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| Principle of Homogeneity of Dimensions |
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| It states that if the dimensions of each term on both the sides of equation are same, then the physical quantity will be correct. |
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| Example, |
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| To check the correctness of v = u + at, using dimensions |
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| Dimensional formula of final velocity v = [LT-1] |
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| Dimensional formula of initial velocity u = [LT-1] |
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| Dimensional formula of acceleration x time, at = [LT-2 x T] |
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| = [LT-1] |
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 Dimensions on both sides of each term is the same. Hence, the equation is dimensionally correct. |
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| The value of a physical quantity can be obtained in some other system, when its value in one system is given using the method of dimensional analysis. |
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| Measure of a physical quantity is given by X = nu, |
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| u - size of unit, |
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| n - numerical value of physical quantity for the chosen unit. |
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| Let u1 and u2 be units for measurement of a physical quantity in two systems and let n1 and n2 be the numerical values of physical quantity for two units. |
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 n1u1 = n2u2 |
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| Let a, b and c be the dimensions of physical quantity in mass, length and time |
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| M1, L1, T1 and M2, L2, T2 are units in two systems of mass, length and time. |
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| This equation is used to find the value of a physical quantity in the second or the new system, when its value in the first or the given system is known. |
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| The principle of homogeneity of dimensional equation is used to derive a relation between various physical quantities. |
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| To derive a physical relation, the dependent factors of a given physical quantity is found. Assuming its dimensions in terms of these factors, the final dimensional equation is written in terms of mass, length and time. Equating the powers of M, L and T on both sides of the dimensional equation, three equations are formed by which, value of unknown powers can be calculated. By substituting these values in the equation, the real form of relation is achieved. |
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| One illustration to establish a relation between different physical quantities. |
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| Let us find an expression for the time period (d) of a simple pendulum. The time period t may depend upon (i) mass m of the bob of the pendulum (whose length is length l) , (iii) acceleration due to gravity g, at the place where the pendulum is responding. (v) angle of swing q. |
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| Combining all the four factors, we get |
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| K - dimensionless constant |
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| Writing dimensionally, |
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| Comparing the powers of like terms on both sides, |
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| So, the equation becomes |
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