Physical pendulum

AP Physics C: Mechanics only, Topic 7.5. EK 7.5.A.1 defines it as a rigid body that undergoes oscillation about a fixed axis. For small amplitudes the C: Mechanics sheet prints

$T_{\text{phys}} = 2\pi\sqrt{\frac{I}{mgd}}$

where $I$ is the rotational inertia about the pivot, not about the center of mass, and $d$ is the distance from the pivot to the center of mass.

Where it comes from. Displace the body and gravity acting at the center of mass supplies a restoring torque, $\tau = -mgd\sin\theta$ (EK 7.5.A.2.i). Apply the small-angle approximation and Newton's second law in rotational form gives $\tau = -mgd\theta = I\alpha$, which is the SHM differential equation $\frac{d^2\theta}{dt^2} = -\omega^2\theta$ (EK 7.5.A.2.ii and iii).

Two traps. Shift $I$ to the pivot with the parallel-axis theorem before substituting; reaching for $I_{\text{cm}}$ is the standard error. And $d$ is the pivot to center of mass distance, not the length of the object: for a uniform rod pivoted at one end, $d = L/2$ while $I = \frac{1}{3}mL^2$.

A simple pendulum is the limiting case. Put all the mass at distance $\ell$ so $I = m\ell^2$ and $d = \ell$, and the formula collapses to $T_p = 2\pi\sqrt{\ell/g}$ (EK 7.5.A.3).