Question

If velocity (V), force (F) and time (T) are chosen as fundamental quantities, express (a) mass and (b) energy in terms of V, F and T.

Answer 1

$\left(a\right)$

Let M = (Some number) $\left(V{)}^a\right(F{)}^b(T{)}^c$

Equating dimensions of both the sides, we get
$M^1{L}^0{T}^0=\left(1\right)\left[L^1{T}^{-1}{]}^a\right[M^1{L}^1{T}^{-2}{]}^b[T^1{]}^c$

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

Get $a=-1,b=1,c=1$.

M = (Some number) $\left(V^{-1}{F}^1{T}^1\right)$.
$\left[M\right]=\left[V^{-1}{F}^1{T}^1\right]$

$\left(b\right)$

Similarly, we can also express energy in terms of V, F and T.
Let [E] = [Some number]$\left[V{]}^a\right[F{]}^b\left[T^c\right]$

$\Rightarrow \left[M{L}^2{T}^{-2}\right]=\left[M^0{L}^0{T}^0\right]\left[L{T}^{-1}{]}^a\right[ML{T}^{-2}{]}^b\left[T^c\right]$

$\Rightarrow \left[M^1{L}^2{T}^{-2}\right]=\left[M^b{L}^{a+b+c}{T}^{a-2b+c}\right]$

$\Rightarrow 1=b;2=a+b+c;-2=-a-2b+c$
Get $a=1;b=1,c=1.$
$\Rightarrow E=\left(Somenumber\right)V^1{F}^1{T}^1$

or $\left[E\right]=\left[V^1\right]\left[F^1\right]\left[T^1\right]$.