Sample lecture note: the simple pendulum

A real note from a physics-flavored lecture, kept here as a living example of what these pages can do. The subject: why a pendulum’s period doesn’t care how heavy the bob is.

The big idea

For small swings, a pendulum is a simple harmonic oscillator in disguise. Gravity provides a restoring torque, and for small angles that torque is proportional to the displacement — which is the defining property of SHM.

Setting it up

Summing torques about the pivot for a bob of mass on a massless rod of length :

This is nonlinear because of the . The standard trick¹ is the small-angle approximation (good to about 1% below ):

The mass cancels on both sides — the period depends only on length and gravity. A heavier bob swings no faster.

Symbols

Symbol	Meaning	Units
	Angular displacement from vertical	rad
	Rod length, pivot to bob	m
	Gravitational acceleration	m/s²
	Angular frequency of oscillation	rad/s
	Period of one full swing	s

Checking it numerically

import numpy as np

g = 9.81  # m/s^2

def period(length_m: float) -> float:
    """Small-angle period of a simple pendulum."""
    return 2 * np.pi * np.sqrt(length_m / g)

for L in (0.25, 1.0, 4.0):
    print(f"L = {L:>4} m  ->  T = {period(L):.2f} s")

Quadrupling the length doubles the period — exactly the scaling the formula promises.

Linearization around an equilibrium shows up everywhere — circuits, control theory, economics. The pendulum is just the friendliest place to meet it. ↩