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UNITS AND MEASUREMENT 7 mass, one dimension in length, and –2 otherwise rounding errors can build up. For example, the reciprocal of 9.58, calculated (after dimensions in time. The dimensions in all other rounding off) to the same number of significant base quantities are zero. Note that in this type of representation, the figures (three) is 0.104, but the reciprocal of magnitudes are not considered. It is the quality 0.104 calculated to three significant figures is 9.62. However,ifwehadwritten1/9.58=0.1044 of the type of the physical quantity that enters. Thus, a change in velocity, initial velocity, and then taken the reciprocal to three significant average velocity, final velocity, and speed are figures, we would have retrieved the original all equivalent in this context. Since all these value of 9.58. quantities can be expressed as lengt–1time, This example justifies the idea to retain one their dimensions are [L]/[T] or [L T ]. more extra digit (than the number of digits in the least precise measurement) in intermediate 1.5 DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS steps of the complex multi-step calculations in order to avoid additional errors in the process The expression which shows how and which of of rounding off the numbers. the base quantities represent the dimensions of a physical quantity is called the dimensional 1.4 DIMENSIONS OF PHYSICAL QUANTITIES formula of the given physical quantity. For example, the dimensional formula of the volume The nature of a physical quantity is described is [M° L T°], and that of speed or velocity is by its dimensions. All the physical quantities -1 –2 represented by derived units can be expressed [M° L T ]. Similarly, [M° L T ] is –3e dimensional formula of acceleration and [M L T°] that of in terms of some combination of seven mass density. fundamental or base quantities. We shall call An equation obtained by equating a physical these base quantities as the seven dimensions quantity with its dimensional formula is called of the physical world, which are denoted with the dimensional equation of the physical square brackets [ ]. Thus, length has the quantity. Thus, the dimensional equations are dimension [L], mass [M], time [T], electric current the equations, which represent the dimensions [A], thermodynamic temperature [K], luminous of a physical quantity in terms of the base quantities. For example, the dimensional intensity [cd], and amount of substance [mol]. The dimensions of a physical quantity are the equations of volume [V], speed [v], force [F] and powers (or exponents) to which the base mass density [ ρ] may be expressed as quantities are raised to represent that [V] = [M L T ] quantity. Note that using the square brackets [v] = [M L T ] [ ] round a quantity means that we are dealing –2 with ‘the dimensions of’ the quantity. [F] = [M L–3 0 ρ ] = [M L T ] In mechanics, all the physical quantities can be written in terms of the dimensions [L], [M] The dimensional equation can be obtained and [T]. For example, the volume occupied by from the equation representing the relations an object is expressed as the product of length, between the physical quantities. The dimensional formulae of a large number and breadth and height, or three lengths. Hence the wide variety of physical quantities, derived from dimensions of volume are [L] × [L] × [L] = [L] = [L ]. As the volume is independent of mass and time, the equations representing the relationships it is said to possess zero dimension in mass [M°], among other physical quantities and expressed in terms of base quantities are given in zero dimension in time [T°] and three Appendix 9 for your guidance and ready dimensions in length. reference. Similarly, force, as the product of mass and 1.6 DIMENSIONAL ANALYSIS AND ITS acceleration, can be expressed as Force = mass × acceleration APPLICATIONS = mass × (length)/(time) 2 The recognition of concepts of dimensions, which The dimensions of force are [M] [L]/[T] = 2 guide the description of physical behaviour is [M L T ]. Thus, the force has one dimension in of basic importance as only those physical Reprint 2026-27