Question

If momentum $(P)$, area $(A)$ and time $(T)$ are taken to be the fundamental quantities then the dimensional formula for energy is :

• A. $\left[ PA ^{-1} T ^{-2}\right]$
• B. $\left[ PA ^{1 / 2} T ^{-1}\right]$
• C. $\left[ P ^{2} AT ^{-2}\right]$
• D. $\left[ P ^{1 / 2} AT ^{-1}\right]$

Answer 1

Answer: (B)

$\left[ PA ^{1 / 2} T ^{-1}\right]$

Answer 2

Let $[ E ]=[ P ]^{ x }[ A ]^{ y }[ T ]^{ z }$
$ML ^{2} T ^{-2}=\left[ MLT ^{-1}\right]^{ x }\left[ L ^{2}\right]^{ y }[ T ]^{z}$
$ML ^{2} T ^{-2}= M ^{ x } L ^{ x +2 y } T ^{- x +2}$
$\rightarrow x =1$
$\rightarrow x +2 y =2$
$1+2 y =2$
$y =\frac{1}{2}$
$\rightarrow- x + z =-2$
$-1+ z =-2$
$z =-1$
$[ E ]=\left[ PA ^{1 / 2} T ^{-1}\right]$