Calculate simple pendulum period, frequency, energy, and speed. Includes exact large-angle correction and energy distribution analysis.
The **Simple Pendulum Calculator** provides a complete analysis of a point-mass pendulum swinging from a rigid, massless string. Unlike basic calculators that only use the small-angle approximation T = 2π√(L/g), this tool computes the exact period with higher-order correction terms, quantifies the error from the small-angle approximation, and shows how energy converts between kinetic and potential forms throughout the swing.
A simple pendulum is one of the most fundamental systems in classical mechanics. Despite its simplicity — just a mass on a string in a gravitational field — it exhibits rich physics including isochronism (constant period at small angles), nonlinear oscillation at large angles, and parametric resonance when the pivot oscillates. Pendulums have been used for centuries in clocks, seismometers, and gravimeters.
This calculator handles multiple scenarios: compute the period from length and gravity, determine the required length for a specific period, or extract gravitational acceleration from observed oscillations. Presets cover everything from wall clocks to Foucault pendulums to hypothetical Mars and Moon pendulums, making it a comprehensive exploration tool for pendulum physics.
The small-angle approximation (sin θ ≈ θ) is excellent below about 15° but introduces significant error for larger swings. At 30°, the error is 1.7%; at 45° it is 4%; at 90° it exceeds 18%. This calculator shows exactly when the simple formula breaks down and by how much.
The energy analysis is valuable for understanding conservation of mechanical energy. Total energy (KE + PE) remains constant — you can visually track how potential energy at the top converts to kinetic energy at the bottom. The velocity visualization confirms that maximum speed occurs at the lowest point, where all potential energy has converted to kinetic energy.
Small-angle period: T₀ = 2π√(L/g) Exact period (series): T = T₀[1 + (1/4)sin²(θ₀/2) + (9/64)sin⁴(θ₀/2) + (25/256)sin⁶(θ₀/2) + …] Maximum speed: v_max = √[2gL(1 − cos θ₀)] Height at angle θ: h = L(1 − cos θ) Maximum PE: mgh = mgL(1 − cos θ₀) (equals max KE) Variables: L = length (m), g = gravitational acceleration (m/s²), θ₀ = release angle (rad), m = mass (kg)
Result: 2.0064 s exact period
At L = 1 m: T₀ = 2π√(1/9.81) = 2.0061 s (small-angle). With θ₀ = 10°, the series correction factor is 1.00019, giving T_exact = 2.0064 s. The small-angle error is only 0.019% — negligible at 10°.
Galileo Galilei first studied pendulum motion around 1602, allegedly inspired by watching a chandelier swing in the Pisa cathedral. He discovered that the period is independent of amplitude (approximately true for small angles) and independent of the bob mass. Christiaan Huygens built the first pendulum clock in 1656, achieving unprecedented accuracy of about 15 seconds per day.
The seconds pendulum played a role in defining the meter. In 1791, French scientists considered defining the meter as the length of a seconds pendulum at 45° latitude. They ultimately chose 1/10,000,000 of the distance from the equator to the pole, but the coincidence that the seconds pendulum is about 1 meter long persists.
The exact period of a simple pendulum is an elliptic integral: T = 4√(L/g) × K(sin(θ₀/2)), where K is the complete elliptic integral of the first kind. The series expansion used in this calculator converges quickly for angles below about 120° but becomes slow near 180°.
At exactly θ₀ = 180°, the period is infinite — the pendulum takes infinite time to reach the unstable equilibrium at the top. This is a manifestation of the separatrix in phase space between oscillatory and rotational motion. Near this critical angle, the motion is extremely sensitive to initial conditions.
Two pendulums connected by a spring exhibit normal modes: in-phase oscillation (both swing together) and anti-phase oscillation (they swing opposite). Energy transfers back and forth between the pendulums in a phenomenon called beats. A driven pendulum (forced by periodic external torque) can exhibit resonance when the driving frequency matches the natural frequency, and at high amplitudes, it transitions to chaotic motion — one of the simplest physical systems to show deterministic chaos.
In Newton's second law, both the gravitational force (mg sin θ) and inertia (m × acceleration) are proportional to mass. The mass cancels out of the equation of motion: d²θ/dt² = -(g/L) sin θ. This means a heavy and light pendulum of the same length have identical periods.
The approximation sin θ ≈ θ (in radians) is within 1% for angles up to about 23°. At 30° the error is ~1.7%, at 45° ~4%, at 60° ~8%, and at 90° ~18%. For precision time-keeping or large swings, use the exact period formula.
A seconds pendulum has a period of exactly 2 seconds (1 second per half-swing). At standard gravity (g = 9.80665 m/s²), the required length is L = g/(4π²) × T² = 0.9937 m ≈ 1 meter. This is not a coincidence — the original definition of the meter was approximately the length of a seconds pendulum.
If given enough energy, a pendulum can go over the top. This requires an initial velocity v > 2√(gL) at the bottom. However, it is no longer oscillating — it rotates continuously. The transition between oscillation and rotation is at exactly E = 2mgL (the pendulum barely reaches the top with zero velocity).
Pendulums are isochronous at small angles — the period is nearly constant regardless of amplitude. This makes them excellent timekeepers: even as the pendulum slowly loses energy to friction, its period barely changes. The escapement mechanism gives the pendulum a tiny push each swing to compensate for friction.
Air resistance causes damping: the amplitude decreases exponentially over time. For small, heavy bobs in air, damping is minimal (quality factor Q > 1000). Lighter bobs or pendulums in viscous fluid damp quickly. Damping also slightly increases the period: T_damped = T/√(1 − 1/(4Q²)).