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 thes three three simultaneous equations and three unknown 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)" ":. L^(-2)(hG)/(V^(3))`
or `L=h^(1//2)G^(1//2)V^(-3//2)`
Put (A) in (1) to get `V=h^(1//2)G^(1//2)V^(-3//2)T^(-1)`
`:.T=h^(1//2)G^(1//2)V^(-5//2)`
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) rArr 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)`
:. `M=h^(1/2) G^(1/2) V^(1/2) ` ...(C)