Pendulum Simulator

For a simple pendulum swinging through small angles, the period is T = 2π√(L/g), where L is the length in metres and g ≈ 9.81 m/s². A 1 m pendulum on Earth gives T = 2π√(1/9.81) ≈ 2.01 s. Length is measured from the pivot to the centre of mass of the bob. The formula assumes a point mass on a massless string and small angle (under about 15°). For larger swings, the period grows slightly because of the breakdown of the small-angle approximation. Otherwise, plug in length and gravity, and you've got your period.

T = 2π√(L/g). The period grows with the square root of length, so doubling L doesn't double T — it multiplies it by √2 ≈ 1.414. To get a 2-second period (one tick per second on a grandfather clock), you need L = g(T/2π)² ≈ 0.99 m. That's why pendulum clocks tend to be about a metre tall. A child's swing 4 m long has T ≈ 4 s. The dependence on length explains why longer swings feel slower, and shorter ones snap back quickly. Mass and amplitude don't enter the formula at all.

Rearrange T = 2π√(L/g) to L = g(T/2π)². Suppose you want a pendulum with a 1.5 s period: L = 9.81 × (1.5/2π)² ≈ 0.559 m. This reverse-solving is exactly what clockmakers do when designing pendulum-driven timepieces — start with the desired tick rate, work back to find the rod length. The formula assumes a simple pendulum, which is an idealisation; real physical pendulums (with extended bodies) need a slightly different formula involving the moment of inertia, but the simple version is close enough for most applications.

No, the period of a simple pendulum is independent of mass. Whether you swing a 10-gram earring or a 10-kilogram bowling ball on the same length string, the period stays the same: T = 2π√(L/g). This is because gravity scales the restoring force in proportion to mass, while inertia also scales with mass, so the two effects cancel exactly. Galileo first noticed this property and used it to argue that all objects fall with the same acceleration. In real pendulums, air drag and string mass introduce tiny mass-dependent corrections, but the leading behaviour is genuinely mass-free.

Stronger gravity means a faster pendulum. T = 2π√(L/g) shows the period decreases as g increases. So a pendulum that ticks once per second on Earth would swing more slowly on the Moon (g ≈ 1.62 m/s²): the same length there gives T ≈ 4.93 s instead of 2.01 s. This is actually how scientists once measured local gravity variations — by timing precise pendulums in different places. Even within Earth, g changes slightly with latitude and altitude, so a pendulum clock calibrated in London runs slightly off in Quito. The relationship is square-root, so the effect is moderate but measurable.

T = 2π√(L/g) is exact only in the limit of zero amplitude. For finite swings, the true period grows slightly — about 0.7% longer at 30° amplitude, around 1.7% at 45°, and noticeably more beyond. The small-angle approximation works because sin(θ) ≈ θ in radians for small θ, which linearises the equation of motion into simple harmonic. Above 15° or so, the proper period requires elliptic integrals. For lab work and clock design, staying below 5° keeps the simple formula extremely accurate. So the calculator's answer is best trusted when amplitude is genuinely small.

A damping simulator shows how amplitude decays over time when air resistance and pivot friction sap energy. Damped oscillation follows x(t) = A exp(−γt) cos(ωt), where γ is the damping coefficient. The amplitude shrinks exponentially while the period stays roughly the same (slightly longer in heavily damped cases). You'll see the pendulum's swing get smaller each cycle until it stops. By varying the damping parameter, students can explore the difference between underdamped, critically damped, and overdamped regimes — concepts that show up everywhere from car suspensions to electronic circuits.