PX285 - C4 - simple pendulum

consider a simple pendulum bearing a mass, $m$ , at the end of a string of length, $l$ , inclined at an angle, $θ$ , to the vertical
the kinetic energy:

T = \frac{1}{2} I {\dot{θ}}^{2} = \frac{1}{2} m l^{2} {\dot{θ}}^{2}

where, $I$ is the moment of inertia of the pendulum about the pivot

the potential energy:

V = - m g l \cos θ

applying the euler-lagrange equation:

\begin{aligned} - (m g l \sin θ + m l^{2} \ddot{θ}) & = 0 \\ - \frac{g}{l} \sin θ & = \ddot{θ} \end{aligned}

let, $ω^{2} = \frac{- g}{l}$ , and using the small angle approximation:

\ddot{θ} \approx - ω^{2} θ

this is a simple harmonic relation, where $ω = \sqrt{\frac{g}{l}}$ is the angular frequency
the solution is in the form:

θ (t) = A \sin (ω t + ϕ)

where, $A$ and $ϕ$ are arbitrary amplitude and phase difference respectively, which can be determined given two boundary conditions