A sphere of radius 1 mm and density 5500 kg/m³ falls at terminal speed through a liquid of density 1000 kg/m³ and viscosity 1 Pa·s. Take g = 10 m/s². If the rate of heat generation by viscous dissipation is Nπ × 10⁻⁷ W, find N.
Physics · Properties of solid and fluid · viscosity and terminal velocity
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Question

A sphere of radius 1 mm and density 5500 kg/m³ falls at terminal speed through a liquid of density 1000 kg/m³ and viscosity 1 Pa·s. Take g = 10 m/s². If the rate of heat generation by viscous dissipation is Nπ × 10⁻⁷ W, find N.

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

G1 Physics · Accepted Answer

First calculate terminal speed: vₜ = 2r²(ρₛ−ρₗ)g/ (9η). Using r = 10⁻³ m, ρₛ−ρₗ = 4500 kg/m³, g = 10, η = 1, we obtain vₜ = 0.01 m/s. Stokes drag is F(viscous) = 6πηrvₜ = 6π×1×10⁻³×10⁻² = 6π×10⁻⁵ N. Power dissipated as heat: P = F(viscous)vₜ = 6π×10⁻⁵×10⁻² = 6π×10⁻⁷ W. Hence, N = 6.