---
title: "Finite-Amplitude Pendulum Interactive Lab | Sandbox Physics"
description: "Does a bigger swing change a pendulum's period? It does. A bigger swing takes longer than the small-angle formula predicts, by an amount given by an elliptic integral. At a 90-degree amplitude the period is about 18 percent longer. As the swing approaches the very top, the period grows sharply."
canonical_url: "https://sandboxphysics.com/labs/finite-pendulum/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/finite-pendulum/"
last_modified: "2026-10-25"
---

M062 · Nonlinear oscillations / measuring the period

# Finite-Amplitude Pendulum

**Does a bigger swing change a pendulum's period?** It does. A bigger swing takes longer than the small-angle formula predicts, by an amount given by an elliptic integral. At a 90-degree amplitude the period is about 18 percent longer. As the swing approaches the very top, the period grows sharply.

Drag the bob and compare large swings with a small-angle reference. Measure the longer period, then try damping or full rotations.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The model is a point bob on a rigid rod with a fixed pivot in uniform gravity. The rod can carry tension or compression, so unlike a string it never goes slack. A viscous torque at the pivot is optional. The elliptic-integral period describes the undamped orbit at the starting energy; with damping, the interval between crossings changes as energy is lost. RK4 integration of the power, step halving and the signed rod force show the limits of the model and of the numerics. The diagnostic readings separate kinetic energy, gravitational energy and accumulated heat. Scene lengths are rescaled to fit, and the bead size, supports and face markings are schematic. The gray bob follows the undamped small-angle solution even where it does not apply, so it may be inaccurate for large swings. The gray reference is offset slightly in depth for visibility.

Physics tutorial

## When amplitude becomes part of the clock

**Background** The restoring torque is proportional to the sine of the angle. Near the bottom that is almost linear, but on large swings the bob lingers near its turning points.

**Why it matters** OpenStax section 15.4 gives the pendulum equation and the limit of the small-angle approximation. The prediction for finite swings comes from the energy integral, and the complete elliptic integral in it is evaluated independently with NIST DLMF 19.8.5.

Start with the essentials

- **Focus question**: How much timing error does the small-angle approximation create?
- **One-sentence intuition**: Keep two things apart: the period predicted for the undamped orbit, and the period actually measured between two successive crossings.

### Core mathematical model

#### Position and velocity in the lab frame

$$
\mathbf r=L(\sin\theta,0,-\cos\theta),\quad \mathbf v=L\dot\theta(\cos\theta,0,\sin\theta)
$$

The vertical axis points up. The velocity is tangent to the path. The gray bob is a separate small-angle reference.

#### Nonlinear equation of motion

$$
\ddot\theta+2\zeta\omega_0\dot\theta+\omega_0^2\sin\theta=0,\quad \omega_0=\sqrt{g/L}
$$

The viscous damping torque is proportional to the moment of inertia and to the angular velocity.

#### Period of an undamped swing

$$
T=\frac{4}{\omega_0}K(k),\quad k=\sin\frac{\theta_{\max}}2
$$

The turning angle comes from the energy, and the release may include an initial velocity.

#### Elliptic integral from a separate algorithm

$$
K(k)=\frac{\pi}{2\operatorname{AGM}(1,\sqrt{1-k^2})}
$$

The prediction uses its own numerical algorithm, not the integrator that computes the motion.

#### Energy and dissipated heat

$$
E=\tfrac12mL^2\dot\theta^2+mgL(1-\cos\theta),\quad E+Q=E_0
$$

Heat is integrated from the viscous power, so any mismatch left in the balance is still an independent check.

#### Rod force, with its sign

$$
F_{\mathrm{rod}}=m(g\cos\theta+L\dot\theta^2)
$$

Positive force pulls toward the pivot; negative force pushes away. A rigid rod supports both.

### Common difficulties

#### A prediction is not a measurement

**Typical misconception** The period shown at release should already be a measurement.

**Better mental model** The prediction is available at once. A measurement has to wait until the bob passes the bottom twice in the same direction.

#### Damping changes orbit energy

**Typical misconception** One elliptic-integral period describes every damped cycle.

**Better mental model** The fixed reference uses the starting energy. As the swing shrinks, later crossing intervals change.

#### A rod can push

**Typical misconception** Any release from a high angle must make the support go slack.

**Better mental model** That is true for a string, which can only pull. This apparatus uses a rigid rod on purpose.

### Run the experiment

1. 01

   #### Calibrate near the bottom

   Acquire the full record for Small-angle clock, then open Release and numerical settings to change the bob mass.

   **What to observe:** The period stays the same, while energy and force grow in proportion to the mass.

2. 02

   #### Increase the swing

   Compare Large swing and Near the upright barrier with the gray reference bob. Orbit the apparatus or use Swing plane.

   **What to observe:** The harmonic reference pulls ahead, and the measured interval matches the longer period from the elliptic integral.

3. 03

   #### Switch from swinging to rotating

   Choose Complete rotations, then reverse the initial angular velocity.

   **What to observe:** The angle shown within one turn jumps back, but the unwrapped angle stays continuous, and so does the timing of full turns in the same direction.

4. 04

   #### Separate physics from numerical error

   Open Recording and diagnostics, acquire a damped record, then halve the maximum numerical step under Release and numerical settings.

   **What to observe:** The damping loss remains, while the independent energy mismatch and the difference between the two runs shrink as the step gets smaller.

Physical conclusion

Mass sets energy and rod force, while length, gravity and orbit energy set the period.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)