IB Physics HLTopic 1 — Motion, Forces & EnergyPaper 1 & 2Viscous drag on a sphere~8 min read
Stokes’ Law
Drop a ball bearing into honey and it sinks slowly; drop it into water and it drops fast. The difference is viscous drag — the resistance a fluid puts up as an object moves through it. For a small sphere moving slowly, that drag follows a clean equation called Stokes’ law, built from just the sphere’s size, its speed, and the fluid’s thickness.
📘 What you need to know
Viscous drag is the frictional force between an object and a fluid, opposing their relative motion
For a small sphere moving slowly through a fluid, the drag is given by Stokes’ law: Fd = 6πηrv
η is the fluid’s viscosity — how thick or resistant to flow it is (units N s m−2 or Pa s)
The drag grows with the sphere’s radiusr and its speedv through the fluid
A low-viscosity fluid (like water) flows easily; a high-viscosity one (like honey) resists flowing
Stokes’ law only applies to a small sphere moving at low speed (smooth, streamlined flow)
Viscous drag
Viscous drag is defined as:
Viscous drag
The frictional force between an object and a fluid that opposes their relative motion
It’s the same resistive idea as surface friction, but now the “surface” is a fluid — a liquid or a gas. When the fluid is air, this drag is just air resistance. As the sphere pushes through, it has to shove fluid out of the way, and the fluid pushes back. The faster it goes and the bigger it is, the more drag it feels.
Stokes’ law equation
For a small sphere moving slowly through a fluid, the viscous drag is:
Stokes’ lawFd = 6πηrv
where Fd is the viscous drag force (N), η is the fluid’s viscosity (N s m−2, also written Pa s), r is the radius of the sphere (m), and v is its speed through the fluid (m s−1). The drag is directly proportional to all three of η, r, and v — double any one of them and the drag doubles.
As the sphere moves through the fluid (velocity v, green), the fluid streams smoothly around it and resists its motion with a viscous drag Fd (red), given by 6πηrv.
What viscosity means
Viscosity (η) is a measure of how much a fluid resists flowing — you can think of it as the fluid’s “thickness.” A fluid with low viscosity pours easily, like water; one with high viscosity pours slowly and reluctantly, like honey or ketchup. The higher the viscosity, the more drag a moving object feels, which is exactly why the ball bearing sinks so much more slowly through honey than through water.
Viscosity is a property of the fluid at a given temperature. The size of the drag force also depends on the object’s speed, its size, and its shape — Stokes’ law captures the specific case of a slow-moving sphere.
Stokes’ law comes with fine print: it only works for a small sphere moving slowly, where the fluid flows past it in smooth, orderly layers (called laminar or streamlined flow). Speed the sphere up, or make it big and irregular, and the flow becomes turbulent — swirling and chaotic — and the neat 6πηrv relationship breaks down. In an exam, if you’re given a sphere and a slow speed, Stokes’ law is your tool.
WE 1
A small bead of radius 0.50 mm moves through water at a steady speed of 0.010 m s−1. The viscosity of water is 1.0 × 10−3 Pa s. Calculate the viscous drag on the bead.
Step 1 — list the quantities in SI units
r = 0.50 mm = 5.0 × 10⁻⁴ m
v = 0.010 m/s, η = 1.0 × 10⁻³ Pa s
Step 2 — apply Stokes’ lawFd = 6πηrv
= 6π × (1.0×10⁻³) × (5.0×10⁻⁴) × 0.010
Fd = 9.4 × 10⁻⁸ NConvert the radius to metres first — 0.50 mm is 5.0 × 10⁻⁴ m, not 0.50 m. Tiny drag, as you’d expect for a tiny bead.
🛠️ Using Stokes’ law
Convert to SI units: radius in metres, speed in m s−1, viscosity in Pa s.
Check it’s a sphere at low speed — only then does Stokes’ law apply.
Substitute intoFd = 6πηrv, or rearrange for the unknown.
To find speed:v = Fd ÷ (6πηr). To find radius:r = Fd ÷ (6πηv).
WE 2
A bead of radius 1.5 mm falls through a thick oil of viscosity 0.80 Pa s. At one instant the viscous drag on it is 1.1 × 10−4 N. Calculate the bead’s speed at that instant.
Step 1 — rearrange Stokes’ law for speedv = Fd ÷ (6πηr)Step 2 — convert and substitute
r = 1.5 mm = 1.5 × 10⁻³ m
v = (1.1×10⁻⁴) ÷ (6π × 0.80 × 1.5×10⁻³)v = 4.9 × 10⁻³ m s⁻¹A slow crawl — just millimetres per second — which is exactly the low-speed regime where Stokes’ law holds.
💡 Top tips
Convert the radius to metres. Millimetre-sized beads are common here — 1.5 mm is 1.5 × 10−3 m.
Watch for diameter vs radius. If a question gives the diameter, halve it before substituting.
Drag rises with speed. As a falling sphere speeds up, the drag grows — the basis of terminal velocity.
Stokes’ law is for slow spheres only. Fast or non-spherical objects need a different treatment.
Quick recap: viscous drag opposes an object’s motion through a fluid. For a small, slow-moving sphere it’s given by Stokes’ law, Fd = 6πηrv, growing with the fluid’s viscosity, the sphere’s radius, and its speed. Viscosity measures how much a fluid resists flowing — low for water, high for honey.
⚠ Common mistakes
Leaving the radius in mm instead of converting to metres
Using the diameter where the equation needs the radius
Applying Stokes’ law to fast or non-spherical objects, where the flow is turbulent and the equation fails
Forgetting the 6π factor, or the factor of π entirely
Mixing up viscosity units — N s m−2 and Pa s are the same thing
Viscous drag grows as a sphere speeds up. So a sphere falling through a fluid keeps accelerating only until the drag (plus buoyancy) balances its weight — after which it falls at a steady terminal velocity. Understanding what holds an object up in a fluid comes next: Buoyancy, and how it combines with drag to set that terminal speed.
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