Question

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

• A. $[p^1\, A^{-1}\, t^{-1}]$
• B. $[p^2\, A^{1}\, t^{1}]$
• C. $[p^1\, A^{-1/2}\, t^{1}]$
• D. $[p^1\, A^{1/2}\, t^{-1}]$

Answer 1

Answer: (D)

$[p^1\, A^{1/2}\, t^{-1}]$

Answer 2

Let, energy $E = k p ^{ a } A ^{ b } t ^{ c } \,\,$...(i) where $k$ is a dimensionless constant of proportionality Equating dimensions on both sides of (i), we get $\left[ M L ^{2} T ^{-2}\right]=\left[ M L T ^{-1}\right]^{ a }\left[ M ^{0} L ^{ 2 } T ^{0}\right]^{ b }\left[ M ^{0} L ^{0} T \right]^{ c }$ $=\left[ M ^{ a } L ^{ a + 2 b } T ^{- a + c }\right]$ Applying the principle of homogeneity of dimensions, we get $a=1 \ldots$ (ii) $a + 2 b = 2$...(iii) $- a + c =- 2$...(iv) On solving eqs. (ii), (iii) and (iv), we get $a =1, b =\frac{1}{2}, c =-1$ $\therefore[ E ]=\left[ p ^{1} A ^{1 / 2} t ^{-1}\right]$