Suppose you design your system of dimensions and take velocity (V). Planck constant (h) and gravitational constant (G) as the fundamental quantities . What will be the dimensions of length (L), mass (M) and time (T) in this system ?

Explanation

We know , `[V] = [LT^(-1)] ….(1) " " [h] = [ML^(-2) T^(-1)] " "…(2) " "[G] = [M^(-1) L^3 T^(-2)]" "…(3)`
Now , we have these three simultaneous equations and three unknowns i.e., M , L and T .
We just have to solve these three equations to find out the values of M, L and T in terms of V, H and G .
Let us reduce these three equations in two equations and two unknowns .
`(2) xx (3) ` gives, `hG = L^5 T^(-3)" "...(4)`
Now we have (1) and (4) in L and T only. Cubing equation (1) and dividing in by (4) will eliminate T.
`(V^3)/(hG) = (L^3 T^(-3))/(L^5 T^(-3)) = L^(-2) therefore L^(-2) (hG)/(V^3)`
or `L= h^(1//2) G^(1//2) V^(-3//2)" ".....(A)`
Put (A) in (1) , to get , `V = h^(1//2) G^(1//2) V^(-3//2) T^(-1)`
`therefore T = h^(1//2) G^(1//2) V^(-5//2)" "........(B)`
Put (A) and (B) in (2) to get
`h = M . (h^(1//2) G^(1//2) V^(-3//2))^2 (h^(1//2) G^(1//2) V^(-5//2))^(-1) implies h = M. h^1 G^1 V^(-3) h^(-1//2) G^(-1//2) V^(-5//2) = M h^(1//2) G^(1//2) V^(-1//2)`
`therefore M = h^(1/2) G^(1/2) V^(1/2) ......(C)`