Revision notes on unit and dimensions
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This document discusses units and dimensions in physics. It defines units as standards for measuring physical quantities so that measurements can be compared. Several systems of units are described, including the CGS, FPS, and MKS systems. The document also defines dimensional analysis, which determines the fundamental units of mass, length, and time that make up a physical quantity. Examples of dimensions, units, and derived mechanical and electrical physical quantities are provided. Principles for dimensional homogeneity and types of measurement errors are also summarized.
Revision notes on unit and dimensions
- 1. Revision Notes on Unit and Dimensions In order to make the measurement of a physical quantity we have, first of all, to evolve a standard for that measurement so that different measurements of same physical quantity can be expressed relative to each other. That standard is called a unit of that physical quantity. System of Units:- (a) C.G.S (Centimeter-Grand-Second) system. (b) F.P.S. (Foot-Pound-Second) system. (c) M.K.S. (Meter-Kilogram--Second) system. (d) M.K.S.A. (Meter-Kilogram-Second-Ampere) unit. Dimensional Formula:- Dimensional formula of a physical quantity is the formula which tells us how and which of the fundamental units have been used for the measurement of that quantity. How to write dimensions of physical quantities:- (a) Write the formula for that quantity, with the quantity on L.H.S. of the equation. (b) Convert all the quantities on R.H.S. into the fundamental quantities mass, length and time. (c) Substitute M, L and T for mass, length and time respectively.
- 2. (d) Collect terms of M,L and T and find their resultant powers (a,b,c) which give the dimensions of the quantity in mass, length and time respectively. Characteristics of Dimensions:- (a) Dimensions of a physical quantity are independent of the system of units. (b) Quantities having similar dimensions can be added to or subtracted from each other. (c) Dimensions of a physical quantity can be obtained from its units and vice-versa. (d) Two different physical quantities may have same dimensions. (e) Multiplication/division of dimensions of two physical quantities (may be same or different) results in production of dimensions of a third quantity. PHYSICAL QUANTITY SYMBOL DIMENSION 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 q none Radian None Mechanical Physical Quantities (derived) PHYSICAL QUANTITY SYMBOL DIMENSION MEASUREMENT UNIT UNIT Area A L2 square meter m2 Volume V L3 cubic meter m3
- 3. velocity v L/T meter per second m/sec angular velocity w T-1 radians per second 1/sec acceleration a LT-2 meter per square second m/sec2 angular acceleration a T-2 radians per square second 1/sec2 Force F MLT-2 Newton Kg m/sec2 Energy E ML2T-2 Joule Kg m2/sec2 Work W ML2T-2 Joule Kg m2/sec2 Heat Q ML2T-2 Joule Kg m2/sec2 Torque t ML2T-2 Newton meter Kg m2/sec2 Power P ML2T-3 watt or joule/sec Kg m2/sec3 Density D or ρ ML-3 kilogram per cubic meter Kg/m3 pressure P ML-1T-2 Newton per square meter Kg m-1/sec2 impulse J MLT-1 Newton second Kg m/sec Inertia I ML2 Kilogram square meter Kg m2 luminous flux f C lumen (4Pi candle for point source) cd sr
- 4. illumination E CL-2 lumen per square meter cd sr/m2 entropy S ML2T-2K- 1 joule per degree Kg m2/sec2K Volume rate of flow Q L3T-1 cubic meter per second m3/sec kinematic viscosity n L2T-1 square meter m2/sec per second dynamic viscosity m ML-1T-1 Newton second per square meter Kg/m sec specific weight g ML-2T-2 Newton per cubic meter Kg m-2/sec2 Electrical Physical Quantities (derived) Electric current I QT-1 Ampere C/sec emf, voltage, potential E ML2T-2Q-1 Volt Kg m2/sec2C resistance or impedance R ML2T-1Q-2 ohm Kgm2 /secC2 Electric conductivity s M-2L-2TQ2 mho secC2/Kg m3 capacitance C M-1L-2T2 Q2 Farad sec2C2/Kgm2 inductance L ML2Q-2 Henry Kg m2 /C2 Current density J QT-1L-2 ampere per square meter C/sec m2 Charge density r QL-3 coulomb per cubic meter C/m3 magnetic flux, B MT-1Q-1 weber per Kg/sec C
- 5. Magnetic induction square meter magnetic intensity H QL-1T-1 ampere per meter C/m sec magnetic vector potential A MLT-1Q-1 weber/meter Kg m/sec C Electric field intensity E MLT-2Q-1 volt/meter or newton/coulomb Kg m/sec2 C Electric displacement D QL-2 coulomb per square meter C/m2 permeability m MLQ-2 henry per meter Kg m/C2 permittivity, e T2Q2M-1L-3 farad per meter sec2C2/Kgm3 dielectric constant K M0L0T0 None None frequency f or n T-1 Hertz sec-1 angular frequency W T-1 radians per second sec-1 Wave length l L Meters M Principle of homogeneity:- It states that “ the dimensional formulae of every term on the two sides of a correct relation must be same.” Types of error:- (a) Constant errors:- An error is said to be constant error if it affects, every time, a measurement in a similar manner.
- 6. (b) Systematic errors:- Errors which come into existence by virtue of a definite rule, are called systematic errors. (c) Random error or accidental error:- Error which takes place in a random manner and cannot be associated with a systematic cause are called random or accidental errors. (d) Absolute error:- Relative Error:- Percentage Error:-