Question

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

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

Answer 1

Answer: (A)

Let $S=k E^{a} V^{b} T^{c}$
where $k$ is a dimensionless constant.
Writing the dimensions on both sides, we get
$\left[M^{1} L^{0} T^{-2}\right]=\left[M L^{2} T^{-2}\right]^{a}\left[L T^{-1}\right]^{b}[T]^{c}$
$=\left[M^{a} L^{2 a+b} T^{-2 a-b+c}\right]$
Applying principle of homogeneity of dimensions,
we get, $a=1 \ldots(i)$
$2 a+b=0 \ldots(i i) $
$-2 a-b+c=-2 \ldots(i i i)$
Adding $(i i)$ and $(i i i)$, we get
$c=-2$
From $(i i), b=-2 a=-2$
$\therefore S=k E V^{-2} T^{-2} $
or $[S]=\left[E V^{-2} T^{-2}\right]$