Simple Pendulum

A simple pendulum is an idealized model consisting of a point mass suspended from a massless, non-elastic string fixed to a pivot point. When displaced by a small angle and released, it oscillates back and forth, executing simple harmonic motion. The restoring force is directly proportional to displacement along the arc, bringing the pendulum back to equilibrium.

The period of a simple pendulum depends only on two factors: its length and the acceleration due to gravity. It is completely independent of the mass of the bob or the maximum displacement. For small displacements less than approximately 15 degrees, the period remains nearly constant regardless of amplitude.

At equilibrium, gravitational force and string tension balance each other. When displaced, the gravitational force resolves into radial and tangential components. The radial component counters tension, while the tangential component creates a restoring torque that brings the pendulum back to equilibrium.

The motion of a simple pendulum is determined solely by its period, which depends on length and gravitational acceleration. Mass has no effect on the restoring torque or the equation of motion. Two pendulums with different bob masses but identical lengths will oscillate identically when displaced by the same angle.

For small displacements, a simple pendulum behaves identically to a simple harmonic oscillator. The restoring force is directly proportional to displacement, satisfying the fundamental condition for harmonic motion. This relationship allows the period equation to be derived using simple harmonic motion principles.

Simple pendulums have numerous practical applications in daily life, including pendulum clocks, playground swings, and fishing line sinkers. These applications rely on the predictable, periodic motion of pendulums. The independence of period from mass makes pendulums reliable for timekeeping and other mechanical systems.

The restoring torque equals the tangential component of gravitational force multiplied by the string length. This torque acts to rotate the pendulum back toward equilibrium. The magnitude of this torque is proportional to the angular displacement, creating the restoring force characteristic of oscillations about an equilibrium position.