CP2 – Viscosity by Falling Ball | Edexcel A Level Physics

Theory & Background

Stokes' Law, terminal velocity and the determination of viscosity

Objectives

Determine the viscosity η of glycerol using a falling steel ball bearing
Understand the forces acting on a sphere falling through a viscous fluid
Apply Stokes' Law and plot v against r² to calculate η from the gradient
Evaluate sources of uncertainty and assess the accuracy of your result

Forces on a Falling Sphere

When a ball falls through a viscous liquid three forces act on it: its weight W downward, the upthrust U upward (Archimedes' principle), and the viscous drag F upward (opposing motion). Initially the ball accelerates, but as speed increases, drag increases until the net force is zero — the ball then falls at constant terminal velocity.

At terminal velocity the forces balance:

W = U + F mg = ρ_fluid · V · g + 6πηrv ρ_ball · (4/3)πr³ · g = ρ_fluid · (4/3)πr³ · g + 6πηrv

where r = radius of ball, v = terminal velocity, η = dynamic viscosity, ρ = density

Stokes' Law & Deriving η

Rearranging for terminal velocity v gives the key equation:

v = 2r²(ρ_ball − ρ_fluid)g / 9η ∴ v ∝ r²

Plotting v (y-axis) against r² (x-axis) gives a straight line through the origin.
Gradient m = 2(ρ_ball − ρ_fluid)g / 9η, therefore η = 2(ρ_ball − ρ_fluid)g / 9m

This linear relationship is the basis of the experiment — you measure terminal velocity for balls of different radii, then use the gradient of the v vs r² graph to find η.

Conditions for Stokes' Law

Stokes' Law F = 6πηrv only applies when:

The flow around the ball is laminar (not turbulent) — requires slow speeds and small balls
The ball is spherical and rigid
The ball is far from the container walls (wall correction may be needed)
The fluid is Newtonian — viscosity doesn't change with flow speed

Glycerol is highly viscous which ensures laminar flow for small steel ball bearings, making it ideal for this experiment.

Known Values (Glycerol, 25°C)

ρ_glycerol = 1261 kg m⁻³ ρ_steel = 7800 kg m⁻³ η_glycerol (accepted) = ~0.934 Pa·s at 25°C g = 9.81 m s⁻²

Note: viscosity of glycerol is highly temperature-dependent. Results vary significantly with temperature.

Step-by-Step Procedure

Follow these steps in the Simulation tab to collect your data

Equipment

Tall measuring cylinder of glycerol · Steel ball bearings (3–5 different radii) · Micrometer screw gauge · Ruler · Stopwatch or light gates · Magnet (to retrieve balls) · Electronic balance

Select a ball size

Use the radius slider in the Simulation tab to select a ball bearing radius. In the real experiment you would measure the diameter with a micrometer at three points and calculate the mean radius.

Mark measurement zone

Two rubber bands mark the start and end of the measurement zone — set so the ball has already reached terminal velocity before crossing the upper band. The distance L between bands is fixed at 0.200 m.

Drop the ball and record time

Click Drop Ball. The ball falls through the glycerol. The timer records the time t for the ball to travel between the two bands. Realistic noise is added to simulate real timing uncertainty.

Repeat 3 times

Repeat the drop 3 times at the same radius to obtain a mean time and assess random uncertainty.

Record the reading

Click Record Reading. The mean time, terminal velocity v = L/t̄, and r² are all calculated automatically.

v = L / t̄ = 0.200 / t̄

Repeat for 5 different radii

Change the ball radius and repeat steps 3–5. You need at least 5 different radii for a reliable graph.

Analyse results

Go to the Graph & Analysis tab. Plot v against r² and use the gradient to calculate η.

η = 2(ρ_ball − ρ_fluid)g / 9m

Controls

Ball material

ρ = 7800 kg m⁻³

Liquid

η ≈ 0.934 Pa·s · ρ = 1261 kg m⁻³

Ball radius r 1.0 mm

        0.5 mm2.5 mm
      

Drop 1

— s

Drop 2

— s

Drop 3

— s

Mean t

—

Drops at this radius: 0/3

Readings recorded: 0/5 minimum

Live derived values

r (m) —

r² (m²) —

v = L/t̄ (m/s) —

Results Data Table

Auto-filled from simulation. You need at least 5 different radii for a reliable graph.

#	r/ mm	r/ m	r²/ m²	t₁/ s	t₂/ s	t₃/ s	t̄/ s	v = L/t̄/ m s⁻¹	Range/ s	%U in t
No data yet — go to the Simulation tab and collect readings.

Graph & Analysis

Plot of terminal velocity v (y-axis) against r² (x-axis). Gradient m gives η = 2(ρ_ball − ρ_fluid)g / 9m

Calculated η

— Collect ≥5 readings first

Graph Parameters

Gradient m—

R² (fit quality)—

Accepted η (25°C)0.934 Pa·s

% Difference—

Uncertainty

Avg %U in t—

%U in v—

%U in η (approx)—

Note: %U in v ≈ %U in t (since v = L/t and L is precisely known). Error bars shown in y-direction.

Interpretation

Complete your experiment to see the analysis here.

Discussion Questions

Write your answers below. Click "Show Model Answer" when ready to compare.

Question 1

Explain why a ball falling through glycerol reaches a terminal velocity. Include all three forces in your answer.

Initially the ball accelerates because its weight exceeds the upthrust plus viscous drag. As speed increases, Stokes' drag (F = 6πηrv) increases proportionally with v. When weight = upthrust + drag, the net force is zero and acceleration becomes zero — the ball falls at constant terminal velocity. Weight and upthrust are constant throughout; only drag changes with speed.

Question 2

The graph of v against r² should be a straight line through the origin. Explain why, using the equation derived from Stokes' Law.

From Stokes' Law at terminal velocity: v = 2r²(ρ_ball − ρ_fluid)g / 9η. This is of the form v = mr² where m = 2(ρ_ball − ρ_fluid)g / 9η is a constant (since ρ_ball, ρ_fluid, g and η are all constant). Therefore v is directly proportional to r², giving a straight line through the origin with gradient m. The gradient can then be used to calculate η.

Question 3

State two conditions that must be satisfied for Stokes' Law to be valid in this experiment, and explain how the experimental setup ensures each is met.

1. Laminar flow — the flow around the ball must not be turbulent. This is ensured by using small ball bearings (r < 2.5 mm) falling slowly through highly viscous glycerol. The small radius and high viscosity keep the Reynolds number well below 1. 2. The ball must be far from container walls — wall effects can significantly increase drag. Using a wide cylinder (diameter ≫ ball diameter) minimises this error. A correction factor can also be applied if the cylinder diameter is known.

Question 4

Suggest why the viscosity of glycerol is highly temperature-dependent, and explain how this affects the reliability of your result.

Viscosity arises from intermolecular forces resisting flow. At higher temperatures, molecules have greater kinetic energy and move more freely, reducing the resistance to flow — so viscosity decreases significantly. For glycerol, a few degrees of temperature change can alter η by 10–20%. If the temperature of the glycerol is not measured and controlled throughout the experiment, different readings may be taken at slightly different temperatures, introducing a systematic error into the v vs r² graph and an unreliable value of η. The experiment should be conducted at a constant, measured temperature.

Question 5

Your measured value of η differs from the accepted value of 0.934 Pa·s. Identify two sources of systematic error that could cause this and suggest how each could be reduced.

1. Wall effect — the presence of the cylinder walls slows the ball beyond what Stokes' Law predicts for an infinite fluid, causing terminal velocity to be lower than expected, leading to an overestimate of η. This can be reduced by using a wider cylinder or applying the Ladyzhenskaya wall correction factor. 2. Ball not at terminal velocity — if the measurement zone begins before the ball has reached terminal velocity, the measured speed will be lower than terminal velocity, overestimating η. This can be reduced by ensuring the upper measurement band is placed well below the liquid surface, or by using light gates to verify constant velocity.