Physical Pendulum Calculator

Compute the oscillation period, reduced length, angular frequency, and moment of inertia for a rigid body swinging about a pivot.

Use consistent SI units. For custom shapes, provide I directly or use preset examples below.
? Uniform rod (pivot at end): m=1kg, L=1m → d=0.5m, I=0.333 kg·m²
? Solid disc (pivot on rim): m=2kg, R=0.2m → d=0.2m, I=0.12 kg·m²
⭕ Thin hoop (pivot on circumference): m=1.5kg, R=0.3m → d=0.3m, I=0.27 kg·m²
? Rectangular plate (edge): m=1.2kg, width=0.4m → d=0.2m, I≈0.064 kg·m²
⚙️ Demo: m=0.8kg, d=0.35m, I=0.18 kg·m²
Local computation & animation: All calculations and canvas animation run in your browser. No data is sent to any server.

Physical Pendulum Theory & Formula Derivation

A physical pendulum (or compound pendulum) is any rigid body that oscillates about a fixed horizontal axis not passing through its center of mass. Unlike a simple pendulum (point mass on a massless string), a physical pendulum’s period depends on its moment of inertia and the distance from the pivot to the center of mass. The restoring torque is provided by gravity: τ = – m g d sinθ, leading to the differential equation I α = – m g d θ for small angles. Hence, the angular frequency ω = √(m g d / I), and the period T = 2π √( I / (m g d) ).

$$ T = 2π \sqrt{\frac{I}{m g d}} \quad , \quad L_{\text{eq}} = \frac{I}{m d} = \frac{k^2}{d} $$

Where k = radius of gyration, d = distance from pivot to CM.

The reduced length Leq = I/(m d) corresponds to the length of an equivalent simple pendulum having the same period. This concept is extensively used in gravimeters and seismometers. The physical pendulum also illustrates the parallel axis theorem: I = Icm + m d², which makes experimental determination of g possible.

How to Use the Calculator: Step-by-Step

  • Enter basic parameters: Mass (kg), distance from pivot to center of mass (m), and moment of inertia about the pivot (kg·m²). If you know the inertia about the CM, apply parallel axis theorem manually or let the calculator assist via examples.
  • Choose a preset shape: Click on example buttons — uniform rod, disc, hoop, or rectangular plate — they automatically fill correct I and d for that geometry.
  • Adjust g: Standard gravity 9.81 m/s², but you can modify for different planets or experimental conditions.
  • Compute: The tool instantly returns period (T), angular frequency (ω), equivalent simple pendulum length, and the torque constant.
  • Visualize: The canvas shows a schematic pendulum with pivot (red), CM (blue), and the rigid body representation to reinforce concepts.

Real-World Applications & Engineering Relevance

Case Study: Kater's Reversible Pendulum

Kater's pendulum is a physical pendulum with two adjustable pivots, allowing precise determination of local gravitational acceleration without knowing the mass distribution. Our calculator can model each pivot configuration: by locating the periods equalization point, the reduced length Leq equals the distance between pivots. Geodesists and metrology labs use such pendulums to measure g with high accuracy. This tool replicates the theoretical period for any arbitrary rigid body, aiding experimental design.

Other applications include: clock escapements (period regulation), bridge vibration analysis (modelling piers as compound pendulums), robotic swing-up control (inverted pendulum variants), and sports equipment design (tennis racket sweet spot relates to percussion center – a concept linked to physical pendulum oscillations).

Derivation Walkthrough (Small Angle Approximation)

Starting from torque τ = I α = – m g d sinθ. For small θ (θ < 10°), sinθ ≈ θ, giving I d²θ/dt² = – m g d θ → d²θ/dt² + (m g d / I) θ = 0, which is SHM with ω² = m g d / I. Therefore, T = 2π/ω = 2π √(I/(m g d)). No approximation is required for the restored results because the calculator assumes harmonic regime; if large amplitudes, error is minimal for most teaching contexts. The formula is exact for the linearized model.

Special Cases & Comparison with Simple Pendulum

If the entire mass is concentrated at a point distance L from the pivot, then I = m L², and T = 2π √(L/g) — the simple pendulum formula. For a uniform rod pivoted at one end, I = (1/3) m L², d = L/2, so T = 2π √( (2L)/(3g) ) ≈ 0.816 * 2π√(L/g). The calculator automatically handles these variations. For a physical pendulum, the period can be minimized for a given mass by choosing d such that I is optimized (this occurs when d = k, radius of gyration).

Object / Shape Pivot Location Moment of Inertia I (about pivot) d (CM to pivot) Period T (example: m=2kg, g=9.81)
Uniform Rod (L=1.0m) At one end ⅓ m L² = 0.667 kg·m² 0.5 m 2π √(0.667/(2*9.81*0.5)) = 1.64 s
Solid Sphere (R=0.2m) On surface ²⁄₅ mR² + mR² = 7/5 mR² = 0.112 kg·m² 0.2 m 2π √(0.112/(2*9.81*0.2)) = 1.06 s
Thin Circular Hoop (R=0.3m) Rim point 2 m R² = 0.27 kg·m² 0.3 m 2π √(0.27/(1.5*9.81*0.3)) ≈ 1.56 s

Frequently Asked Questions

A simple pendulum assumes a point mass suspended by a massless string, while a physical pendulum is any rigid body oscillating about a pivot. The period of a physical pendulum depends on the moment of inertia and mass distribution.

The standard formula uses small-angle approximation (<10°). For larger amplitudes, the period increases slightly (elliptic integrals). However, this calculator serves typical physics and engineering needs where small oscillations are assumed.

Presets are based on exact theoretical formulas for uniform density objects. For real objects with non-uniform mass, you can manually input the measured inertia. The tool gives high numeric precision using double-precision arithmetic.

Reduced length L_eq = I/(m d) defines the length of a simple pendulum with the same period. It allows comparison between different rigid pendulums and is critical in gravimeter design and Kater’s pendulum.

In theory, the ideal pendulum assumes no friction at the pivot. Real-world bearings introduce damping; this calculator gives the natural undamped period. For precision engineering, consider additional corrections.

Physics validation & expert background – This tool implements the classical formula T = 2π √(I/(mgd)) as derived from Euler-Lagrange equations. References: University Physics (OpenStax), Feynman Lectures on Physics Vol I, and Mechanics by Landau & Lifshitz. Last update: May 2026.

Authoritative sources: Encyclopedia Britannica: Physical Pendulum; Physics Hypertextbook; NIST – pendulum gravimetry.