Question

If the velocity of light $c$, universal gravitational constant $G$ and Planck's constant $h$ are chosen as fundamental quantities. The dimensions of mass in the new system is :

• A. $\left[h^1{c}^1{G}^{-1}\right]$
• B. $\left[h^{-1/2}{c}^{1/2}{G}^{1/2}\right]$
• C. $\left[h^{1/2}{c}^{1/2}{G}^{-1/2}\right]$
• D. $\left[h^{1/2}{c}^{-1/2}{G}^1\right]$

Answer 1

Answer: (C)

Say dimensional formale of mass is $H^x{C}^y{G}^z$
$M^1={\left(M{L}^2{T}^{-1}\right)}^x\left(L^{-1}\right){\left(M^{-1}{L}^3{T}^{-2}\right)}^Z$
$M^1{L}^0{T}^0=M^{x-z}{L}^{2x+y+3z}{T}^{-x-yz}$
on comparing both side
$x-z=1$
$2x+y+3z=0$
$-x-y-2z=0$
On solving above equations we get
$x=\frac{1}{2}y=\frac{1}{2}z=\frac{-1}{2}$