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

Question:

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

1. $\left[\mathrm{PA}^{-1} \mathrm{~T}^{-2}\right]$

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

3. $\left[\mathrm{P}^{2} \mathrm{AT}^{-2}\right]$

4. $\left[\mathrm{P}^{1 / 2} \mathrm{AT}^{-1}\right]$

Correct Option: , 2

Solution:

Let $[\mathrm{E}]=[\mathrm{P}]^{\mathrm{x}}[\mathrm{A}]^{\mathrm{y}}[\mathrm{T}]^{\mathrm{z}}$

$\mathrm{ML}^{2} \mathrm{~T}^{-2}=\left[\mathrm{MLT}^{-1}\right]^{x}\left[\mathrm{~L}^{2}\right]^{y}[\mathrm{~T}]^{\mathrm{z}}$

$\mathrm{ML}^{2} \mathrm{~T}^{-2}=\mathrm{M}^{\mathrm{x}} \mathrm{L}^{x+2 \mathrm{y}} \mathrm{T}^{-\mathrm{x}+\mathrm{z}}$

$\rightarrow \mathrm{x}=1$

$\rightarrow \mathrm{x}+2 \mathrm{y}=2$

$1+2 y=2$

$\mathrm{y}=\frac{1}{2}$

$\rightarrow-x+z=-2$

$-1+z=-2$

$\mathrm{z}=-1$

$[\mathrm{E}]=\left[\mathrm{PA}^{1 / 2} \mathrm{~T}^{-1}\right]$