pendulums

Pendulums are devices in which a mass is suspended from a pivot so that it can swing about an axis under the influence of gravity. The simplest form consists of a point mass m attached to a light string or rod of length L. When displaced from its lowest point and released, it oscillates back and forth along an arc.

For small angular displacements, the motion approximates simple harmonic motion with period T ≈ 2π sqrt(L/g), where g is the local acceleration due to gravity. The small-angle approximation applies best for swing angles under about 15 degrees. In more general cases the period depends on the distribution of mass and the distance from the pivot to the center of mass. A physical or compound pendulum has period T = 2π sqrt(I/(m g d)), where I is the moment of inertia about the pivot and d is the distance from the pivot to the center of mass.

Energy in an ideal pendulum is conserved, with potential energy at the top converting to kinetic energy at the bottom. Real pendulums experience damping from air resistance and friction at the pivot, causing the amplitude to decay over time.

Variations include the Foucault pendulum, which demonstrates Earth's rotation, and torsion or gyroscopic pendulums used in gravimetry. Pendulums are foundational in timekeeping, notably in clocks, where escapements regulate motion. They also serve in metrology to measure gravity, in educational demonstrations of harmonic motion, and in seismology for sensing ground motion.