If momentum ( $p$ ), area ( $A$ ) and time ( $t$ ) are taken to be fundamental quantities, then ener

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If momentum ( $p$ ), area ( $A$ ) and time ( $t$ ) are taken to be fundamental quantities, then energy has the dimensional formula\

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(d) : Let, energy $E=k p^a A^b t^c$ where $k$ is a dimensionless constant of proportionality. Equating dimensions on both sides of (i), we get $ \left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight]=\left[\mathrm{MLT}^{-1}ight]^a\left[\mathrm{M}^0 \mathrm{~L}^2 \mathrm{~T}^0ight]^b\left[\mathrm{M}^0 \mathrm{~L}^0 \mathrm{~T}ight]^c=\left[\mathrm{M}^a \mathrm{~L}^{a+2 b} \mathrm{~T}^{-a+c}ight] $ Applying the principle of homogeneity of dimensions, we get $$ \begin{align*} & a=1  	ag{ii}\ & a+2 b=2  	ag{iii}\ & -a+c=-2 	ag{iv} \end{align*} $$ On solving eqs. (ii), (iii) and (iv), we get $$ a=1, b=\frac{1}{2}, c=-1 \quad 	herefore \quad[E]=\left[p^1 A^{1 / 2} t^{-1}ight] $$