Small Angle Approximation Verifier

Physics Oscillations and Waves • Simple Harmonic Motion (shm) Basics

Written by STEM Calculators Team Published March 6, 2026 Updated March 6, 2026

For a simple pendulum of length \(L\), the small-angle approximation \(\sin\theta\approx\theta\) leads to SHM with period \[ T_{\text{approx}} \approx 2\pi\sqrt{\frac{L}{g}}. \] The exact period at amplitude \(\theta_0\) is \[ T_{\text{exact}} = 4\sqrt{\frac{L}{g}}\,K(k),\qquad k=\sin\!\left(\frac{\theta_0}{2}\right), \] where \(K\) is the complete elliptic integral of the first kind. This calculator compares both and reports the percent error. Includes an interactive error-vs-angle plot (zoom + pan) and a slow pendulum animation.

Pendulum swing animation

Schematic animation at amplitude \(\theta_0\). Uses the computed periods (approx/exact) to set the timing (slow by default).

Not to scale. Angle shown in degrees. Timing uses the selected mode.

Interactive error vs. angle plot

Plots percent error \(\%\text{error}=\frac{T_{\text{exact}}-T_{\text{approx}}}{T_{\text{exact}}}\times 100\%\) versus \(\theta_0\). Axes include units: \(\theta_0\) in degrees (°), error in percent (%). Mouse wheel/trackpad to zoom, drag to pan.

Tip: on narrow screens, scroll horizontally to see the full plot.

Small-angle check: \(\sin\theta\) vs \(\theta\)

Visual comparison in radians. For small \(\theta\), the curves overlap closely.

Axes: \(\theta\) (rad) and \(\sin\theta\) (unitless).

Enter values and click “Calculate”.

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Theory

A simple pendulum consists of a mass on a light rod/string of length \(L\) swinging under gravity. Using the angular displacement \(\theta(t)\), the exact equation of motion is \[ \theta'' + \frac{g}{L}\sin\theta = 0. \] This equation is nonlinear because of \(\sin\theta\), so its period depends on the amplitude \(\theta_0\).

Small-angle approximation. When \(\theta\) is small (in radians), the Taylor expansion \[ \sin\theta = \theta - \frac{\theta^3}{6} + \cdots \] implies \(\sin\theta \approx \theta\). Substituting into the equation of motion gives the linear SHM equation \[ \theta'' + \frac{g}{L}\theta = 0, \] whose solutions are sinusoidal with angular frequency \(\omega=\sqrt{g/L}\). The corresponding period is \[ T_{\text{approx}} = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{L}{g}}. \] This is the famous amplitude-independent period used for clock pendulums and many introductory physics problems.

Exact period and elliptic integrals. For finite amplitudes, energy conservation leads to an integral expression for the period: \[ T_{\text{exact}} = 4\sqrt{\frac{L}{g}}\,K(k), \qquad k=\sin\!\left(\frac{\theta_0}{2}\right), \] where \(K(k)\) is the complete elliptic integral of the first kind. This calculator evaluates \(K(k)\) numerically using a fast and accurate AGM method: \[ K(k)=\frac{\pi}{2\,\mathrm{AGM}\!\left(1,\sqrt{1-k^2}\right)}. \] As \(\theta_0\to 0\), \(k\to 0\) and \(K(k)\to \pi/2\), recovering \(T_{\text{approx}}\).

Error behavior. The small-angle approximation underestimates the period at larger amplitudes because the restoring torque is weaker than the linear model predicts near larger angles (since \(\sin\theta < \theta\) for \(\theta>0\)). A useful comparison is the percent error \[ \%\text{error}=\frac{T_{\text{exact}}-T_{\text{approx}}}{T_{\text{exact}}}\times 100\%. \] For small angles like \(5^\circ\), the error is typically well below \(0.1\%\). As the amplitude grows (e.g., \(30^\circ, 60^\circ\)), the error becomes noticeable and the “clock accuracy” assumption breaks down.

Lissajous-style oscilloscopes are not needed here; instead, the key idea is that linearization (\(\sin\theta\approx\theta\)) changes the physics by removing amplitude dependence. More advanced topics include physical pendulums (distributed mass), damping/forcing, and chaotic motion in double pendulums, where the small-angle SHM model is insufficient.

Frequently Asked Questions

What specifically is the small-angle approximation?

The small-angle approximation substitutes the expression sin(x) with merely x when the angle is small, which enormously simplifies the governing differential equation.

Why does the pendulum equation lose accuracy at large angles?

At large swing angles, the physical restoring force is strictly proportional to the sine of the angle, not the angle itself, breaking the linear requirement for pure SHM.

At what angle does the approximation generally fail?

Engineers often safely use the approximation up to roughly 15 degrees, above which typical percentage error quickly climbs beyond 1 percent.