= Simple pendulum
{title2=$\ddot\theta+(g/\ell)\sin\theta=0$}
A point mass constrained to a fixed-length massless arm under uniform gravitational <acceleration> is a simple pendulum. With no drag its angular equation is $\ddot\theta+(g/\ell)\sin\theta=0$. The <small-angle approximation> yields <simple harmonic motion>, whereas the full restoring force is nonlinear. Adding drag gives a <damped pendulum>.
For an ideal point mass $m$ swinging in a vertical plane, the <torque> about the pivot is $-mg\ell\sin\theta$. Dividing by its <moment of inertia> $m\ell^2$ gives the displayed equation. Both the inertia and the gravitational <torque> scale with $m$, explaining why the point mass cancels. This derivation and the small-amplitude period are given in https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums[Ling, Sanny and Moebs, University Physics, pendulums].
Released from rest at a small angular <amplitude> $\theta_0$, the linearized solution is $\theta(t)\simeq\theta_0\cos(\sqrt{g/\ell}\,t)$, with <oscillation period> $T_0=2\pi\sqrt{\ell/g}$. For $g=9.81\,\mathrm{m/s^2}$, lengths one and four metres give $T_0\approx2.0061\,\mathrm s$ and $4.0121\,\mathrm s$. This same scaling follows from <dimensional analysis>; the dynamical calculation supplies the coefficient $2\pi$.
Finite angular <amplitude> changes the exact ideal period. <Conservation of energy> gives $\dot\theta^2=(2g/\ell)(\cos\theta-\cos\theta_0)$ for release from rest. Integrating a quarter oscillation and substituting $\sin(\theta/2)=\sin(\theta_0/2)\sin\phi$ gives
$$
T(\theta_0)=4\sqrt{\frac\ell g}\,K\!\left(\sin\frac{\theta_0}2\right),\qquad
\frac{T(\theta_0)}{T_0}=\frac2\pi\int_0^{\pi/2}\frac{d\phi}{\sqrt{1-\sin^2(\theta_0/2)\sin^2\phi}},\quad 0<\theta_0<\pi.
$$
Here $K$ is the <complete elliptic integral of the first kind>. The <small-angle approximation> gives a ratio tending to one; at $10^\circ,60^\circ,90^\circ$, the ratios are approximately $1.00191,1.07318,1.18034$. Thus almost constant period at small <amplitude> has a quantitative limit. As $\theta_0\to\pi$, the period diverges, connecting to the <pendulum separatrix approach takes infinite time>.
\Image[/pendulum-period-scaling.png]
{title=Length scaling and finite-amplitude period corrections for an ideal simple pendulum}
These predictions assume the specified ideal system. Drag changes the <mechanical energy>; a massive support changes the <moment of inertia>. The distinction is useful in experiments: a change in period can test an approximation rather than merely indicate an arithmetic mistake.
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