Question

If force, time and velocity are treated as fundamental quantities, then dimensional formula of energy will be

• A. $\left[FTV\right]$
• B. $\left[F{T}^{2}V\right]$
• C. $\left[FT{V}^2\right]$
• D. $\left[F{T}^2{V}^2\right]$

Answer 1

The correct option is A $\left[FTV\right]$

We know,
$\left[F\right]=\left[ML{T}^{-2}\right]$
$\left[t\right]=\left[T\right]$
$\left[v\right]=\left[L{T}^{-1}\right]$
and $\left[E\right]=\left[M{L}^2{T}^{-2}\right]$

Using dimensional analysis:
$\left[E\right]=C\left[F{]}^a\right[t{]}^b[v{]}^c$
$\left[M{L}^2{T}^{-2}\right]=C\left[ML{T}^{-2}{]}^a\right[T{]}^b[L{T}^{-1}{]}^c$
$\left[M{L}^2{T}^{-2}\right]=C\left[M^a{L}^{a+c}{T}^{-2a+b-c}\right]$

Comparing the powers of same physical quantities:
$a=1a+c=2b-c-2a=-2\Rightarrow c=1\Rightarrow b=1$

$\therefore \left[E\right]=\left[FTV\right]$