Question

If the velocity of light (c), gravitational constant (G), and Plancks' constant (h) are chosen as fundamental units, then find the dimensions of mass in new system.

Answer 1

Let $m\propto \left[c^x{G}^y{h}^z\right]$

or $m=K\left[c^x{G}^y{h}^z\right]$

By substituting the dimension of each quantity in both the sides,

$\left[M^1{L}^0{T}^0\right]=K\left[L{T}^{-1}{]}^x\right[M^{-1}{L}^3{T}^{-2}{]}^y[M{L}^2{T}^{-1}{]}^z$
$M=\left[M^{-y+z}{L}^{x+3y+2z}{T}^{x-2y-z}\right]$

By equating the power of M, L and T in both the sides :

$-y+z=1,x+3y+2z=0,-x-2y-z=0$

By solving above three equations,

$x=1/2,y=-1/2,andz=1/2.$

$\therefore m\propto c^{1/2}{G}^{-1/2}{h}^{1/2}$