PX155 - D10 - pendulums and SHM

mass on a light string displaced from vertical by $θ$ :
PX155 - A2 - newton's second law: $F = m a = - m g \sin θ$ (restoring force)
- acceleration around arc:

\frac{d^{2} s}{d t^{2}} = - g \sin θ

$a = \ddot{s}$ , but $s = l θ$
$∴ m l \ddot{θ} = - m g \sin θ$

\frac{d^{2} θ}{d t^{2}} = - \frac{g}{l} \sin θ

- using small angle approximation:

\frac{d^{2} θ}{d t^{2}} \approx - \frac{g}{l} θ

- this is the equation of SHM with $ω^{2} = \frac{g}{l}$ :

∴ T = 2 π \sqrt{\frac{l}{g}}