3 separate trials for 3 different pendulum lengths (27.5, 37.5 and 50cm) were performed. The pendulum was constructed using of a piece of lightweight twine tied vertically to a ASUS Zenfone 10 (172.0g, 14.65cm height, 6.81cm width) on the lower end and a ruler on the top end. The ruler was weighed down using a textbook, allowing the twine and smartphone to hang off the edge of a desk. The smartphone was raised and released from a 30 degree angle and allowed to swing until rest. Data$^{[1]}$ was gathered using the phyphox app and the phone’s accelerometer.
Simplifying assumptions:
Neglecting drag, we can approximate the maximum speed at the bottom of the pendulum using conservation of mechanical energy:
$K_i+U_{Gi}=0+mg(L-Lcos\frac{\pi}{6})=\frac{1}{2}mv_{max}^2+0=K_f+U_{Gf}$
$$ \begin{equation} v_{max}=\sqrt{2gL(1-cos\frac{\pi}{6})} \end{equation} $$
Similarly, we can use Newton’s 2nd law to get the same expression
$F_{net_t}=-mgsin\theta =ma_t \rightarrow a_t=-gsin\theta$
$a_t=\alpha L=\frac{d \omega}{dt}L=\frac{\omega d \omega}{d\theta}L$
$-\frac{g}{L}sin\theta d\theta=\omega d \omega$
Integrating both sides, we get