Question

If $C$ the velocity of light, $h$ Planck's constant and $G$ Gravitational constant are taken as fundamental quantities, then the dimensional formula of mass is:

• A. $h^{-1/2}{G}^{-1/2}{C}^0$
• B. $h^{1/2}{C}^{1/2}{G}^{-1/2}$
• C. $h^{-1/2}{C}^{1/2}{G}^{-1/2}$
• D. $h^{-1/2}{C}^{-1/2}{G}^{-1/2}$

Answer 1

Answer: (B)

Dimensional formula for, $h=\left[M{L}^2{T}^{-1}\right]$

Dimensional formula for, $G=\left[M^{-1}{L}^3{T}^{-2}\right]$

Dimensional formula for, $C=\left[L{T}^{-1}\right]$

Let mass be represented as: $m=h^a{G}^b{C}^c$

$\therefore$ $\left[M\right]=$ $[M{L}^2{T}^{-1}{]}^a$ $[M^{-1}{L}^3{T}^{-2}{]}^b$ $[L{T}^{-1}{]}^c$

$\left[M\right]=$ $\left[M^{a-b}{L}^{2a+3b+c}{T}^{-(a+2b+c)}\right]$

On comparing we get:

$a-b=1$ .........(1)

$2a+3b+c=0$ .........(2)

$a+2b+c=0$ .........(3)

Subtracting (3) from (2) we get:

$a+b=0$

$⟹b=-a$ ...........(4)

Solving (1) and (4):

$a=\frac{1}{2}$ $b=-\frac{1}{2}$

From (3),

$\frac{1}{2}+2\times \frac{-1}{2}+c=0$

$⟹c=\frac{1}{2}$

$⟹$ $\left[m\right]=\left[h^{1/2}{G}^{-1/2}{C}^{1/2}\right]$