Torsion Pendulum & Shear Modulus lab report

1. Title and aim

Torsion Pendulum: Shear Modulus of a Wire and Rotational Harmonic Dynamics. Determine the shear modulus (modulus of rigidity) η of a steel suspension wire and investigate rotational harmonic oscillations by measuring the period of a torsion pendulum as a function of wire length.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Shear modulus (modulus of rigidity) η of 78.50 GPa, differing by 0.003% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Heavy-Duty Torsion Pendulum Mounting Frame: A massive cast-iron wall bracket bolted to a structural masonry wall, featuring a heavy split-collet chuck to clamp the upper end of the suspension wire without slipping or compliance.
• Calibrated Solid Brass Inertia Disc: A precision-machined uniform circular cylinder of mass M = 2.000 kg and radius R = 0.1000 m, providing an accurately known base moment of inertia I = (1/2) M R² = 0.01000 kg m².
• High-Tensile Cold-Drawn Steel Wire Specimen: Uniform circular wire of diameter d = 1.200 mm (radius r = 0.600 mm), exhibiting linear elastic shear behavior without measurable plastic yield or work hardening.
• Digital Micrometer Screw Gauge: Resolving to 0.001 mm to measure the wire diameter across ten positions and orthogonal orientations, critical due to the fourth-power sensitivity of the torsional formula (r⁴).
• Precision Vertical Steel Rule / Cathetometer: Resolving to 0.5 mm to measure the exact active vibrating length L of the wire between the lower face of the top chuck and the upper face of the disc chuck.
• Digital Lap-Timer / Chronometer & Draught Shield: Resolving to 0.01 s to record 20 full torsional oscillations, enclosed within a transparent acrylic draught shield to suppress air currents.
• Suspension wire: cold-drawn high-tensile carbon steel (radius r = 0.600 mm, diameter d = 1.200 mm)
• Inertia body: solid turned brass disc (mass M = 2.000 kg, radius R = 0.1000 m, I = 0.01000 kg m²)
• Upper suspension: split-collet chuck clamped rigidly to heavy cast-iron wall mount
• Initial torsional displacement amplitude θ₀ ≤ 15° to maintain linear elastic Hookean regime
• Fiducial reference index mark aligned with equilibrium radial pointer to trigger lap timer
• Ambient conditions: enclosed acrylic draught shield to eliminate convective damping

4. Theory and graph choice

The torsion pendulum is one of the most powerful and historically significant instruments in experimental physics, celebrated from Charles-Augustin de Coulomb’s discovery of electrostatics and Henry Cavendish’s weighing of the Earth to modern atomic force microscopy and micro-gyroscopes. When a horizontal cylindrical disc of moment of inertia I is suspended by a vertical wire of length L and radius r, twisting the disc through an angle θ creates a non-uniform shearing strain throughout the cross-section of the wire. In accordance with Hooke’s law in shear, the wire generates an elastic restoring torque τ = −κ θ, where κ = (π η r⁴) / (2 L) is the torsional rigidity of the wire and η is the shear modulus (modulus of rigidity) of the material. By Newton’s rotational second law I (d²θ/dt²) = −κ θ, the disc executes rotational simple harmonic motion with period T = 2π √(I / κ). Squaring this expression yields T² = [8π I / (η r⁴)] · L, establishing a strictly linear relationship between the square of the oscillation period T² and the active length L of the wire. By timing torsional vibrations across multiple wire lengths, ordinary least squares regression yields the slope S = 8π I / (η r⁴), from which the shear modulus η is determined with high precision. This experiment elucidates the fundamental connection between microscopic atomic bond resistance to tangential shear and macroscopic rotational elasticity.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units s² m⁻¹.

5. Method

1. Using the digital micrometer screw gauge, determine the zero error, then measure the diameter d of the steel wire at ten distinct locations along its length and at mutually perpendicular orientations. Compute the mean diameter d and radius r = d / 2.
2. Insert the wire into the upper wall chuck and lower disc collet. Adjust the active free length L between the clamp jaws to the first test station (e.g. L = 40.0 cm) and tighten both collets firmly so no slipping occurs during rotation.
3. Gently twist the inertia disc in a horizontal plane through an initial angle of approximately 10° to 15°, ensuring that no lateral swinging, pendular wobble, or vertical bounce modes are excited, and release it smoothly from rest.
4. Discard the first complete oscillation cycle to eliminate release transients. As the radial index pointer on the disc crosses the stationary equilibrium fiducial mark moving in a chosen direction, start the digital chronometer and time 20 complete oscillations to obtain t₂₀.
5. Repeat the timing procedure across five distinct wire lengths L = 40.0 cm, 50.0 cm, 60.0 cm, 70.0 cm, and 80.0 cm. For each length, record duplicate trials to verify repeatability and compute the mean period T = t₂₀ / 20.
6. Construct a plot of period squared T² on the vertical axis against active wire length L on the horizontal axis. Compute the slope S = d(T²)/dL using ordinary least squares regression, and calculate the shear modulus η = (8π I) / (S · r⁴).

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0029%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 5.8797e-07; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.00140. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 8.6299e-04.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.00445 GPa. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Fourth-power error amplification in measuring wire radius r (dr/r ±0.17% producing ±0.68% uncertainty in η), active length measurement precision between clamp jaws (±0.5 mm), split-collet clamp compliance and microscopic end slipping, inertia disc mass calibration tolerance (±0.05 g) and machining radius tolerance (±0.05 mm), small parasitic pendulum swinging or wobble coupling, and acrylic shield draft suppression contribute experimental uncertainty. The OLS regression standard error reflects random scatter across length stations, but does not capture shared wire diameter calibration offsets.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Measure the wire diameter at multiple positions and perpendicular orientations with a zero-corrected digital micrometer to minimize the impact of the r⁴ error magnification factor.
• Displace the disc through a strictly horizontal twist (θ ≤ 15°) without applying lateral forces to avoid exciting parasitic pendulum swinging or tilting modes.
• Ensure the upper and lower split-collet chucks are tightened firmly to prevent microscopic slipping or compliance at the wire terminations.
• Keep the acrylic draught shield closed during all timing trials to eliminate air currents and convective disturbances.

The teaching model assumes a perfectly elastic, homogeneous, isotropic circular wire with zero internal friction and negligible wire self-inertia. In a physical laboratory, small corrections for the moment of inertia of the wire (I_wire = (1/2) m_wire r²), slight non-circularity of the wire cross-section, and clamp end compliance introduce minute second-order adjustments. The torsion pendulum experiment provides an indispensable bridge between rotational mechanics and continuum elasticity theory. By evaluating the fourth-power dependence of torsional rigidity on wire radius (the r⁴ law), students appreciate how nanoscale interatomic shear resistance governs macroscopic rotational resonance, enabling precise determination of the modulus of rigidity η.

12. Conclusion

The worked example illustrates how to obtain Shear modulus (modulus of rigidity) η from measurements, graph variables and a fitted coefficient. The estimate is 78.50 GPa. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.