Question

If gravitational constant $\left(G\right),$ planks constant $\left(h\right)$ and speed of light $\left(c\right)$ considered as fundamental physical quantities then find dimensional formula of time

• A. $\left[G^{\frac{1}{2}}{h}^{\frac{1}{2}}{c}^{\frac{-5}{2}}\right]$
• B. $\left[G^1{h}^2{c}^{\frac{-5}{2}}\right]$
• C. $\left[G^{\frac{1}{2}}{h}^{\frac{1}{2}}{c}^{-2}\right]$
• D. $\left[G^2{h}^{\frac{1}{2}}{c}^{-2}\right]$

Answer 1

Answer: (A)

Let
$\left[T\right]={\left[G\right]}^{\alpha }{\left[h\right]}^{\beta }{\left[C\right]}^{\gamma }$ . . . . . . . . . . . . . .(1)
$={\left[M^{-1}{L}^3{T}^{-2}\right]}^{\alpha }\cdot {\left[M{L}^2{T}^{-1}\right]}^{\beta }{\left[L{T}^{-1}\right]}^{\gamma }$
$\Rightarrow -\alpha +\beta =0$
$3\propto +2\beta +\gamma =0$
$-2\alpha -\beta -\gamma =1$
On solving we get
$\alpha =\frac{1}{2},\beta =\frac{1}{2},\gamma =-\frac{5}{2}$
Substitute the values in equation (1), we get
$\left[T\right]=\left[G^{\frac{1}{2}}{h}^{\frac{1}{2}}{C}^{-\frac{5}{2}}\right]$