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.
Physics / Torsion Pendulum & Shear Modulus
FREE TO EXPLOREEXPERIMENT 38 · PHYSICS LABORATORY
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. Explore a virtual apparatus, collect readings and analyse a graph.
CHANGE · OBSERVE · MEASURE
Your virtual workbench.
TIME FOR 20 OSCILLATIONS T₂₀—
MEASUREMENT STATUS
Ready for a virtual observation.
Adjust the apparatus, then take a reading.
Timing is generated from the model; a short on-screen trial represents a longer physical interval. Animation is illustrative.
Simulation assumptions and limits
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.
Know your apparatus.
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.
FOLLOW THE METHOD
A reason behind every step.
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. Suggested independent settings: 40.0, 50.0, 60.0, 70.0, 80.0 cm.
Wire Diameter Metrology with Micrometer Screw Gauge
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.
Why this step matters
Because torsional rigidity depends on r⁴, an uncertainty of only 1% in radius produces a 4% error in the calculated shear modulus η; rigorous averaging suppresses dimensional error.
Active Wire Length Clamping and Alignment
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.
Why this step matters
Ensuring rigid clamping prevents virtual lengthening from clamp compliance and guarantees that all torsional strain is confined strictly to the measured active length L.
Initial Torsional Displacement and Release
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.
Why this step matters
Small angular displacements (≤ 15°) ensure that the maximum shear strain γ = r θ / L remains well below the elastic yield limit of carbon steel, avoiding plastic hysteresis.
Equilibrium Transit Timing of Twenty Oscillations
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.
Why this step matters
Timing at the equilibrium position where angular velocity is maximum minimizes human visual transit trigger error, while accumulating 20 cycles divides trigger latency by 20.
Systematic Length Variation Across Five Stations
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.
Why this step matters
Varying L over a two-fold span provides a broad baseline for linear regression, allowing verification of the proportionality T² ∝ L across diverse resonance frequencies.
Linear Regression Analysis and Shear Modulus Calculation
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).
Why this step matters
Linear regression eliminates zero-point length measurement offsets and averages out random timing scatter, delivering a rigorous metrological evaluation of η.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into a result.
Points are your observations. The fit uses SI x and y values before display scaling.
LINEAR FIT · y = mx + b
T² = \left(\frac{8\pi I}{\eta r^4}\right) L
| # | Suspension Wire Length L (cm) | Time for 20 Oscillations t₂₀ (s) | Wire length L (m) | Period squared T² (s²) | Remove |
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Calculate it yourself.
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