What are the Defects of Dimensional Analysis ?

Defects of Dimensional Analysis

(i) While deriving a formula the proportionality constant cannot be found.

(ii) The formula for a physical quantity depending on more than three other physical quantities cannot be derived. It can be checked only.

(iii) The equations of the type $v=u \pm$ at cannot be derived. They can be checked only.

(iv) The equations containing trigonometrical functions ( $\sin \theta, \cos \theta$, etc), logarithmic functions ( $\log x, \log x^{3}$ etc) and exponential functions $\left(e^{x}, e^{x^{2}} \operatorname{etc}\right)$ cannot be derived. They can be checked only.

Ex. How many dynes are in $20 \mathrm{~N}$ ?

Sol. Dimensional formula of force $(F)=M^{1} L^{1} T^{-2}$

$n_{1}=20$

$\mathrm{n}_{2}=?$

$n_{2}=\left[\frac{M_{1}}{M_{2}}\right]^{a}\left[\frac{L_{1}}{L_{2}}\right]^{b}\left[\frac{T_{1}}{M_{2}}\right]^{c} n_{1}$

$\mathrm{n}_{2}=20\left[\frac{1 \mathrm{~kg}}{\mathrm{gm}}\right]^{11}\left[\frac{1 \mathrm{~m}}{\mathrm{~cm}}\right]^{1}\left[\frac{1 \mathrm{sec} .}{\mathrm{sec}}\right]^{-2}=20\left[\frac{10^{3} \mathrm{gm}}{\mathrm{gm}}\right]^{1}\left[\frac{10^{2} \mathrm{~cm}}{\mathrm{~cm}}\right]^{1}$

$n_{2}=20 \times 10^{5}$

Ans. $\Rightarrow 20 \mathrm{~N}=20 \times 10^{5}$ dyne

Ex. Find the number of ergs in one Joule.

Sol. $ \mathrm{n}_{1}=\mathrm{n}_{2}\left[\frac{\mathrm{M}_{2}}{\mathrm{M}_{1}}\right]\left[\frac{\mathrm{L}_{2}}{\mathrm{~L}_{1}}\right]^{2}\left[\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}\right]^{-2} $

$n_{1}=1 \cdot\left[\frac{1 \mathrm{~kg}}{1 \mathrm{gm}}\right]\left[\frac{1 \mathrm{~m}}{1 \mathrm{~cm}}\right]^{2}\left[\frac{1 \mathrm{sec}}{1 \mathrm{sec}}\right]^{-2}=1\left[\frac{1000 \mathrm{gm}}{1 \mathrm{gm}}\right]\left[\frac{100 \mathrm{~cm}}{1 \mathrm{~cm}}\right]^{+2}$

$n_{1}=10^{3} \cdot 10^{4}=10^{7}$

$\therefore 1$ Joule $=10^{7}$ erg.

Ex. Value of acceleration due to gravity is $9.8 \mathrm{~m} / \mathrm{sec}^{2}$. Find its value in $\mathrm{km} / \mathrm{hr}^{2}$

Sol. $ \mathrm{n}_{1}=\mathrm{n}_{2}\left[\frac{\mathrm{L}_{2}}{\mathrm{~L}_{1}}\right] \cdot\left[\frac{\mathrm{T}_{2}}{\mathrm{~T}_{1}}\right]^{-2} $

Given $\quad \mathrm{n}_{2}=9.8$

therefore, $\mathrm{n}_{1}=9.8\left[\frac{1 \mathrm{~m}}{1 \mathrm{~km}}\right] \cdot\left[\frac{1 \mathrm{sec}}{1 \mathrm{hr}}\right]^{-2}=9.8\left[\frac{1 \mathrm{~m}}{1000 \mathrm{~m}}\right] \cdot\left[\frac{1 \mathrm{sec}}{60 \times 60 \mathrm{sec}}\right]^{-2}$

$n_{1}=9.8\left[\frac{1}{1000} \times 60 \times 60 \times 60 \times 60\right]=98 \times 36 \times 36=127008$

$\therefore \quad \mathrm{g}=127008 \mathrm{~km} / \mathrm{hr}^{2}$

Ex. Density of oil is $0.8 \mathrm{gm} / \mathrm{cm}^{3}$. Find its value in MKS system.

Sol. $n_{1}=n_{2} \cdot\left[\frac{M_{2}}{M_{1}}\right]\left[\frac{L_{2}}{L_{1}}\right]^{-3}=0.8 \cdot\left[\frac{1 \mathrm{gm}}{1 \mathrm{~kg}}\right] \cdot\left[\frac{1 \mathrm{~cm}}{1 \mathrm{~m}}\right]^{-3}$

$n_{1}=0.8 \cdot\left[\frac{1 \mathrm{gm}}{1000 \mathrm{gm}}\right] \cdot\left[\frac{1 \mathrm{~cm}}{100 \mathrm{~cm}}\right]^{-3} \quad$ or $\quad n_{1}=0.8 \times 10^{3}$

$\therefore \quad$ density of oil is $0.8 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ in MKS system.

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