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

Solution

1. Assume the mass $M$ is proportional to $c^x{G}^y{h}^z$, so $[M]=[L{T}^{-1}{]}^x[M^{-1}{L}^3{T}^{-2}{]}^y[M{L}^2{T}^{-1}{]}^z$.
2. Equating the powers of $M$, $L$, and $T$ on both sides gives: $-y+z=1$, $x+3y+2z=0$, and $-x-2y-z=0$.
3. Solving this system of linear equations yields $x=1/2$, $y=-1/2$, and $z=1/2$.
4. Substituting these values back into the expression gives $M\propto c^{1/2}{G}^{-1/2}{h}^{1/2}=\sqrt{\frac{hc}{G}}$.
5. Hence the answer is (A).