Let us now consider a mass suspended from a string. If it is displaced slightly and then released, the simple pendulum also performs an oscillation. In this case, the restoring force is gravity acting on the center of mass of the pendulum.
For the following derivation, we assume that the string has negligible mass. This allows us to treat the center of mass as coinciding with the mass at the end of the string, and all forces act at that point. The pendulum mass moves back and forth along the arc of a circular path between two turning points.
Such an idealized system is often called a mathematical pendulum, because its equation of motion can be solved relatively easily.
The gravitational force $\vec{F}_\mathrm{G}$ can be decomposed into two components: the restoring force $\vec{F}_\mathrm{R}$, which acts tangentially along the arc, and a component $\vec{F}_\mathrm{S}$, which pulls along the string. While the magnitude of $\vec{F}_\mathrm{G}$ remains constant, the magnitudes of the other components change over time as the displacement angle $\varphi(t)$ varies.
The acceleration along the circular path is caused by the restoring force $F_\mathrm{R} = F_\mathrm{G} \sin\varphi(t)$, and can be expressed as $m\ddot{x}(t) = ml\ddot{\varphi}(t)$.
Equating both sides yields:
$$
mg \sin\varphi(t) = -ml\ddot{\varphi}(t)
$$
As with the spring pendulum, this leads to a second-order differential equation, but now the angle $\varphi(t)$ is the function of interest. For small angles, we can use the approximation $\sin\varphi \approx \varphi$. Substituting into the equation above gives:
$$
g\varphi(t) = -l\ddot{\varphi}(t)
$$
This differential equation can be solved using the harmonic ansatz:
$$
\varphi(t) = A \sin(\omega t)
$$
Substituting into the equation yields the angular frequency:
$$
\omega = \sqrt{\frac{g}{l}}
$$
From this, the period of oscillation is:
$$
T = 2\pi \sqrt{\frac{l}{g}}
$$
Without the small-angle approximation, the differential equation is no longer analytically solvable. Instead, it leads to an infinite series expansion, which becomes more accurate as more terms are included.