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How to determine the formula if physical quantities are given

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For this we should know the dependence of the physical quantity on other quantities and consider it as a product type of the dependence.

Q)Consider a simple pendulum, having a bob attached to a string, that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length (l), mass of the bob (m) and acceleration due to gravity (g). Derive the expression for its time period using method of dimensions.

The dependence of time period T on the quantities l, g and m as a product may be written as

T=kl^xg^ym^z

where k is dimensionless constant and x, y and z are the exponents. By considering dimensions on both sides, we have

[L^0 M^0 T^1 ]=[L ]^x [L T^{-2} ]^y [M ]^z=L^{x+y}T^{-2y}M^z

On equating the dimensions on both sides, we have

x + y = 0; –2y = 1; and z = 0 

T=kl^{\frac{1}{2}}g^{\frac{-1}{2}}

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Safeer PP

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