To check the dimensional correctness of the equations, let's find the dimensions of each physical quantity involved: Dimension of time period ($T$) = $[M^0 L^0 T^1]$ Dimension of radius/distance ($R$) = $[M^0 L^1 T^0]$ Dimension of mass ($M$) = $[M^1 L^0 T^0]$ Dimension of universal gravitational constant ($G$) = $[M^{-1} L^3 T^{-2}]$

Now, let's check the dimensions of the right-hand side (RHS) of the first option, $T = 2\pi \sqrt{\frac{R^3}{GM}}$ (ignoring the dimensionless constant $2\pi$): $\text{Dimension of RHS} = \sqrt{\frac{[L]^3}{[M^{-1} L^3 T^{-2}][M]}}$ $= \sqrt{\frac{[L^3]}{[M^0 L^3 T^{-2}]}}$ $= \sqrt{[T^2]} = [T^1]$

Since the dimension of the LHS ($[T]$) is equal to the dimension of the RHS ($[T]$), the equation $T = 2\pi \sqrt{\frac{R^3}{GM}}$ is dimensionally correct.