Applications of Dimensional Analysis question from NEET 2025

Answer: (a)

1. Excess pressure $\Delta P \propto \dfrac{S}{r}$; Bernoulli $\frac{1}{2}\rho v^2 = \Delta P$ gives $v \propto \sqrt{\dfrac{S}{\rho r}}$, so $a = -\frac{1}{2}$.
2. Volume outflow: $-4\pi r^2\dfrac{dr}{dt} = A v \propto A\sqrt{\dfrac{S}{\rho}}\,r^{-1/2}$.
3. $dt \propto \dfrac{1}{A}\sqrt{\dfrac{\rho}{S}}\,r^{5/2}dr$; integrating from $R$ to 0: $T \propto S^{-1/2}A^{-1}\rho^{1/2}R^{7/2}$.
4. So $\alpha = -\frac{1}{2}$, $\beta = -1$, $\gamma = \frac{1}{2}$, $\delta = \frac{7}{2}$: the answer is (a).