Physics · Lesson 19

Viscosimeter

Fluid Mechanics Stokes' Law Terminal Velocity Drag Force

Drop a steel ball into a column of fluid and time its fall. The slower it falls, the thicker the fluid. This simple technique — Stokes' viscosimetry — lets engineers measure the viscosity of everything from water to motor oil to honey, using only a ball, a ruler, and a stopwatch.

Who Was George Gabriel Stokes?

George Gabriel Stokes (1819–1903) was an Irish mathematician and physicist at Cambridge University. In 1851 he derived the drag force on a sphere moving through a viscous fluid — what we now call Stokes' Law. His work is foundational to fluid mechanics, meteorology (how raindrops fall), and particle physics (how droplets behave in bubble chambers). Robert Millikan later used Stokes' Law in his famous oil-drop experiment to measure the charge of an electron.

Stokes' Law & Terminal Velocity

When a sphere falls through a viscous fluid, three forces act on it: gravity (down), buoyancy (up), and drag (up). At terminal velocity, these forces balance and the sphere moves at a constant speed.

v_t = 2r²(ρ_s − ρ_f)g / 9η

v_t = terminal velocity (m/s)

r = sphere radius (m)

ρ_s = sphere density (kg/m³)

ρ_f = fluid density (kg/m³)

g = 9.8 m/s²

η = dynamic viscosity (Pa·s)

Rearranged to find viscosity from a measured terminal velocity: η = 2r²(ρ_s − ρ_f)g / (9v_t)

Fluid	Viscosity (Pa·s)	Density (kg/m³)
Water (20°C)	0.001	998
Motor oil (SAE 30)	0.10	880
Glycerol	1.40	1260
Honey	10.0	1400

Interactive Viscosimeter

Select a fluid and sphere, then drop the sphere to observe terminal velocity.

Terminal v—

Viscosity η—

Fall Time—

Real-World Applications

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Engine Oil Grades SAE ratings (5W-30, 10W-40) specify viscosity at cold and hot temperatures. Too thick → hard to start; too thin → poor lubrication.

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Blood Viscosity Blood is a non-Newtonian fluid. High viscosity (thick blood) increases stroke risk. Anticoagulants reduce viscosity to protect circulation.

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Food Industry Ketchup, sauces, and dressings must have the right viscosity to pour well. Rheology labs measure consistency at every production batch.

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Industrial Quality Control Paint, adhesives, lubricants, and polymers are all tested for viscosity to ensure consistent application and performance in the field.

Practice Problems

Use v_t = 2r²(ρ_s−ρ_f)g / 9η and Stokes drag F = 6πηrv.

Easy1. Terminal velocity is reached when the drag force plus buoyancy equals the gravitational force. True or False?

Hint: At terminal velocity the net force is zero — gravity = drag + buoyancy.

Easy2. Which fluid has the highest viscosity?

Hint: Honey η ≈ 10 Pa·s, far higher than glycerol (1.4), motor oil (0.1), or water (0.001).

Medium3. Stokes' Law: F = 6πηrv. A sphere of r = 0.002 m falls at v = 0.01 m/s in glycerol (η = 1.4 Pa·s). What is the drag force? (Answer in scientific notation ×10⁻⁴ N, 2 sig figs.)

Hint: F = 6π × 1.4 × 0.002 × 0.01 = 6π × 2.8×10⁻⁵ ≈ 5.28×10⁻⁴ N. Enter 5.28 (value ×10⁻⁴).

Medium4. The same sphere falls in honey (100× more viscous than glycerol). Compared to glycerol, its terminal velocity would be approximately:

Hint: v_t ∝ 1/η. If η increases 100×, terminal velocity decreases 100×.

Challenge5. A steel sphere (r = 1 mm = 0.001 m, ρ_s = 7800 kg/m³) falls in glycerol (η = 1.4 Pa·s, ρ_f = 1260 kg/m³). g = 9.8 m/s². Calculate terminal velocity in m/s (3 sig figs).

Hint: v_t = 2×(0.001)²×(7800−1260)×9.8 / (9×1.4) = 2×10⁻⁶×6540×9.8/12.6 ≈ 0.0102 m/s