Question

If the units of velocity of light '$C$', Gravitational constant '$G$' and Planck's Constant '$h$' are taken as fundamental units, the dimensional formula for Mass in the new system will be :

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

Answer 1

Answer: (C)

$\left[C^{\frac{1}{2}}{G}^{-\frac{1}{2}}{h}^{\frac{1}{2}}\right]$

$M\propto C^{\alpha }{G}^{\beta }{h}^{\gamma }$

$M=M^1{L}^0{T}^0,C=L{T}^{-1},G=M^{-1}{L}^3{P}^{-2}$

$h=M^1{L}^2{T}^{-1}$

$M^1\propto \left(L{T}^{-1}{)}^{\alpha }\right(M^{-1}{L}^3{T}^{-2}{)}^{\beta }(M^1{L}^2{T}^{-1}{)}^{\gamma }$

$-\beta +\gamma =1$

$\alpha +3\beta +2\gamma =0$ $\Rightarrow \alpha =\frac{1}{2}=\gamma$

$-\alpha -2\beta -\gamma =0$

$\beta =-\frac{1}{2}$