If the speed of light `c ( = 3xx10^8m//s),` Planck's constant `h(=6.6xx10^(-34)J - s)` and gravitational constant `G( = 6.67xx10^(-11)mks units)` be chosen as fundamental units, find out the dimensions and value of unit of mass.

Explanation

`Let m =k c^x h^y G^z`
`[M^1 L^0 T^0]`
`= [LT^(-1)]^x [ML^2 T^(-1)]^y [M^(-1) l^3 T^(-2)]^z …. (i)`
`=M^(y-z). L^(x+2y +3z) T^(-x-y-2z)`
Applying the principle of homogeneity of
dimensions.
y - z = 1 ….(ii)
x + 2y + 3z = 0 .....(iii)
-x y -2z =0 .....(iv)
Adding (iii) and (iv), we get
y +z = 0 .....(v)
Adding (ii) and (v) `2 = 1, y= 1//2, `
From (v) `z = -y = -1//2`
From (iii), `x = -2y - 3z = -1 + 3//2 = 1//2`
:. from (i),
`m = kc ^(1//2) h^(1//2) G^(-1//2)`
` = ksqrt((ch)/(G))`
Taking k =1 ,
`m =sqrt((3xx10^8xx6.6xx10^(-34))/(6.67xx10^(-11))`
`~~10^(-7)kg`