s [M1L2T–1] Coefficient of Linear Kelvin-1 [M0L0T0–1] Mechanical equivalent of Heat (J) Joule/Calorie [M0L0T0] Vander wall’s constant (a) Newton-m4 [ML5T–2] Vander wall’s constant (b) m3 [M0L3T0] Coefficient of thermal Expansion () Quantities having Same Dimensions Dimension 0 0 –1 Quantiy [M L T ] Frequency, angular frequency, angular velocity, velocity gradient and decay constant [M1L2T–2] Work, internal energy, potential energy, kinetic energy, torque, moment of force [M1L–1T–2] Pressure, stress, Young’s modulus, bulk modulus, modulus of rigidity, energy density 1 1 –1 [M L T ] Momentum, impulse [M0L1T–2] Acceleration due to gravity, gravitational field intensity 1 1 –2 [M L T ] Thrust, force, weight, energy gradient [M1L2T–1] Angular momentum and Planck’s constant [M1L0T–2] Surface tension, Surface energy (energy per unit area), spring constant 0 0 0 [M L T ] Strain, refractive index, relative density, angle, solid angle, distance gradient, relative permittivity (dielectric constant), relative permeability etc [M0L2T–2] 2 –2 –1 [ML T  ] [M0L0T1] Latent heat and gravitational potential Thermal capacity gas constant, Boltzmann constant and entropy l / g, m / k, R / g , where I= length g = acceleration due to gravity, m = mass, k = spring constant, R= Radius of earth [M0L0T1] L/R, LC, RC where L= inductance, R= resistance, C= capacitance [ML2T–2] I 2 Rt, q2 v2 2 t, Vlt, qV, LI2, , CV where I = C R current, t= time, q= charge, L= inductance, C= capacitance, R=Resistance Units & Measurements Application of Dimensional Analysis (iii) Deducing relations among the Physical Quantities If one knows the dependency of a (i) Convert a physical quantity from one system physical quantity on other quantities and if the to other The measure of a physical quantity is dependency is of the product type, then using the nu = constant If a physicall quantity X has dimensional method of dimensional analysis, relation between the formula MaLb T c and if (derived) units of that quantities can be derived. physical quantity in two systems are M1aLb1T1c and Ma2Lb2 T2c respectively and n1 and n2 be the numerical Limitation of Dimensional Analysis : value in the two systems respectively, then N1 u1  n 2 u 2  n1 M1aLb1T1c  n2 Ma2Lb2 T2c a b M  L   T   n2  n1  1   1   1   M2   L 2   T2  c where M1 L1 and T1 are fundamental units of mass, length and time in the first (known) system and M2, L2 and T2 are fundamental units of mass, length and time in the second (unknown) system, Thus knowing the values of fundamental units in two systems and numerical value in one system, the numerical value in other system may be evaluted. (ii) To check the dimensional correctness of a given physical relation This is based on the ‘principle of homogeneity’. According to this principle the dimensions of each term on both sides of an equation must be the same. If X  A  BC 2  DEF , Then according to principle of homogeneity X  A  BC 2  DEF If the dimensions of each term on both sides are same, the equation is dimensionally correct, otherwise not. A dimensionally correct equation may or may not be physically correct. The dimensional method is a simple and a very convenient way of finding the dependence of physical quantity on other quantities but it has its own limitations. Some of which are listed as follows: (i) In more complicated situations, it is often not easy to find out the factors on which a physical quantity will depend, In such cases, one has to make a guess which may or may not work. (ii) This method gives no information about the dimensionless constant which has to be determined either by experiment or by a complete mathematical derivation. (iii) This method is used only if a physical quantity varies as the product of other physical quantities. It fails if a physical quantity depends on the sum or difference of two quantities. (iv) This method will not work if a quantity depends on another quantity as sin or cos of an angle, i.e. if the dependence is on a trigonometric function. Similar is the case with logarithmic and exponential function. The method works only if the dependence is on proper functions only. (v) This method does not give a complete information in cases where a physical quantity depends on more than three quantities, because by equating the powers of M,L and T, we can obtain only three equations for the exponents.