| GyanCentral - The hub for engineering and law students - IIT-JEE, AIEEE, BITSAT, CLAT, AILET - 2012: Dimensions Preparatory material Sep 30th 2012, 20:07 | GyanCentral - The hub for engineering and law students - IIT-JEE, AIEEE, BITSAT, CLAT, AILET - 2012 | | The most comprehensive career education and test preparation forums in india. | | | Dimensions Preparatory material Sep 30th 2012, 19:17 By international agreement a small number of physical quantities such as length, time etc. are chosen and assigned standards. These quantities are called 'base quantities' and their units as 'base units'. All other physical quantities are expressed in terms of these 'base quantities'. The units of these dependent quantities are called 'derived units'. The standard for a unit should have the following characteristics. (a) It should be well defined. (b) It should be invariable (should not change with time) (c) It should be convenient to use (d) It should be easily accessible The 14th general conference on weights and measures (in France) picked seven quantities as base quantities, thereby forming the International System of Units abbreviated as SI (System de International) system. Base quantities and their units The seven base quantities and their units are | Base quantity | Unit | Symbol | | Length | Metre | M | | Mass | Kilogram | Kg | | Time | Second | Sec | | Electric current | Ampere | A | | Temperature | Kelvin | K | | Luminous intensity | Candela | Cd | | Amount of substance | Mole | Mole |
| Quantity | Unit | Symbol | Express in base units | | Force | newton | N | Kg-m/sec2 | | Work | joules | J | Kg-m2/sec2 | | Power | watt | W | Kg-m2/sec3 | | Pressure | pascal | Pa | Kg m-1/S2 |
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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 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. seconds 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 10Ns. (4) Not more than one solid's is used i.e. all units of numerator written together before the '/' sign and all in the denominator written after that. i.e. It is 1 ms -2 or 1 m/s -2 not 1m/s/s. (5) Punctuation marks are not written after the unit e.g. 1 litre = 1000 cc not 1000 c.c. It has to be borne in mind that SI system of units is not the only system of units that is followed all over the world. There are some countries (though they are very few in number) which use different system of units. For example: the FPS (Foot Pound Second) system or the CGS (Centimeter Gram Second) system. Dimensions The unit of any derived quantity depends upon one or more fundamental units. This dependence can be expressed with the help of dimensions of that derived quantity. In other words, the dimensions of a physical quantity show how its unit is related to the fundamental units. To express dimensions, each fundamental unit is represented by a capital letter. Thus the unit of length is denoted by L, unit of mass by M. Unit of time by T, unit of electric current by I, unit of temperature by K and unit of luminous intensity by C. Remember that speed will always remain distance covered per unit of time, whatever is the system of units, so the complex quantity speed can be expressed in terms of length L and time T. Now,we say that dimensional formula of speed is LT -1. We can relate the physical quantities to each other (usually we express complex quantities in terms of base quantities) by a system of dimensions. Dimension of a physical quantity are the powers to which the fundamental quantities must be raised to represent the given physical quantity. | Dimensions | | Example Density of a substance is defined to be the mass contained in unit volume of the substance. Hence, [density] = ([mass])/([volume]) = M/L3 = ML-3 So, the dimensions of density are 1 in mass, -3 in length and 0 in time. Hence the dimensional formula of density is written as [ρ]= ML-3T0 It is to be noted that constants such as ½ π, or trigonometric functions such as "sin wt" have no units or dimensions because they are numbers, ratios which are also numbers. |
| Applications of Dimensions | | The limitations are as follows: (i) If dimensions are given, physical quantity may not be unique as many physical quantities have the same dimension. For example, if the dimensional formula of a physical quantity is [ML2T-2] it may be work or energy or even moment of force. (ii) Numerical constants, having no dimensions, cannot be deduced by using the concepts of dimensions. (iii) The method of dimensions cannot be used to derive relations other than product of power functions. Again, expressions containing trigonometric or logarithmic functions also cannot be derived using dimensional analysis, e.g. s = ut + 1/3at2 or y = a sin theta or P= P0e (–Mgh)/RT cannot be derived. However, their dimensional correctness can be verified. (iv) If a physical quantity depends on more than three physical quantities, method of dimensions cannot be used to derive its formula. For such equations, only the dimensional correctness can be checked. For example, the time period of a physical pendulum of moment of inertia I, mass m and length l is given by the following equation. T = 2∏√(I/mgl) (I is known as the moment of Inertia with dimensions of [ML2] through dimensional analysis), though we can still check the dimensional correctness of the equation (Try to check it as an exercise). (v) Even if a physical quantity depends on three Physical quantities, out of which two have the same dimensions, the formula cannot be derived by theory of dimensions, and only its correctness can be checked e.g. we cannot derive the equation. |
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