Question:
If the velocity of light c, Planck’s constant h and gravitational constant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities.
Solution:
Principle of homogeneity is used for solving this problem.
[h] = [ML2T1] [c] = [LT-1] [G] = [M-1L3T-2]
Let m = kcahbGc
Solving the above we get,
m = kc1/2h1/2G-1/2 = k√ch/G
Let L = kcahbGc
Solving the above we get,
L = kc-3/2h1/2G1/2 = k√hG/c3
Let T = cahbGc
Solving the above we get,
L = kc-5/2h1/2G1/2 = k√hG/c5