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 ?

Suppose you design your system of dimensions and take velocity (V). Pl

1.	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 ?
Answer» Solution :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)