A spherical balloon has surface tension S, outlet area A, gas density ρ and initial radius R. Gas escapes and the balloon shrinks to zero radius in time T. Let v(r) ∝ rᵃ and T ∝ Sᵅ Aᵝ ρᵞ Rᵟ. Identify (a, α, β, γ, δ).
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A spherical balloon has surface tension S, outlet area A, gas density ρ and initial radius R. Gas escapes and the balloon shrinks to zero radius in time T. Let v(r) ∝ rᵃ and T ∝ Sᵅ Aᵝ ρᵞ Rᵟ. Identify (a, α, β, γ, δ).

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G1 Physics · Accepted Answer

Step 1: The balloon’s excess pressure scales as ΔP ∝ S/r. By energy balance for the exiting gas, (1/2)ρv² ∝ ΔP. Step 2: Hence v(r) ∝ [S/(ρr)]¹ᐟ² = S¹ᐟ²ρ⁻¹ᐟ²r⁻¹ᐟ², so a = −1/2. Step 3: Spherical volume V = (4/3)πr³, so dV/dt = 4πr²(dr/dt). Gas leaves through area A with volume rate Av. Step 4: In magnitude, 4πr²|dr/dt| = Av ∝ A S¹ᐟ²ρ⁻¹ᐟ²r⁻¹ᐟ². Step 5: Therefore dt ∝ (ρ/S)¹ᐟ² r⁵ᐟ² dr/A. Step 6: Integrate radius from 0 to R: T ∝ S⁻¹ᐟ² A⁻¹ ρ¹ᐟ² R⁷ᐟ². Final answer: a = −1/2, α = −1/2, β = −1, γ = 1/…