In the physical pendulum, the approximation of an infinitely thin, massless string—as used in the simple pendulum—is no longer applied. Instead, as shown in the original figure, the connection between the pendulum mass and the suspension point A has a finite thickness and mass.
In the general case, the pendulum consists of a three-dimensional object with arbitrary shape and mass distribution. As a result, the center of mass (CM) shifts from the end of the string toward the suspension point by a certain distance.
Rather than working with forces, we now deal with torques. To simplify the derivation, we can apply the substitutions outlined in the translation–rotation analogy: the mass $m$ becomes a moment of inertia $I$, and the force $F$ becomes a torque $D$.
This transforms the equation of motion into:
$$
mgd \sin\varphi = -I \ddot{\varphi}
$$
Here, $d$ is the distance between the suspension point and the center of mass.
Solving this differential equation (again using the small-angle approximation $\sin\varphi \approx \varphi$) yields:
$$
\omega = \sqrt{\frac{mgd}{I}}
$$
From this, the period of oscillation is:
$$
T = 2\pi \sqrt{\frac{I}{mgd}}
$$