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Applying the principle of homogeneity of dimensions, determine which one is correct. where $\mathrm{T}$ is time period, $\mathrm{G}$ is gravitational constant, $M$ is mass, $r$ is radius of orbit.
$\mathrm{T}^2=\frac{4 \pi^2 \mathrm{r}}{\mathrm{GM}^2}$
$\mathrm{T}^2=4 \pi^2 \mathrm{r}^3$
$\mathrm{T}^2=\frac{4 \pi^2 \mathrm{r}^3}{G M}$
$\mathrm{T}^2=\frac{4 \pi^2 \mathrm{r}^2}{G M}$
Solution
According to principle of homogeneity dimension of $LHS$ should be equal to dimensions of RHS so option $(3)$ is correct.
$\mathrm{T}^2=\frac{4 \pi^2 \mathrm{r}^3}{\mathrm{GM}}$
${\left[\mathrm{T}^2\right]=\frac{\left[\mathrm{L}^3\right]}{\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right][\mathrm{M}]}}$
(Dimension of G is $\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]$ )
$\left[\mathrm{T}^2\right]=\frac{\left[\mathrm{L}^3\right]}{\left[\mathrm{L}^3 \mathrm{~T}^{-2}\right]}=\left[\mathrm{T}^2\right]$
