52, 57; CP4, CP5MaterialsCore

Core Practicals 4–5: Viscosity and Young Modulus

Both experiments use carefully measured geometry, but their physical models differ. A falling sphere tests viscous drag; a stretched wire tests elastic stress and strain.

40 min 4.2 Solid Materials

What you'll be able to do

Determine viscosity from terminal motion
Measure Young modulus from an elastic gradient
Evaluate squared-diameter uncertainty and model limitations

CP4: use a terminal-speed force balance

Measure a small sphere’s radius and density, and the density and temperature of the liquid. Let it fall through a sufficiently wide transparent column. Place timing marks below the initial accelerating region, then verify steady speed using successive equal-distance intervals. Find v from timed separation divided by travel time.

At terminal speed, weight equals upthrust plus Stokes drag. Use the force balance to infer viscosity. Stokes’ law assumes a small spherical object moving slowly with laminar flow; wall proximity, turbulence, bubbles or an unsteady speed can make the simple model unsuitable. Temperature matters because viscosity changes with it.

\eta=\frac{2r^2(\rho_s-\rho_f)g}{9v}

CP5: measure elastic extension

Measure the test wire’s original length and its diameter at several positions and orientations with a zero-checked micrometer. Apply a small preload to remove slack, then add known loads in steps while measuring extension with a suitable fine scale or travelling microscope.

Stay within the linear elastic region and check unloading recovery. A reference wire helps detect support movement or thermal effects. For a graph of force F against extension ΔL, gradient is EA/L; hence E equals gradient times original length divided by cross-sectional area. Secure the clamp and use a catch tray for loads.

E=\frac{FL}{A\Delta L},\qquad A=\frac{\pi d^2}{4}

Evaluate geometry and control variables

Viscosity depends on radius squared and Young modulus depends on inverse diameter squared, so a fractional size uncertainty contributes twice that amount to the final estimate in the usual propagation model. Repeated geometry readings reveal variation but cannot correct a micrometer zero offset without a separate check.

A longer wire gives a larger measurable extension at the same stress. A longer timed liquid segment reduces fractional timing uncertainty only if the sphere is already moving steadily throughout it. Match improvements to the actual limitation rather than extending the apparatus indiscriminately.

Work through the reasoning

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.

Example 1A sphere has radius 0.50 mm, density difference 6800 kg m⁻³ and terminal speed 0.020 m s⁻¹. Estimate viscosity with g=9.81 m s⁻².

\eta=2(0.50\times10^{-3})^2(6800)(9.81)/[9(0.020)]

\eta=0.185

Pa s.

AnswerAbout 0.19 Pa s, if the Stokes assumptions hold.

Example 2A 2.0 m wire of area

2.0\times10^{-7}

m² has F against extension gradient

2.0\times10^4

N m⁻¹. Find E.

E=(\text{gradient})L/A

E=(2.0\times10^4)(2.0)/(2.0\times10^{-7})

Answer

2.0\times10^{11}

Pa.

Equation recap

\eta=\frac{2r^2(\rho_s-\rho_f)g}{9v}

Equation recap

E=\frac{FL}{A\Delta L},\qquad A=\frac{\pi d^2}{4}

Equation recap

Common mistakes to avoid

Timing the accelerating part of the fall as terminal motion.

Verify equal-distance travel times have settled.

Calculating wire area from diameter as though it were radius.

Use πd²/4.

Key takeaways

Use a full terminal-speed force balance.
Use elastic gradients to measure Young modulus.
Small geometry uncertainties can dominate both experiments.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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