Pendulum Period Calculator

A simple pendulum is an idealized model consisting of a point mass (bob) attached to a massless, inextensible string or rod, free to swing from a fixed pivot under gravity. Real pendulums approximate this model when the string is much longer than the bob and the oscillation angle is small. The period depends only on the length of the string and the local gravitational acceleration, not on the mass of the bob or the amplitude (for small angles).

In the simple pendulum formula T = 2*pi*sqrt(L/g), mass does not appear. Both the restoring force (the component of gravity pulling the bob back toward rest) and the inertia (resistance to motion) are proportional to mass, so the mass cancels out. Galileo reportedly observed this isochronism around 1602 by timing swinging chandeliers against his own pulse. It is a consequence of the equivalence of gravitational and inertial mass.

The formula T = 2*pi*sqrt(L/g) is exact only for infinitesimally small oscillation angles. For small angles (typically below about 15 degrees), sin(theta) is very close to theta in radians, and the formula gives a very good approximation. For larger amplitudes the period is longer than this formula predicts. At 45 degrees, the true period is about 4% longer; at 90 degrees, it is about 18% longer. For precise large-angle calculations an elliptic integral is required.

A Foucault pendulum is a long, heavy pendulum free to swing in any direction. As it swings, the plane of oscillation appears to rotate relative to the floor. This rotation is caused by Earth's rotation beneath the pendulum. At the North or South Pole it completes one full rotation in 24 hours; at other latitudes the rotation rate is slower. The first public demonstration was by Leon Foucault at the Pantheon in Paris in 1851, providing direct visible proof that Earth rotates.

A pendulum clock relies on the isochronism of small pendulum swings to keep time. A classic 'seconds pendulum' has a period of exactly 2 seconds (one second per half-swing), requiring a length of approximately 0.994 m at standard gravity. The escapement mechanism converts the pendulum's swing into the controlled rotation of clock gears. Temperature changes affect the length of the pendulum rod and thus the period, which is why precision clocks use compensating pendulums made of materials with low thermal expansion.