Torsion Pendulum Calculator
Enter moment of inertia I and torsion constant κ to compute period T=2π√(I/κ), frequency f and angular frequency ω. I=1 kg·m², κ=1 → T≈6.283 s; watch hairspring I=1e-7, κ=1e-5 → T≈0.628 s (1.59 Hz).
Input Data
Results
At a glance:A torsion pendulum: a rigid body suspended below an elastic thin filament, oscillating under the filament's torsional restoring torque τ=−κθ as angular simple harmonic motion, with period T=2π√(I/κ), where I is the body's moment of inertia about the suspension axis and κ the filament's torsion constant (N·m/rad). Unlike a physical pendulum, the oscillation does not depend on gravity — it relies only on the elastic restoring torque — so it can be used in zero-gravity environments such as vacuum or space. Angular frequency ω=√(κ/I). History: Coulomb invented the torsion balance in 1777 to measure tiny forces; Cavendish used a torsion pendulum in 1798 to measure the gravitational constant G; Huygens invented the hairspring-balance wheel in 1675 to make watches keep accurate time. Classic example: I=1 kg·m², κ=1 N·m/rad → T=2π≈6.283 s; watch hairspring I=1e-7, κ=1e-5 → T=0.628 s (1.59 Hz). Applications: (1) watch hairspring-balance wheel — the timing core; (2) torsion balance — measuring tiny forces (Coulomb, gravity); (3) moment-of-inertia measurement — back-calculate I from T; (4) Cavendish experiment — measuring G; (5) vacuum balances — zero-g measurement.
Formula
Period: T = 2π·√(I/κ)
Angular freq: ω = 2π/T = √(κ/I)
Frequency: f = 1/T = (1/2π)·√(κ/I)
Restoring torque: τ = −κ·θ (Hooke's law)
Torsion constant: κ = π·G·r⁴/(2L) (round filament)
$$T = 2\pi\sqrt{\frac{I}{\kappa}}, \quad \omega = \sqrt{\frac{\kappa}{I}}$$How to Use
- Enter moment of inertia I (kg·m²) and torsion constant κ (N·m/rad).
- The tool computes T=2π√(I/κ), frequency f and angular frequency ω.
- Common: I=1, κ=1 → T=2π≈6.283 s; I=0.01, κ=0.1 → T≈1.987 s.
Case Studies
Cavendish measurement of G
In 1798 Cavendish used a torsion pendulum to measure G=6.674e-11 N·m²/kg². Two lead balls (m=158 kg) attracted the small balls (m=729 g), force F=GmM/r²≈1.7e-7 N.
Pendulum I=7.3e-5 kg·m², κ≈1e-8 N·m/rad → T=2π√(I/κ)≈17 minutes, the tiny force accumulating an observable deflection.
Cavendish's precision ~1%; modern torsion measurements reach 0.01% (2018 CODATA G=6.67430e-11, uncertainty 0.0022%).
Watch hairspring-balance timing
Mechanical watch balance I≈1e-7 kg·m², hairspring κ≈1e-5 N·m/rad → T=2π√(1e-7/1e-5)=0.628 s (1.59 Hz).
Each period the escapement releases energy once; 1.59 Hz×2 half-periods=28800 beats/hour (vph). High-beat 36000 vph (5 Hz) resists shocks better.
Hairspring material sets accuracy: iron-nickel (Nivarox) temperature coefficient <1 ppm/°C, error <2 s/day. Quartz (32768 Hz) far exceeds mechanical.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.