Pendulum
A pendulum is a mass (called a bob) suspended from a fixed point by a string or rod that oscillates back and forth under the influence of gravity. For small angular displacements (less than about 15°), a simple pendulum exhibits simple harmonic motion, and its period depends only on its length and the local gravitational acceleration, not on mass or amplitude. Pendulums have historically been used in clocks and are fundamental to understanding oscillatory systems.
Key Formula
T = 2π × √(L / g)
LaTeX: T = 2\pi\sqrt{\frac{L}{g}}
| Symbol | Meaning | Unit |
|---|---|---|
| T | Period of oscillation | s |
| L | Length of the pendulum | m |
| g | Acceleration due to gravity | m/s² |
Worked Example
Problem
A simple pendulum has a length of 2.5 m. Find its period and frequency at a location where g = 9.8 m/s². Also find the length required for a period of exactly 2.0 s.
Solution
Part 1 — T = 2π√(L/g) = 2π√(2.5/9.8) = 2π√(0.2551) = 2π × 0.505 ≈ 3.17 s. Frequency f = 1/T ≈ 0.315 Hz. Part 2 — Rearrange: L = g(T/2π)² = 9.8 × (2.0/2π)² = 9.8 × (0.3183)² = 9.8 × 0.1013.
Answer
T ≈ 3.17 s, f ≈ 0.315 Hz. For T = 2.0 s, L ≈ 0.993 m ≈ 1.0 m.
Period of a simple pendulum for various lengths (g = 9.8 m/s²)
| Length (m) | Period T (s) | Frequency f (Hz) | Angular freq ω (rad/s) | Application |
|---|---|---|---|---|
| 0.25 | 1.00 | 1.00 | 6.28 | Metronome beat |
| 0.993 | 2.00 | 0.50 | 3.14 | Clock seconds pendulum |
| 1.0 | 2.01 | 0.50 | 3.13 | Grandfather clock |
| 9.8 | 6.28 | 0.159 | 1.00 | Large hall pendulum |
| 0.10 | 0.635 | 1.57 | 9.89 | Small demonstration |
Interactive Tools
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Related Terms
Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and always directed toward it. The motion follows a sinusoidal pattern over time, characterized by constant amplitude, frequency, and period in the absence of damping. SHM is the basis for understanding pendulums, springs, sound waves, and alternating electric circuits.
Spring-Mass System
A spring-mass system consists of a mass attached to an ideal spring that obeys Hooke's Law, where the restoring force is proportional to the displacement from equilibrium. When displaced and released, the mass oscillates in simple harmonic motion with a period that depends on the mass and the spring constant, but not on amplitude. This system is the canonical model for oscillatory behavior in physics and engineering.
From Latin "pendulum" meaning "hanging thing," derived from "pendere" (to hang). Galileo Galilei first noted the isochronous property (constant period) of pendulums around 1602 by observing a swinging lamp in the Pisa Cathedral.