State Stokes' law. Derive an expression for the terminal velocity of a small sphere falling through a viscous liquid.

1. 1 · State Stokes' law. A small sphere of radius r moving with speed v in a fluid of viscosity η feels a viscous force F = 6πηrv.
2. 2 · Draw the forces. A sphere in a liquid: W downward; B and F upward.
3. 3 · Write each force. W = (4/3)πr³ρg; B = (4/3)πr³σg; F = 6πηrv.
4. 4 · Condition for terminal velocity. F grows with speed until W = B + F; then the speed v₀ stays constant.
5. 5 · Solve. 6πηrv₀ = (4/3)πr³(ρ − σ)g, so v₀ = 2r²(ρ − σ)g / 9η.
6. 6 · Real life. Raindrops reach terminal velocity, so they fall gently; a parachute uses the same idea.
7. 7 · Close in one line. Terminal velocity grows with the square of the radius.

Scoring tip: Always include the buoyant force. Leaving it out gives ρ in place of (ρ − σ) and loses a mark.