When a sphere falls through a viscous fluid, it eventually reaches a constant velocity called terminal velocity, at which the drag force exactly balances gravity. Stokes’ law predicts that terminal velocity is proportional to the square of the sphere’s radius. By measuring terminal velocity for spheres of different radii falling through glycerol, you can verify this relationship and determine the viscosity of glycerol — a genuine derived physical quantity. This is an excellent IB Physics IA.
This practical is suitable for IB Diploma Physics HL.
Background Theory
At terminal velocity, the net force on the sphere is zero. Balancing gravity, upthrust, and viscous drag:
6πηrv = (4/3)πr³(ρₛ − ρₑ)g
Rearranging for terminal velocity v:
v = 2r²(ρₛ − ρₑ)g / 9η
Where η = dynamic viscosity of the fluid (Pa s), r = radius of sphere (m), ρₛ = density of sphere, ρₑ = density of fluid, g = gravitational field strength. This predicts v ∝ r²: a log-log plot of v against r should give a straight line with gradient = 2, confirming the power law. The viscosity η can then be calculated from the gradient of a v vs r² graph.
Variables
- Independent variable (IV): Radius of steel sphere r (mm) — e.g. 0.5, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0 mm (measure diameter with micrometer and halve)
- Dependent variable (DV): Terminal velocity v (m s⁻¹) — measured by timing the sphere between two marked lines in the fluid
- Controlled variables (CV): Same fluid (glycerol at constant temperature), same measuring cylinder, drop point at centre of cylinder, temperature (viscosity of glycerol is very temperature sensitive)
Equipment
- Steel ball bearings of different radii (0.5–3.0 mm)
- Micrometer screw gauge (±0.005 mm)
- Tall measuring cylinder (at least 30 cm tall) filled with glycerol
- Two rubber bands or tape markers placed a fixed distance apart on the cylinder (e.g. 20 cm apart, starting well below the surface)
- Ruler (±0.5 mm)
- Digital stopwatch (±0.01 s)
- Tweezers or a release mechanism to drop spheres centrally
- Thermometer (glycerol viscosity changes significantly with temperature)
- Magnet to retrieve steel ball bearings
Safety
Safety
⚠️ Glycerol is non-toxic but very slippery — clean up any spills immediately to prevent falls. Take care not to drop ball bearings on feet. Dispose of glycerol into the appropriate waste disposal bottles provided.
Method
- Measure the diameter of each ball bearing using the micrometer at least three times and calculate the mean radius r.
- Fill the measuring cylinder with glycerol. Mark two horizontal lines with rubber bands: the upper line should be at least 5 cm below the surface (to allow terminal velocity to be reached), and the lower line 20 cm below the upper.
- Record the temperature of the glycerol.
- Using tweezers, drop a ball bearing centrally into the glycerol above the upper marker.
- Start the stopwatch as the ball crosses the upper marker and stop when it crosses the lower marker. Record the time t.
- Calculate v = distance / time (m s⁻¹).
- Retrieve the ball using a magnet. Repeat each sphere size at least five times and calculate a mean v.
- Repeat for all sphere sizes. Check temperature before each set of measurements.
Results Table
| r (mm) | r² (mm²) | v₁ (m s⁻¹) | v₂ (m s⁻¹) | v₃ (m s⁻¹) | v₄ (m s⁻¹) | v₅ (m s⁻¹) | Mean v (m s⁻¹) |
|---|---|---|---|---|---|---|---|
| 0.50 | 0.25 | ||||||
| 0.75 | 0.56 | ||||||
| 1.00 | 1.00 | ||||||
| 1.50 | 2.25 | ||||||
| 2.00 | 4.00 | ||||||
| 2.50 | 6.25 | ||||||
| 3.00 | 9.00 |
Analysis
1. Plot mean v (y-axis) against r² (x-axis). Stokes’ law predicts a straight line through the origin.
2. Calculate the gradient of the best-fit line: gradient = 2(ρₛ − ρₑ)g / 9η
3. Rearrange to find viscosity: η = 2(ρₛ − ρₑ)g / (9 × gradient)
Using: ρₛ (steel) ≈ 7800 kg m⁻³, ρₑ (glycerol) ≈ 1260 kg m⁻³, g = 9.81 m s⁻²
4. Compare your calculated η to the literature value for glycerol at your measured temperature (typically 0.9–1.5 Pa s at 20–25 °C).
5. Also plot ln(v) against ln(r). The gradient of this log-log plot gives the power n in v ∝ rⁿ. Theory predicts n = 2.
Discussion Points
- Why does terminal velocity increase with sphere radius? Explain in terms of the balance of forces.
- Why is glycerol used rather than water? What happens if the fluid is not viscous enough?
- Why is it important to drop the sphere centrally and why must terminal velocity be reached before the timing zone?
- Why does temperature affect the results so significantly? What is the physical reason viscosity decreases with temperature?
- What is the Reynolds number for your largest sphere, and does it fall within the range where Stokes’ law is valid (Re < 0.1)?
Guidance
IA Guidance
The dual graphical approach (v vs r² and ln-ln plot) and the calculation of a real physical property (viscosity) make this a standout IB Physics IA. To score highly:
- Research Design: Justify your radius range — spheres too small produce velocities too slow to time accurately; spheres too large may exceed the Stokes’ law regime. Show Reynolds number calculations to justify your range.
- Data Analysis: Include error bars based on the spread of your five measurements per radius. Propagate uncertainty in r (from micrometer) through to uncertainty in r² and v. Calculate δη from the uncertainty in the gradient.
- Conclusion: State η ± δη and compare to the literature value at your measured temperature. Confirm the power law using your ln-ln plot gradient.
- Evaluation: Discuss the effect of the cylinder walls (a correction factor is needed if r is not much smaller than the cylinder radius) and the sensitivity of glycerol viscosity to temperature as the main sources of systematic error.
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