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

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

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

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

Correct Option: , 3

Solution:

(3) Energy, $E \propto A^{a} T^{b} P^{c}$

or,                        $\quad E=k A^{a} T^{b} P^{c}$                    $\ldots$ (i)

where $k$ is a dimensionless constant and $a, b$ and $c$ are the exponents.

Dimension of momentum, $P=M^{1} L^{1} T^{-1}$

Dimension of area, $A=L^{2}$

Dimension of time, $T=T^{1}$

Putting these value in equation (i), we get

$M^{1} L^{2} T^{-2}=M^{c} L^{2 a+c} T^{b-c}$

by comparison

$c=1$

$2 a+c=2$

$b-c=-2$

$c=1, a=1 / 2, b=-1$

$\therefore E=A^{1 / 2} T^{-1} P^{\mathrm{l}}$