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Theme A — Space, Time and Motion

A.2.9 Stoke’s Law — IB Physics HL Revision Notes & Topical Questions

5 min read 11 parts Lesson 20 of 63 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Lesson summary

A.2.9 Stoke's Law — IB PHYSICS SL notes: State Stokes' law for the drag on a small sphere in a viscous fluid and use it to find terminal speed.Viscous drag = the frictional force between an object and a fluid that opposes the motion between them.In air the drag is air resistance.…

  • Size of the force
  • Laminar flow
  • Dimension check
  • Temperature
  • Small, smooth sphere
  • Large fluid

Part 1 of 9Viscous drag

1Viscous drag
Viscous drag the frictional force between an object and a fluid that opposes the motion between them.
  • In air the drag is air resistance.
2Size of the force
  • Size of the force it depends on the speed, size and shape of the object.
3Laminar flow
  • Laminar flow at low speed the fluid flows smoothly past the object and the drag is proportional to the speed.

Part 2 of 9Stokes' law

Equation
  • Fd = 6πηrv for a small sphere.
  • Fd is the viscous drag force in N.
  • η is the fluid viscosity in Pa s (N s m−2).
  • r is the radius of the sphere in m.
  • v is the speed of the sphere through the fluid in m s−1.
Dimension check
  • η has unit kg m−1 s−1, so ηrv has unit kg m s−2, a force.

Part 3 of 9Viscosity

1Viscosity
  • Viscosity how much a fluid resists flowing, its "thickness".
  • Water flows more easily than ketchup, so it has lower viscosity.
2Flow rate
  • Flow rate the rate of flow is inversely proportional to the coefficient of viscosity.
3Values
  • Values honey about 10 Pa s, water about 1 × 10−3 Pa s and air about 2 × 10−5 Pa s.
4Temperature
  • Temperature the viscosity of liquids falls sharply as temperature rises, so warm syrup pours more easily.

Part 4 of 9Conditions for validity

1Small, smooth sphere
  • Small, smooth sphere the law applies to a small smooth sphere.
2Laminar flow
  • Laminar flow the speed must be low, with a small Reynolds number.
3Large fluid
  • Large fluid there are no wall effects and the viscosity is uniform.
4High speed
  • High speed flow becomes turbulent and drag varies roughly as v2, so the formula fails.
5Where it works
  • Where it works a ball bearing falling slowly in oil, tiny droplets in mist and sediment grains in water.

Part 5 of 9Terminal speed of a sphere

Forces at terminal speed
  • Forces at terminal speed the weight Ws balances the drag plus the buoyancy force, Ws = Fd + Fb.
Weight of the sphere
  • Ws = (4/3)πr3ρsg.
Buoyancy
  • Fb = (4/3)πr3ρfg, the weight of the fluid displaced.
Balance
  • (4/3)πr3ρsg = 6πηrv + (4/3)πr3ρfg.
Result
  • Result vT = 2r2(ρs − ρf)g / (9η).

Part 6 of 9What terminal speed depends on

Radius
  • Radius vT is directly proportional to r2, so doubling the radius gives four times the speed.
Viscosity
  • Viscosity vT is inversely proportional to η.
Density difference
  • Density difference a larger ρs − ρf gives a greater speed.
Rising spheres
  • Rising spheres if ρs is less than ρf the sphere rises, as air bubbles do in water.

Part 7 of 9Motion before terminal speed

At release
  • At release the speed is zero, so the drag is zero and the initial acceleration is /mass.
    • (weight − upthrust)
As speed grows
  • As speed grows the drag rises, the resultant falls and the acceleration decreases to zero.
v–t graph
  • v–t graph it rises with a decreasing gradient and approaches a horizontal asymptote at vT.

Part 8 of 9Measuring viscosity

1
Drop small ball bearings of known radius and density into a tall transparent tube of the liquid.
2
Time the fall between two marks well below the surface so that terminal speed has been reached.
3
Calculate vT and then η = 2r2(ρs − ρf)g / (9vT).
4
A graph of vT against r2 is a straight line through the origin.
  • Uses
    • sedimentation of particles in water and blood, droplets in clouds and the Millikan oil-drop experiment.

Part 9 of 9More from the specification

1Stoke's Law for Viscous Drag
  • For a small spherical particle of radius r moving through a viscous fluid at velocity v under laminar flow, the viscous resistive force acting on it is given by Stoke's law: Fd = 6 * pi * eta * r * v,
  • where eta is fluid viscosity.
2Terminal Velocity Equilibrium
  • At terminal velocity, the downward weight of a falling sphere is exactly equal in magnitude to the upward sum of viscous drag and buoyant upthrust,
  • resulting in zero net force and constant linear velocity.
3Momentum and Newton's Second Law
  • An object traveling at constant velocity maintains constant linear momentum (p = mv);
    • according to Newton's second law, a non-zero resultant force is required to produce a time-rate of change of momentum.
4Circular Motion and Centripetal Acceleration
  • An object moving in a circle at constant speed has constant angular speed but constantly changing direction, experiencing a centripetal acceleration a = omega2 * r directed toward the circle centre.
5Normal reaction
  • The contact force perpendicular to a surface.
6Reaction force
  • Force from a support pushing up on an object.
7Contact force
  • Force requiring touching objects.
8Normal
  • Line at 90° to a surface.
9Tension
  • Force transmitted along a string or rope.
10Centripetal force
  • Net force toward the centre, mv2/r.
11Static friction
  • Friction between surfaces at rest relative to each other; fs ≤ μsFN.
12Circular motion
  • Motion along a circular path with a force towards the centre.
13Kinetic energy
  • K = ½mv2, energy of motion.
14Angular speed
  • Magnitude of the angular velocity.
15Centripetal acceleration
  • Ac = v2/r toward the centre.
16Momentum
  • P = mv, a vector.
17Velocity
  • Rate of change of displacement.
18Newton
  • Unit of force, kg m s-2.
19Constant
  • Not changing.
20Collision
  • Short interaction with large internal forces.
21Linear
  • A graph that is a straight line through the origin, so resistance is constant.
22Impulse
  • J = FΔt = Δp.
    What the symbols mean: F = force (N) · Δt = time the force acts (s) · Δp = change in momentum (kg m s−1); FΔt is the impulse (N s)

Key terms

TermMeaning
Viscous dragFrictional force between an object and a fluid opposing their relative motion
Viscosity (η)Measure of a fluid's resistance to flow, in Pa s
Stokes' lawFd = 6πηrv, the drag on a small sphere moving slowly through a viscous fluid
Laminar flowSmooth flow of a fluid in parallel layers
Terminal speedConstant speed reached when drag plus upthrust balances the weight
Reynolds numberQuantity comparing inertial and viscous effects in a flow

Avoid these mistakes

1
Using Stokes' law for fast or turbulent flow — it applies only to laminar flow at low speed.
2
Forgetting the buoyancy force in the force balance — include upthrust with the drag.
3
Using the density of the sphere for buoyancy — buoyancy uses the fluid density.
4
Thinking vT is proportional to r — it is proportional to r2.
5
Mixing up the sphere radius with the diameter — use the radius r in 6πηrv.

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