Units and Dimensions Uses of Dimensional Analysis

Uses of Dimensional Analysis There are five primary uses of Dimensional Analysis 1 .) Derivation of units 2.) Conversion of units 3.) Checking the correctness of a formula 4.) Principle of Homogeneity 5.) Derivation of a Physical Formula. 1 .) Derivation of units-By writing the Dimensional formula of a physical quantity, we will be instantly able to write its S.I. (and C.G.S) units. For example, the dimensional formula of Momentum as derived earlier was MLT-1 and thererfore its S.I. unit will be However, sometimes the SI, may have another name, for example, in the case of work done, VW and therefore the work done has an SI unit kgm2s2 but this S.I. unit has an special name joules (or J) Question for Practice :-Derive the s.l. Units of Universal Gravitational Constant (G)' , Power, Permittivity of Free Space and Torque.

2.) Conversion of Units The size of a measured Physical quantity always remains constant. If the magnitude of the Physical Quantity is 'N' and if the unit in which it is being represented is U', then the product 'NU' is always constant Keeping the above concept in mind we can easily convert the magnitude of a physical quantity from one system of unit to another. If a Physical Quantity has a dimensional formula MaibTe and the units of Mass, length and time in the two system of units are M1-4,7, and M2,L2,T2 then NM=N2U2 1 L111 N1(MfLhTE)=N2(MSLT) N2=N,(M Therefore, ,so RE The following examples will clear the concepts

Illustration 1.3 Convert gravitational constant (G) from CGS to MKS system. lustration 11 Convert Newton into dyne. Sol. The Newton is the Sl unit of force and has dimensional Sol Joule: SI system, erg: CGS system formula(MILIT2], ie, as], b= 1, c=-2. So IN-lkgms ntustration 1.2 Convert 1 joule to ergs. Work Forcex DistnMass celeration x Length Sol. The dimensional formula of Gis LT Dimensions of work [MLT1 ie, a --1, b=3, c =-2. Length ngu = Mass x-, X Length (Time)2 CGS system CGSsystem My=1 g SI system My=1kg L1 = 1 m SI system My=1kg =1m T-1s a =1, b = 2, c=-2. L2 = 1 cm 25 OW SI system CGS system M1 = 1 k Here N1 = 1, N2=? Value of G in CGS unit 6.67 x 10 cm'gs" cm S cm S = 6.67 x 10 1000 g 100 cm1 10g] L102cm][S lcm cm s 6.67x 10- G -6.67x 10 MKS units IN:10 Dyne So1 J 10' erg

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Nikhil Mishra

Nikhil Mishra is an engineer and has been a Faculty of Physics in some of the most premier institutes of the country and has been involved

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Tushar Sonar

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