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Q.
If the energy, $E = G^p \,h^q \,c^r$, where $G$ is the universal gravitational constant, $h$ is the Planck’s constant and $c$ is the velocity of light, then the values of $p$, $q$ and $r$ are, respectively
AIIMSAIIMS 2010Physical World, Units and Measurements
Solution:
$E=G^{p}\,h^{q}\,c^{r} \ldots\left(i\right)$
$\left[M^{1}L^{2}T^{-2}\right]=\left[M^{-1}L^{3}T^{-2}\right]^{p}\,\left[ML^{2}T^{-1}\right]^{q}\,\left[LT^{-1}\right]^{r}$
$=\left[M^{-p+q}\,L^{3p+2q+r}\,T^{-2p-q-r}\right]$
Applying principle of homogeneity of dimensions, we get
$-p+q=1 \ldots\left(ii\right)$
$3p + 2q + r = 2 \ldots\left(iii\right)$
$- 2 p - q - r = -2 \ldots\left(iv\right)$
Adding $\left(iii\right)$ and $\left(iv\right)$,
we get $p + q = 0 \ldots\left(v\right)$
Adding $\left(ii\right)$ and $\left(v\right)$,
we get $q=\frac{1}{2}$
From $\left(ii\right)$, we get $p=q-1$
$=\frac{1}{2}-1=-\frac{1}{2}$
From $\left(ii\right)$, we get $-\frac{3}{2}+1+r=2$, $r=\frac{5}{2}$