Planck's constant ( $h$ ), speed of light in vacuum (c) and Newton's gravitational constant $(G)$ ar

Question

Planck's constant ( $h$ ), speed of light in vacuum (c) and Newton's gravitational constant $(G)$ are three fundamental constants. Which of the following combinations of these has the dimension of mass?

Professor PC Thomas Classes · Answer

(b) : According to question, $m \propto h^{p} c^{q} G^{r}$ \\ $m=k h^{p} c^{q} G^{r}$ \\ Writing dimensions of physical quantities on both sides,\\ $\left[\mathrm{M}^{1} \mathrm{~L}^{0} \mathrm{~T}^{0}ight]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-1}ight]^{p}\left[\mathrm{LT}^{-1}ight]^{q}\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}ight]^{r}$\\ Applying the principle of homogeneity of dimensions, we get $p-r=1 \quad$...(i), $2 p+q+3 r=0 \quad \ldots$ (ii),$-p-q-2 r=0 \ldots$ (iii)\\ Solving eqns. (ii) and (iii), we get $p+r=0 \Rightarrow p=-r$. Using (i), $2p=1 \Rightarrow p=\frac{1}{2}, r=-\frac{1}{2}$\\ From (iii), $q = -p-2r = -\frac{1}{2} - 2(-\frac{1}{2}) = \frac{1}{2}$\\ $	herefore m = K \sqrt{\frac{h c}{G}}$