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 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 ∝ S^α A^β ρ^γ R^δ then:

पर्याय

• `a = 1/2, α = 1/2, β = -1, γ = +1, δ = 3/2`

• `a = -1/2, α = -1/2, β = -1, γ = -1/2, δ = 5/2`

• `a = -1/2, α = -1/2, β = -1, γ = 1/2, δ = 7/2`

• `a = 1/2, α = 1/2, β = -1/2, γ = 1/2, δ = 7/2`

Answer 1

`a = -1/2, α = -1/2, β = -1, γ = 1/2, δ = 7/2`

Explanation:

Surface tension S = `F/l`

[S] = [M¹ T⁻²]

T ∝ S^α A^β ρ^γ R

[S] = [M¹ T⁻²]^α [L²]^β [M¹L⁻³]^γ [L]^δ

= [M]^{α + γ} [L]^2β−3γ+δ [T]^−2α

Comparing the powers, we get

α + γ = 0, 2β − 3γ + δ = 0, −2α = 1

`α = -1/2 and γ = -α = 1/2`

2β − 3γ + δ = 0

`2β − 3(1/2) + δ = 0`

`2β − 3/2 + δ = 0`

`2β + δ = 3/2`

`β = −1 and δ = 7/2` satisfy the above equation.