Question

Two solid spherical balls of radius r₁ and r₂ (r₂ < r₁), of density σ are tied up with a long string and released in a viscous liquid column of lesser density ρ and coefficient of viscosity η, with the string just taut as shown. The terminal velocity of spheres is ______.

Options

• `2/9("r"_2^2"g")/eta`(σ - ρ)

• `2/9("r"_1^2"g")/eta`(σ - ρ)

• `2/9(("r"_1^3+"r"_2^3))/("r"_1-"r"_2)((sigma-rho)"g")/eta`

• `2/9(("r"_1^3+"r"_2^3))/("r"_1-"r"_2)((sigma-rho)"g")/eta`

Answer 1

Two solid spherical balls of radius r₁ and r₂ (r₂ < r₁), of density σ are tied up with a long string and released in a viscous liquid column of lesser density ρ and coefficient of viscosity η with the string just taut as shown. The terminal velocity of spheres is `bb(2/9(("r"_1^3+"r"_2^3))/("r"_1-"r"_2)((sigma-rho)"g")/eta)`.

Explanation:

At terminal velocity net force is zero

`6pieta("r"_1+"r"_2)"V"_"T"+4/3pi("r"_1^3+"r"_2^3)rho"g"`

= `4/3pi("r"_1^3+"r"_2^3)`σg

= v = `((6-rho)4/3("r"_1^3+"r"_2^3)g)/(6eta("r"_1+"r"_2))`

= `2/9(("r"_1^3+"r"_2^3))/("r"_1-"r"_2)((sigma-rho)"g")/eta`