A sphere of radius 3.800 × 10^-4 m and density 7800 kg m⁻³ falls at terminal speed 0.003995437 m s⁻¹ in a liquid of density 1200 kg m⁻³. Use g = 9.81 m s⁻². Assume slow laminar flow, a wide vessel and Stokes drag F_d = 6πηrv. The sphere volume is 4πr³/3. Determine the dynamic viscosity η, predict the terminal speed for radius 2r under the same model, and explain two experimental precautions.

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η = 0.52 Pa s; doubled-radius terminal speed 0.01598 m s⁻¹, provided Stokes flow still applies. At terminal speed the acceleration is zero, so the downward weight balances both buoyancy and viscous drag. The driving force after buoyancy is (ρ_s − ρ_l)Vg, not simply the sphere's weight. Substitute V = 4πr³/3 into (ρ_s − ρ_l)Vg = 6πηrv. Cancelling common factors gives η = 2r²(ρ_s − ρ_l)g/(9v). Using the measured speed gives η = 0.52 Pa s. The units kg m⁻¹ s⁻¹ are equivalent to pascal-seconds, as required for dynamic viscosity. For unchanged fluid and sphere material, v_terminal is proportional to r². Doubling radius predicts 0.01598 m s⁻¹, but this prediction is conditional on laminar Stokes flow still being valid. Time the sphere only after it has travelled far enough to reach constant speed, checking equal transit times over equal intervals. Use a sufficiently wide vessel and keep the sphere away from walls, since wall effects alter the drag.

Measure viscosity with a falling sphere · set 4

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