Q. 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
NTA AbhyasNTA Abhyas 2022
Solution:
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
$\left[T\right]=\left[G^{\frac{1}{2}} h^{\frac{1}{2}} C^{- \frac{5}{2}}\right]$
$\left[T\right]=\left[G\right]^{\alpha }\left[h\right]^{\beta }\left[C\right]^{\gamma }$ . . . . . . . . . . . . . .(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
$\left[T\right]=\left[G^{\frac{1}{2}} h^{\frac{1}{2}} C^{- \frac{5}{2}}\right]$