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The simple pendulum formula $T = 2\pi\sqrt{l/g}$ is valid for small angles ($\theta < 15°$) where $\sin\theta \approx \theta$. Unlike spring systems, the pendulum depends on gravity but is independent of mass and amplitude. In non-inertial frames, replace $g$ with $g_{\text{eff}}$: elevator up ($g + a$), elevator down ($g - a$), free fall ($0$, no oscillation), horizontal acceleration ($\sqrt{g^2+a^2}$).

A compound (physical) pendulum has $T = 2\pi\sqrt{I/(mgl)}$, where $I$ is the MOI about the pivot and $l$ is the COM-to-pivot distance. The equivalent simple pendulum length is $L_{\text{eq}} = I/(ml) = (k^2+l^2)/l$ where $k$ is the radius of gyration. The minimum time period occurs when $l = k$ (pivot distance equals radius of gyration), giving $T_{\min} = 2\pi\sqrt{2k/g}$. For a uniform rod pivoted at one end: $T = 2\pi\sqrt{2L/(3g)}$. The percentage change formula $\Delta T/T = \frac{1}{2}\Delta l/l = -\frac{1}{2}\Delta g/g$ is essential for error analysis and pendulum clock problems.