NEET 2025 · PhysicsExcess pressurePrevious Year Question
A balloon is made of a material of surface tension S and its inflation outlet (from where gas is filled) has small area A. It is filled with a gas of density ρ and takes a spherical shape of radius R. When the gas flows freely out, its radius r changes from R to 0 in time T. If the speed v(r) of gas coming out depends on r as rᵅ and T ∝ Sᵝ Aᵞ Rᵟ ρᵋ, then:
- A.α = −1/2, β = −1/2, γ = −1, δ = 7/2, ε = 1/2✓
- B.α = 1/2, β = 1/2, γ = −1/2, δ = 1/2, ε = 7/2
- C.α = 1/2, β = 1/2, γ = −1, δ = 1, ε = 3/2
- D.α = −1/2, β = −1/2, γ = −1, δ = −1/2, ε = 5/2
Correct Answer
(A) α = −1/2, β = −1/2, γ = −1, δ = 7/2, ε = 1/2
Solution & Explanation
Excess (Laplace) pressure ∝ S/r drives efflux: ½ρv² ~ S/r ⇒ v ∝ (S/(ρr))^{1/2}, so α = −1/2. Volume outflow rate A·v empties the balloon: T ~ R³/(A·v(R)) with v(R) ∝ (S/(ρR))^{1/2}. Dimensional/scaling bookkeeping gives β = −1/2, γ = −1, δ = 7/2, ε = 1/2 (option a).
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