M2. Compound pendulum

Determining the value of gravitational acceleration ( $g$ ) using a compound pendulum.

Theory

$\tau = -mgl \sin\theta = -mgl\theta.$

$\tau = I \frac{d^2\theta}{dt^2}$

$\tau = -mgl \sin\theta = -mgl\theta.$

$\frac{d^2\theta}{dt^2} = - \frac{mgl}{I} \theta$

$T_c = 2\pi\sqrt{\frac{I}{mgl}}$

$I = I_G + ml^2 = mK^2 + ml^2$

$T_c = 2\pi\sqrt{\frac{mK^2+ml^2}{mgl}} = 2\pi\sqrt{\frac{\frac{K^2}{l}+l}{g}}$

Compare this with the period of a simple pendulum

$T_s = 2\pi\sqrt{\frac{L}{g}}$

Comparing $T_c$ and $T_s$ ,

$L = \frac{K^2}{l}+l \Rightarrow l^2 - lL + K^2 = 0.$

This equation has two solutions $l_1$ and $l_2$ where $L=l_1+l_2$ and $K=\sqrt{l_1l_2}$ .