Pendulums (Simple, Physical, Torsional) Cheat Sheet

Key Facts

• For a simple pendulum at small angles, the period is $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length from pivot to bob center.
• The small-angle approximation is $\sin\theta \approx \theta$ when $\theta$ is measured in radians and is usually accurate for small amplitudes.
• The angular frequency of a simple pendulum is $\omega = \sqrt{\frac{g}{L}}$, so $T = \frac{2\pi}{\omega}$.
• For a physical pendulum, the period is $T = 2\pi\sqrt{\frac{I}{mgd}}$, where $I$ is rotational inertia about the pivot and $d$ is the distance from pivot to center of mass.
• The parallel-axis theorem gives $I = I_{\mathrm{cm}} + md^2$ when converting rotational inertia from the center of mass to the pivot.
• For a torsional pendulum, the restoring torque is $\tau = -\kappa\theta$ and the period is $T = 2\pi\sqrt{\frac{I}{\kappa}}$.
• Frequency and period are related by $f = \frac{1}{T}$, and angular frequency is related by $\omega = 2\pi f = \frac{2\pi}{T}$.
• Ideal pendulum period is independent of amplitude only when the oscillations are small enough for simple harmonic motion.

Vocabulary

Simple pendulum

A point mass on a light string or rod that swings under gravity with period $T = 2\pi\sqrt{\frac{L}{g}}$ for small angles.

Physical pendulum

An extended rigid body that swings about a pivot, with its mass distribution included through rotational inertia $I$.

Torsional pendulum

A rotating system that twists back and forth because a wire or support produces restoring torque $\tau = -\kappa\theta$.

Restoring torque

A torque that acts opposite the displacement and tends to return the system to equilibrium.

Rotational inertia

A measure of resistance to angular acceleration, written $I$, that depends on both mass and how far the mass is from the axis.

Small-angle approximation

The rule $\sin\theta \approx \theta$ for $\theta$ in radians, which lets many pendulums be modeled as simple harmonic oscillators.