Question

If energy ($E$), velocity ($V$) and time ($T$) are chosen as the fundamental quantities, the dimensional formula of surface tension will be

• A. $\left[E{V}^{-2}{T}^{-2}\right]$
• B. $\left[E^{-2}{V}^{-2}{T}^{-3}\right]$
• C. $\left[E{V}^{-2}{T}^{-1}\right]$
• D. $\left[E{V}^{-1}{T}^{-2}\right]$

Answer 1

Answer: (A)

$\left[E{V}^{-2}{T}^{-2}\right]$

$S=E^x{V}^y{T}^z$
$S=Force/length\Rightarrow \left[T\right]=\left[M{T}^{-2}\right]$
$E=\frac{1}{2}m{v}^2\Rightarrow \left[E\right]=\left[M{L}^2{T}^{-2}\right]$
$\left[V\right]=\left[L{T}^{-1}\right]$
$\left[T\right]=\left[T\right]$
$\left[M{T}^{-2}\right]=\left[E^x{V}^y{t}^z\right]=\left[M{L}^2{T}^{-2}{]}^x\right[L{T}^{-1}{]}^y[T{]}^z$

Comparing coefficients
$x=1$
$2x+y=0$
$-2x-y+z=-2$

Solving above equations $x=1,y=-2,z=-2$

Substituting in expression of surface tension:
$S=E{V}^{-2}{T}^{-2}$

Hence, correct answer is option A.