Simple Pendulum

Introduction

A simple pendulum — a mass suspended from a fixed point by a light, inextensible string — is one of the most studied systems in all of physics. Its regular, repeating motion is a direct manifestation of simple harmonic motion, and for small angles its period depends only on the length of the string and the local gravitational acceleration, not on the mass or the starting angle. This made the pendulum the basis of accurate timekeeping for centuries.

The Physics Explained

When a pendulum is displaced from its equilibrium position and released, two forces act on the bob: tension along the string (which does no work, only changes direction) and gravity straight down. The component of gravity along the arc — tangential to the swing — provides the restoring force that pulls the bob back toward the centre. This restoring force is proportional to the sine of the angle, making the equation of motion nonlinear in general.

For small angles (typically below about 15°), sin(θ) is very well approximated by θ in radians. This linearises the equation and produces simple harmonic motion: the bob oscillates at a constant frequency, and the period T = 2π√(L/g) regardless of amplitude. This is the isochronous property — equal time for each swing — that Galileo reportedly noticed watching a cathedral lamp sway.

For large angles, the sin(θ) approximation breaks down. The actual period grows longer as the amplitude increases, because the restoring force is weaker than the linear approximation predicts. The pendulum takes slightly more time to complete each swing the harder you pull it. At very large angles the motion departs significantly from simple harmonic behaviour.

In reality, friction at the pivot and air resistance gradually remove energy from the system, causing the amplitude to decay over time — a process called damping. A lightly damped pendulum swings for a long time before stopping; a heavily damped one barely completes a swing.

Key Equations

θ''(t) = −(g/L) · sin(θ) (exact equation of motion)

θ''(t) ≈ −(g/L) · θ (small-angle approximation)

T = 2π · sqrt(L / g) (period, small-angle approximation)

f = 1 / T = (1/2π) · sqrt(g / L) (frequency)

ω = sqrt(g / L) (angular frequency, rad/s)

E_total = m·g·L·(1 − cos(θ_max)) (total energy, from height at max angle)

Key Variables

Symbol	Name	Unit	Meaning
θ	Angle	radians (rad)	Displacement angle from vertical equilibrium
L	Length	m	Distance from pivot to centre of the bob
g	Gravitational acceleration	m/s²	9.8 m/s² near Earth's surface
m	Mass	kg	Mass of the pendulum bob
T	Period	s	Time to complete one full oscillation
f	Frequency	Hz	Number of complete oscillations per second
ω	Angular frequency	rad/s	Rate of oscillation in radians per second
θ_max	Amplitude	radians (rad)	Maximum angle reached during the swing
b	Damping coefficient	kg/s	Controls how quickly energy dissipates

Real-World Examples

Mechanical clocks: Pendulum clocks use the isochronous property to keep time. A longer pendulum swings slower; a one-metre pendulum has a period of almost exactly 2 seconds, making it the basis of the grandfather clock.
Seismographs: Early seismographs used heavy pendulums to detect ground motion. When the ground shakes, the pendulum bob tends to stay still due to inertia while the pivot moves — that relative displacement is recorded.
Foucault pendulum: A pendulum allowed to swing freely for hours traces a rotating path, demonstrating the Earth's rotation beneath it. Installations can be found in science museums around the world.

How the Simulation Works

Three sliders let you set the string length (m), the initial angle (degrees), and the damping coefficient. Pressing Start releases the pendulum from the initial angle. The simulation integrates the exact nonlinear equation of motion — not the small-angle approximation — so you can observe the period-lengthening effect at large angles. The damping slider adds a velocity-proportional damping force, so at high values you can watch the pendulum lose energy and settle toward equilibrium. The period readout compares the simulated period to the small-angle formula, making it easy to see where the approximation breaks down.