Two solid spherical balls of density σ and radii r1 and r2 (< r1) are tied up with a long ideal string and released in a viscous liquid column of lesser density ρ with the string just taut as shown. The tension in the string, when terminal velocity is attained, is

Engineering

Physics

Acceleration

Buoyancy

Viscosity and Terminal Velocity New

Question

Two solid spherical balls of density σ and radii r₁ and r₂ (< r₁) are tied up with a long ideal string and released in a viscous liquid column of lesser density ρ with the string just taut as shown. The tension in the string, when terminal velocity is attained, is

$\frac{2}{3} π (r_{2}^{3} - r_{1}^{3}) (ρ - g) g$

$\frac{4}{3} π (r_{2}^{3} - r_{1}^{3}) (σ - ρ) g$

$\frac{4}{3} {πr}_{1} r_{2} (r_{1} - r_{2}) (σ - ρ) g$

$\frac{4}{3} π (\frac{r_{2}^{4} - r_{1}^{4}}{r_{2} - r_{1}}) (σ - ρ) g$

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$T + \frac{4}{3} {πr}_{2}^{3} σg - 6 {πηr}_{2} v = 0$

$- \frac{4}{3} {πr}_{2}^{3} ρg$

$\frac{4}{3} {πr}_{1}^{3} σg - \frac{4}{3} {πr}_{1}^{3} ρg - 6 {πηr}_{1} v - T = 0$

$\frac{4}{3} π (σ - ρ) g (r_{1}^{3} + r_{2}^{3}) = 6 πη (r_{1} + r_{2}) v$

$\Rightarrow T = \frac{4}{3} {πr}_{1}^{3} (σ - ρ) g - 6 {πηr}_{1} (\frac{4}{3} π (σ - ρ) g) (r_{1}^{3} + r_{2}^{3}) (r_{1} + r_{2})$

$= \frac{4}{3} π (σ - ρ) [\frac{r_{1}^{4} + r_{1}^{3} r_{2} - (r_{1}^{4} + r_{1} r_{2}^{3})}{(r_{1} + r_{2})}]$