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Physics · PYQ NEET UG 2025
हिन्दीA balloon is made of a material of surface tension \( S \) and its inflation outlet (from where gas is filled in it) has small area \( A \). It is filled with a gas of density \( \rho \) and takes a spherical shape of radius \( R \). When the gas is allowed to flow freely out of it, its radius \( r \) changes from \( R \) to \( 0 \) (zero) in time \( T \). If the speed \( v(r) \) of gas coming out of the balloon depends on \( r \) as \( r^a \) and \( T \propto S^\alpha A^\beta \rho^\gamma R^\delta \) then
Correct answer · Option 3
\( a = -\frac{1}{2}, \alpha = -\frac{1}{2}, \beta = -1, \gamma = \frac{1}{2}, \delta = \frac{7}{2} \)
Explanation
Using excess pressure \( P = \frac{4S}{r} \) and Bernoulli's equation \( v = \sqrt{\frac{2P}{\rho}} \), we find \( v \propto r^{-1/2} \), hence \( a = -1/2 \). Using dimensional analysis for time \( T = \int \frac{4\pi r^2 dr}{A v(r)} \), we obtain the correct power exponents.
