What are the limitations of dimensional analysis?

State and explain, with examples, the uses of dimensional equations. What are the limitations of dimensional analysis?

Find the dimensions of linear momentum $\left(p\right)$, Some of the relations involving the above quantities are:

$p\mathit{=}mv\mathit{,}\mathit{ }v\mathit{=}\frac{1}{T}\mathit{,}\mathit{ }P\mathit{=}\frac{F}{A}\mathit{,}\mathit{ }\alpha \mathit{=}\frac{{\omega }_2\mathit{-}{\omega }_1}{t_2\mathit{-}{t}_1}\mathit{,}\mathit{ }r\mathit{=}F\mathit{.}r\mathit{,}\mathit{ }I\mathit{=}m{r}^2\mathit{,}\mathit{ }c\mathit{=}\frac{Q}{m\mathit{.}\theta }$, $\gamma \mathit{=}\frac{V_{\mathit{2}}\mathit{-}{V}_{\mathit{1}}}{V_{\mathit{1}}\mathit{\times }\mathit{(}{\mathit{\theta }}_{\mathit{2}}\mathit{-}{\mathit{\theta }}_{\mathit{1}}\mathit{)}}\mathit{,}\mathit{ }\frac{PV}{T}\mathit{=}R$. The symbols $m\mathit{,}\mathit{ }v\mathit{,}\mathit{ }r\mathit{,}\mathit{ }A\mathit{,}\mathit{ }V\mathit{,}\mathit{ }F\mathit{,}\mathit{ }t\mathit{,}\mathit{ }\theta \mathit{,}\mathit{ }T$  are respectively mass, velocity, distance, area, volume, Force, time, temperature, absolute temperature. 

Find the dimensions of frequency $\left(v\right)$, Some of the relations involving the above quantities are:

$p\mathit{=}mv\mathit{,}\mathit{ }\nu =\frac{1}{T}, P=\frac{F}{A}, \alpha =\frac{{\omega }_2-{\omega }_1}{t_2-t_1}, r=F·r\mathit{,}\mathit{ }I=m{r}^2, c=\frac{Q}{m\mathit{.}\theta }$, $\gamma =\frac{V_2-V_1}{V_1\times \left({\theta }_2-{\theta }_1\right)}, \frac{PV}{T}=R$. The symbols $m\mathit{,}\mathit{ }v\mathit{,}\mathit{ }r\mathit{,}\mathit{ }A\mathit{,}\mathit{ }V\mathit{,}\mathit{ }F\mathit{,}\mathit{ }t\mathit{,}\mathit{ }\theta \mathit{,}\mathit{ }T$  are respectively mass, velocity, distance, area, volume, Force, time, temperature, absolute temperature. 

Find the dimensions of pressure $\left(p\right)$. Some of the relations involving the above quantities are:

$p\mathit{=}mv\mathit{,}\mathit{ }\nu \mathit{=}\frac{\mathit{1}}{T}\mathit{,}\mathit{ }P\mathit{=}\frac{F}{A}\mathit{,}\mathit{ }\alpha \mathit{=}\frac{{\omega }_{\mathit{2}}\mathit{-}{\omega }_{\mathit{1}}}{t_{\mathit{2}}\mathit{-}{t}_{\mathit{1}}}\mathit{,}\mathit{ }r\mathit{=}F\mathit{·}r\mathit{,}\mathit{ }I\mathit{=}m{r}^{\mathit{2}}\mathit{,}\mathit{ }c\mathit{=}\frac{Q}{m\theta }$, $\gamma \mathit{=}\frac{{\mathit{V}}_{\mathit{2}}\mathit{-}{\mathit{V}}_{\mathit{1}}}{{\mathit{V}}_{\mathit{1}}\mathit{\times }({\theta }_{\mathrm{2}}-{\theta }_{\mathrm{1}})}\mathit{,}\mathit{ }\frac{\mathit{P}\mathit{V}}{\mathit{T}}\mathit{=}R$. The symbols $m\mathit{,}\mathit{ }v\mathit{,}\mathit{ }r\mathit{,}\mathit{ }A\mathit{,}\mathit{ }V\mathit{,}\mathit{ }F\mathit{,}\mathit{ }t\mathit{,}\mathit{ }\theta \mathit{,}\mathit{ }T$  are respectively mass, velocity, distance, area, volume, Force, time, temperature, absolute temperature. 

Find the dimensions of angular acceleration $\left(a\right)$ Some of the relations involving the above quantities are:

$p\mathit{=}mv\mathit{,}\mathit{ }\nu \mathit{=}\frac{\mathit{1}}{T}\mathit{,}\mathit{ }P\mathit{=}\frac{F}{A}\mathit{,}\mathit{ }\alpha \mathit{=}\frac{{\omega }_{\mathit{2}}\mathit{-}{\omega }_{\mathit{1}}}{t_{\mathit{2}}\mathit{-}{t}_{\mathit{1}}}\mathit{,}\mathit{ }r\mathit{=}F·r\mathit{,}\mathit{ }I\mathit{=}m{r}^{\mathit{2}}\mathit{,}\mathit{ }c\mathit{=}\frac{Q}{m\mathit{.}\theta }$, $\gamma \mathit{=}\frac{{\mathit{V}}_{\mathit{2}}\mathit{-}{\mathit{V}}_{\mathit{1}}}{{\mathit{V}}_{\mathit{1}}\mathit{\times }({\theta }_{\mathrm{2}}-{\theta }_{\mathrm{1}})}\mathit{,}\mathit{ }\frac{\mathit{P}\mathit{V}}{\mathit{T}}\mathit{=}R$. The symbols $m\mathit{,}\mathit{ }v\mathit{,}\mathit{ }r\mathit{,}\mathit{ }A\mathit{,}\mathit{ }V\mathit{,}\mathit{ }F\mathit{,}\mathit{ }t\mathit{,}\mathit{ }\theta \mathit{,}\mathit{ }T$  are respectively mass, velocity, distance, area, volume, Force, time, temperature, absolute temperature. 

Find the dimensions of Torque $\left(t\right)$, Some of the relations involving the above quantities are:

$p\mathit{=}mv\mathit{,}\mathit{ }\nu \mathit{=}\frac{\mathit{1}}{T}\mathit{,}\mathit{ }P\mathit{=}\frac{F}{A}\mathit{,}\mathit{ }\alpha \mathit{=}\frac{{\omega }_{\mathit{2}}\mathit{-}{\omega }_{\mathit{1}}}{t_{\mathit{2}}\mathit{-}{t}_{\mathit{1}}}\mathit{,}\mathit{ }r\mathit{=}F·r\mathit{,}\mathit{ }I\mathit{=}m{r}^{\mathit{2}}\mathit{,}\mathit{ }c\mathit{=}\frac{Q}{m\mathit{.}\theta }$, $\gamma \mathit{=}\frac{{\mathit{V}}_{\mathit{2}}\mathit{-}{\mathit{V}}_{\mathit{1}}}{{\mathit{V}}_{\mathit{1}}\mathit{\times }({\theta }_{\mathrm{2}}-{\theta }_{\mathrm{1}})}\mathit{,}\mathit{ }\frac{\mathit{P}\mathit{V}}{\mathit{T}}\mathit{=}R$. The symbols $m\mathit{,}\mathit{ }v\mathit{,}\mathit{ }r\mathit{,}\mathit{ }A\mathit{,}\mathit{ }V\mathit{,}\mathit{ }F\mathit{,}\mathit{ }t\mathit{,}\mathit{ }\theta \mathit{,}\mathit{ }T$  are respectively mass, velocity, distance, area, volume, Force, time, temperature, absolute temperature.