Pendulum

A pendulum is a weight suspended from a pivot so it can swing freely under gravity. The simplest form is the simple pendulum, consisting of a point mass m attached to a light, inextensible string or rod of length L, swinging in a vertical plane. A more general form is the physical pendulum, where the body itself has extended shape and rotates about a fixed pivot.

For small angular displacements, the restoring torque leads to simple harmonic motion described by θ'' + (g/L) θ = 0. The period is T = 2π sqrt(L/g). Behavior depends on gravity and length; larger L yields longer period; stronger g yields shorter period. For larger amplitudes, the small-angle approximation breaks down and the exact period increases slightly from the simple formula. In real pendulums, air resistance and pivot friction cause damping, gradually reducing amplitude and energy.

Energy perspective: in the absence of damping, mechanical energy is conserved; energy continuously exchanges between potential energy when elevated and kinetic energy at the bottom of the swing. The lowest point corresponds to maximum speed. Applications and historical notes: pendulums have been used in timekeeping since Galileo’s observations and Huygens’s development of the pendulum clock. The Foucault pendulum demonstrates Earth’s rotation by showing a precession of the swing plane that depends on latitude. Pendulums and their principles are used in seismology, geophysics, and various sensors, including accelerometers and inertial measurement units.

In historical timekeeping, the seconds pendulum—the period of one second—had a length near 0.994 meters at standard gravity.