Stokes' Law & Terminal Velocity
A falling ball speeds up until drag and buoyancy balance its weight.
Lesson at a glanceShow the key points
- A sphere of radius r moving slowly at v through a fluid of viscosity η feels a drag F = 6πηrv (14.22) (form found by dimensions; 6π from full theory).
- A body dropped in a fluid accelerates until drag + buoyancy = weight, then falls at constant terminal velocity.
- For a sphere: v_t = 2r²(ρ − σ)g/(9η) (14.23), with ρ the sphere's density and σ the fluid's.
- Stokes' method: time a ball between equally spaced marks; once the times are equal it is at terminal velocity; η = 2r²(ρ − σ)g t/(9d).
Drop a steel ball into a jar of honey and it sinks at a slow, perfectly steady speed. Drop it in air and it keeps speeding up. Viscous drag, growing with speed, makes the difference.
Drag on a sphere: Stokes’ law
A solid moving through a fluid drags the neighbouring fluid with it, setting up velocity gradients, so viscous forces oppose its motion. For a sphere of radius r moving slowly at speed v through a fluid of viscosity η, assume :
M: 1 = c. L: 1 = a + b − c. T: −2 = −b − c. So a = b = c = 1, and with the full theory giving k = 6π:
Terminal velocity
A sphere dropped into a fluid feels its weight W (down), the buoyant force B (up) and the viscous drag F (up).
- W and B are constant; F grows with speed.
- At first v = 0, so F = 0 and the sphere accelerates under W − B.
- As it speeds up, F grows until F = W − B. The net force is then zero and the sphere falls at a constant terminal velocity.
With the sphere’s density ρ and the fluid’s σ:
A steel ball (ρ = 7800 kg/m³) of radius 1.0 mm falls through glycerine (σ = 1260 kg/m³, η = 1.5 N·s/m²). Find its terminal velocity.
An air bubble of radius 1.0 mm rises through castor oil (σ = 960 kg/m³, η = 0.99 N·s/m²). Find its steady speed (air’s density is negligible).
Measuring viscosity: Stokes’ method
A tall tube of the test liquid sits in a water bath at a known temperature, with three equally spaced marks P, Q, R well below the top. Drop a small metal ball in through a narrow guide tube and time it over PQ and over QR. If the two times differ, the ball hasn’t reached terminal velocity before P; try a smaller ball. When the times are equal, . Measure the ball’s radius (screw gauge) and mass, so ρ = m/(4/3 πr³), and then
- 1Stokes’ law: for a slowly moving sphere.
- 2A falling body reaches terminal velocity when drag + buoyancy = weight.
- 3: bigger, denser balls in thinner liquids fall faster.
- 4Stokes' method: equal times over equal distances ⇒ terminal velocity ⇒ η.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.