Question

If the velocity of light $(c)$, gravitational constant $(G)$ and Planck’s constant $(h)$ are chosen as fundamental units, then the dimensions of mass in the new system is

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

Answer 1

Answer: (C)

Let $m\propto c^{x}G^{y}h^{z}$ or $m=Kc^{x}G^{y}h^{z}$
By substituting the dimensions of each quantity in both sides, we get
$\left[M^{1}L^{0}T^{0}\right]=K\left[LT^{-1}\right]^{x}\left[M^{-1}L^{3}T^{-2}\right]^{y}\left[ML^{2}T^{-1}\right]^{z}$
$=\left[M^{-y+z}L^{x+3y+2z}T^{-x-2y-z}\right]$
By equating the powers of $M$, $L$ and $T$ on both sides, we get
$- y + z= 1$, $x + 3y + 2z = 0$, $- x - 2 y - z = 0$
By solving above three equations, $x = 1/2$, $y = -1/2$ and $z = 1/2$.
$\therefore m\propto c^{1/2}G^{-1/2}h^{1/2}$