A sphere of radius 1 mm moves at terminal speed through a liquid. The density of the sphere exceeds the liquid density by 4500 kg/m³. Take g = 10 m/s². If the sphere falls through a further distance 2 m, the viscous heat produced is Nπ × 10⁻⁵ J. Find N.
Physics · Properties of solid and fluid · viscosity and terminal velocity
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Question

A sphere of radius 1 mm moves at terminal speed through a liquid. The density of the sphere exceeds the liquid density by 4500 kg/m³. Take g = 10 m/s². If the sphere falls through a further distance 2 m, the viscous heat produced is Nπ × 10⁻⁵ J. Find N.

Physics · Properties of solid and fluid · viscosity and terminal velocity

G1 Physics · Accepted Answer

At terminal speed, viscous drag equals the effective weight: F(viscous) = (4πr³/3)Δρg. Substitute: r = 10⁻³ m, Δρ = 4500 kg/m³, g = 10 m/s². Therefore, F(viscous) = (4π/3) ×10⁻⁹ ×4500 ×10. Now, 4500×10×10⁻⁹ = 4.5×10⁻⁵. Thus, F(viscous) = (4π/3)(4.5×10⁻⁵) = 6π×10⁻⁵ N. Heat produced over distance H = 2 m: Q = F(viscous)H = 6π×10⁻⁵×2 = 12π×10⁻⁵ J. Hence, N = 12.