If the velocity of light c, the constant of gravitation G and Plank's constant h be chosen as fundamental units, find the dimensions of mass, length and time in terms of c, G and h.

Explanation

(i) `M propto G^(x)C^(y)h^(z)`
`[M^(1)L^(0)T^(0)]=[M^(-1)L^(3)T^(-2)]^(x)[LT^(-1)]^(y)[ML^(2)T^(-1)]^(z)[M^(1)L^(0)T^(0)]=M^(-x+z)L^(3x+y+2z)T^(-2x-y-z)`
`-x+z=1, 3x+y+2x=0, -2x-y-z=0`
Solving these equations, we get
`x=(-1)/2, y=1/2, z=1/2`
`M=G^((-1)/2)C^(1/2)h^(1/2)`
`M=sqrt((hc)/G)`
(ii) Length `(l) propto G^(x)C^(y)h^(z)`
`[M^(0)L^(1)T^(0)]=[M^(-1)L^(3)T^(2)]^(x)[LT^(-1)]^(y)[ML^(2)T^(-1)]^(z)`
`[M^(0)L^(1)T^(0)]=M^(-x+z)L^(3x+y+2z)T^(-2x-y-z)`
`" "` Applying principle of homogeneity
`-x+z=0, 3x+y+2z=0, -2x-y-z=0`
`" "x=1/2, y=(-3)/2,z=1/2`
From equation (1), `l=G^(1/2)C^((-3)/2)h^(1/2)`
`" Length "(l)=sqrt((Gh)/C^(3))`
(iii) Time `(T) propto G^(x)C^(y)h^(z)`
`[M^(0)L^(0)T^(1)]=[M^(-1)L^(3)T^(-2)]^(x)[LT^(1)]^(4)[ML^(2)T^(-1)]^(z)`
`" "[M^(0)L^(0)T^(1)]=M^(-x+z)L^(3x+y+2z)T^(-2x-y-z)`
Applying principle of homogeneity
`-x+z=0, 3x+y+2z=0, -2x-y-z=1`
`" "`Solving these equations, we get
`" "x=1/2, y=(-5)/2 and z=1/2`
`" "therefore T=G^(1/2)C^((-5)/2)h^(1/2)`
Time `(T)=sqrt((Gh)/C^(5))`