Question

If velocity of light c, Planck’s constant h and gravitational contant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.

Answer 1

c = L T⁻¹
h = Joules second = energy / frequency = M L² T⁻¹
G = Newtons meter² / kg²  = M⁻¹ L³ T⁻²

M = c^k  h^m  G^n  = M^(m-n)  L^(k+2m+3n)  T^(-k-m -2n)

    =>  k+2m+3n = 0,   m - n = 1,    and      - k - m - 2 n = 0
    =>  2m +3n = m + 2n,   m = -n    and   so  n = -1/2,   m = 1/2   and  k = 1/2

                   M = √c  √h / √G

L = c^p  h^q  G^r     = M^(q - r)   L^(p +2q + 3 r)   T^(-p - q - 2 r)

     => q - r = 0  ,    p +2q + 3 r = 1            and  -p - q - 2 r = 0
         =>  p+ 5r =1      ,    p = - 3r      =>  r = 1/2,  p = -3/2 ,  q = 1/2

                   L = √h √G / √c³

T = c^x h^y G^z  =  M^(y-z)    L^(x+2y+3z)    T^(-x-y-2z)

     =>  x+2y+3z = 0 ,          y-z = 0        and      -x-y-2z = 1
       => x = - 5 z          z=1/2    x = -5/2      y = 1/2

                 T =  √h √G / √c⁵