If the velocity of light $c$, Planck constant $h$, and gravitational constant $G$ are taken as fundamental units, then the dimension of mass in this system is proportional to which of the following?

Solution

1. Let the dimension of mass be represented as $[M]=[c{]}^x[h{]}^y[G{]}^z$.
2. Substituting the dimensions $[c]=[L{T}^{-1}]$, $[h]=[M{L}^2{T}^{-1}]$, and $[G]=[M^{-1}{L}^3{T}^{-2}]$, we get $[M]=[L{T}^{-1}{]}^x[M{L}^2{T}^{-1}{]}^y[M^{-1}{L}^3{T}^{-2}{]}^z$.
3. Equating the powers of $M$, $L$, and $T$ on both sides gives the system of equations: $y-z=1$, $x+2y+3z=0$, and $-x-y-2z=0$.
4. Solving these equations yields $x=1/2$, $y=1/2$, and $z=-1/2$.
5. Substituting these values back into the expression, we find $M\propto c^{1/2}{h}^{1/2}{G}^{-1/2}=\sqrt{\frac{hc}{G}}$.
6. Hence the answer is (A).