If velocity of light (c ), gravitational constant (G) and Planck's constant (h) are chosen as fundamental units. What would be the dimensions of acceleration due to gravity ?

Explanation

Let `g =Kc^x G^y h^z …..(i)`
where x, y,z are the dimensions and K is dimensionless constatn. Writing the dimensions in (i)
`[M^0 L^1 T^(-2)] = (LT^(-1))^x (M^(-1) L^3 T^-2)^y`
`(ML^2 T^(-1))^z`
`[M^0 L^1 T^(-2)] = M^(-y +z) L^(x+ 3y +2z) T^(-x-2y-z)`
Applying the principle of homogeneity of dimensions, we get
`- y+z = 0, z=y`
`x+3y +2z = 1` or `x +5y = 1 .....`(ii)
`-x-2y -z =-2` or `-x-3y =-2`...(iii)
Add (ii) and (iii) ,`2y = -1, y= -(1)/(2), z = -(1)/(2)`
From (ii), `x +5(-(1)/(2)) =1`
`x = 1 +(5)/(2) = (7)/(2)`
Put in (i), `g = Kc^(7//2) G^(-1//2) h^(-1//2)`