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PhysicsNEET 2025SI units, dimensions and dimensional analysis

Units and Measurements

A balloon is made of a material of surface tension SS and its inflation outlet (from where gas is filled in it) has a small area AA. It is filled with a gas of density ρ\rho and takes a spherical shape of radius RR. When the gas is allowed to flow freely out of it, its radius rr changes from RR to 0 in time TT. If the speed v(r)v(r) of the gas coming out of the balloon depends on rr as rar^a and T∝SαAβργRδT \propto S^\alpha A^\beta \rho^\gamma R^\delta, then:

Question 24 as printed in the NEET 2025 booklet

  1. A.Option set (1)
  2. B.Option set (2)
  3. C.a=−12, α=−12, β=−1, γ=12, δ=72a = -\tfrac12,\ \alpha = -\tfrac12,\ \beta = -1,\ \gamma = \tfrac12,\ \delta = \tfrac72✓
  4. D.Option set (4)

Solution

The excess pressure 4SR\dfrac{4S}{R} drives the outflow, so v∼Sρrv \sim \sqrt{\dfrac{S}{\rho r}}, giving a=−12a = -\tfrac12. Time to empty ∼volumeAv∼R3AρRS=S−1/2A−1ρ1/2R7/2\sim \dfrac{\text{volume}}{Av} \sim \dfrac{R^3}{A}\sqrt{\dfrac{\rho R}{S}} = S^{-1/2} A^{-1} \rho^{1/2} R^{7/2}: α=−12\alpha = -\tfrac12, β=−1\beta = -1, γ=12\gamma = \tfrac12, δ=72\delta = \tfrac72.

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