Pendulum & Spring Period Reference Cheat Sheet

Key Facts

• The period of a simple pendulum for small angles is $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is length and $g$ is gravitational field strength.
• The period of an ideal mass-spring oscillator is $T = 2\pi\sqrt{\frac{m}{k}}$, where $m$ is mass and $k$ is the spring constant.
• Frequency and period are reciprocals, so $f = \frac{1}{T}$ and $T = \frac{1}{f}$.
• Angular frequency is $\omega = 2\pi f = \frac{2\pi}{T}$.
• For a simple pendulum, increasing the length $L$ increases the period because $T \propto \sqrt{L}$.
• For a spring oscillator, increasing the spring constant $k$ decreases the period because $T \propto \frac{1}{\sqrt{k}}$.
• For a spring oscillator, increasing the mass $m$ increases the period because $T \propto \sqrt{m}$.
• The small-angle pendulum formula works best when the initial angle is about $15^{\circ}$ or less.

Vocabulary

Period

The period $T$ is the time required for one complete cycle of oscillation.

Frequency

Frequency $f$ is the number of cycles per second and is measured in hertz, where $1\text{ Hz} = 1\text{ s}^{-1}$.

Angular frequency

Angular frequency $\omega$ describes how quickly the oscillator moves through its cycle in radians per second.

Simple pendulum

A simple pendulum is an ideal mass on a light string that swings under gravity with period $T = 2\pi\sqrt{\frac{L}{g}}$ for small angles.

Spring constant

The spring constant $k$ measures spring stiffness and appears in Hooke's law as $F = -kx$.

Amplitude

Amplitude is the maximum displacement from equilibrium during an oscillation.