Question

If velocity (V), force (F) and time (T) are chosen as fundamental quantities, express

(i) mass and

(ii) energy

in terms of V, F and T

Answer 1

Let M = (some Number) $\left(V{)}^a\right(F{)}^b(T{)}^c$
Equating dimensions of both the sides
$M^1{L}^0{T}^0=\left(1\right)\left[L^1{T}^{-1}{]}^a\right[M^1{L}^1{T}^{-2}{]}^b[T^1{]}^c$
$M^1{L}^0{T}^0=M^b{L}^{a+b}{T}^{-a-2b+c}$

get a = - 1, b = 1, c = 1

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

Similarly, we can also express energy in terms of V, F, T

Let [E] = [some Number] $\left[V{]}^a\right[F{]}^b[T{]}^c$
$\Rightarrow \left[M{L}^2{T}^{-2}\right]=\left[M^0{L}^0{T}^0\right]\left[L{T}^{-1}{]}^a\right[ML{T}^{-2}{]}^b[T{]}^c$

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

$\Rightarrow$ 1 = b; 2 = a + b ; -2 = -a - 2b + c

get a = 1 ; b = 1 ; c = 1

$\therefore$ E = (some Number) $V^1{F}^1{T}^{1}or\left[E\right]=\left[V^1\right]\left[F^1\right]\left[T^1\right]$