A 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 ρ and takes a spherical shape of ra

Question

A 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 ρ and takes a spherical shape of radius R . When the gas is allowed to flow freely out of it, its radius changes from R to 0 (zero) in time T . If the speed η ( r ) of gas coming out of the balloon depends on r as r α and T ∝ S α A β ρ γ R δ thenNEET 2025

TestHub · Accepted Answer

​ T → Time [ T 1 ] S → Surface Tension S = ℓ F ​ = [ L 1 ] [ M 1 L 1 T − 2 ] ​ = [ M 1 T − 2 ] A → Area [ L 2 ] ρ → density [ M 1 L − 3 ] R → Radius [ L 1 ] With Dimensional Analysis → T ∝ S α A β ρ γ R δ [ M 0 L 0 T 1 ] = [ MT − 2 ] α [ L 2 ] β [ ML − 3 V γ [ L ] δ [ M 0 L 0 T 1 ] = [ M α + γ , L 2 β + δ − 3 γ , T − 2 α ] ⇒ − 2 α = 1 ⇒ α = − 2 1 ​ ⇒ α + γ = 0 2 β + γ − 3 γ = 0 2 β + δ − 3 ( 2 1 ​ ) = 0 2 β + δ = 2 3 ​ ​

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