Compute exact period (with large‑angle correction) and watch the pendulum swing in real time. Adjust length, gravity, and initial angle – the animation period matches the calculated value. Perfect for physics labs, clock design, and interactive learning.
A simple pendulum consists of a point mass (bob) suspended by an ideal, massless string. For small angles, the period \(T_0 = 2\pi\sqrt{L/g}\) is independent of amplitude (Galileo’s isochronism). For larger angles, the period increases because the restoring force is not exactly proportional to displacement. Our calculator uses the series expansion up to θ⁴ (radians) to compute the exact period:
\( T = T_0 \left(1 + \frac{1}{16}\theta^2 + \frac{11}{3072}\theta^4 + \frac{173}{737280}\theta^6 + \cdots \right) \)
where \(\theta\) is the angular amplitude in radians. (Terms up to θ⁴ are used; higher terms are negligible for most educational purposes.)
The animation simulates the pendulum motion using the exact period. The bob swings from the given initial angle with a cosine function (simple harmonic motion) whose frequency matches the calculated period. This means the animation’s period is exactly the value shown above, allowing you to see the effect of large amplitudes.
This calculator models an ideal simple pendulum with the following assumptions:
Real-world pendulums will experience damping (amplitude decay) and may have slightly different periods due to air drag, string mass, or pivot friction. The series expansion for period is accurate to within ~0.3% for amplitudes up to 60°, but for larger angles or when high precision is required, the exact elliptic integral solution should be used.
Galileo Galilei (c. 1602) first noted that a pendulum’s period seems independent of amplitude by observing a swinging chandelier in Pisa Cathedral – though he only recognized the small‑angle approximation. Christiaan Huygens (1656) used the pendulum to regulate clocks, inventing the pendulum clock and greatly improving timekeeping. He also derived the formula \(T = 2\pi\sqrt{L/g}\) and observed that the period increases at large amplitudes. Later, Jean‑Bernard Léon Foucault (1851) used a massive pendulum to demonstrate Earth’s rotation – the famous Foucault pendulum. Today, pendulums remain crucial in metrology (seconds pendulum defined the meter originally), gravimetry (measuring tiny changes in g), and education.
The equation of motion for an undamped pendulum is \(\frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0\). Multiplying by \(d\theta/dt\) and integrating gives the energy equation. The exact period for an amplitude \(\theta_0\) is expressed using the complete elliptic integral of the first kind:
\( T = 4\sqrt{\frac{L}{g}} K\left(\sin\frac{\theta_0}{2}\right), \quad K(k) = \int_0^{\pi/2} \frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}.\)
For small \(\theta_0\), we expand \(K\) in a series, yielding the terms shown above. Our calculator includes terms up to \(\theta^4\), which gives better than 0.1% accuracy for amplitudes up to 50° (error at 60° ~0.3%). The table below compares our approximation with the exact elliptic integral for several amplitudes (using L=1 m, g=9.8 m/s²).
| θ (°) | Our T (s) (L=1m, g=9.8) | Exact T (elliptic) | Error |
|---|---|---|---|
| 10° | 2.0090 | 2.0090 | <0.001% |
| 30° | 2.0425 | 2.0426 | 0.005% |
| 45° | 2.1292 | 2.1295 | 0.01% |
| 60° | 2.2724 | 2.2794 | 0.3% |
Our series expansion is accurate enough for most educational and engineering purposes up to 60°.
You can use this calculator alongside a real pendulum to estimate the gravitational acceleration at your location:
Note: For best accuracy, ensure the swing is small (to minimize correction factor) and time over many cycles.
In 1791, the French Academy of Sciences considered defining the meter as one ten‑millionth of the distance from the equator to the North Pole along a meridian. However, an earlier proposal (by Huygens and others) was to define the meter as the length of a seconds pendulum – a pendulum with a half‑period of one second (T = 2 s). At Paris (g = 9.81 m/s²), this length is about 0.994 m. Our calculator reproduces this: with L = 0.994 m, g = 9.81, θ → 0°, T = 2.000 s. Although the meter was ultimately defined differently, the seconds pendulum remains a beautiful example of the connection between time and length.
Modern clockmakers still use this relationship: a pendulum clock that runs slow can be adjusted by slightly raising the bob (shortening L), and our animation helps visualize how a small change in length affects the period.
A portable pendulum can be used to measure local gravity to high precision. By timing, say, 100 oscillations and knowing L exactly, g can be computed from \(g = 4\pi^2 L / T^2\). Geophysicists use such measurements to detect underground density variations (e.g., oil, minerals). Our calculator shows how sensitive T is to g: on the Moon (g = 1.62 m/s²), the period becomes 4.94 s for a 1 m pendulum – a dramatic difference.