How do you derive $$T=2\pi\sqrt{I/mgl},$$ where $I$ is the moment of inertia and $l$ is the length of the pendulum? Is it even the right formula? How would I derive a compound pendulum formula for a pendulum similar to a clock pendulum with a disc attached to a rod?
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1$\begingroup$ MIT L24 - Pendulums $\endgroup$– FarcherCommented Jun 27, 2023 at 16:20
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$\begingroup$ Be careful. You need specify if $I$ is summed up at the center of mass, or at the pivot. $\endgroup$– John AlexiouCommented Jun 28, 2023 at 8:38
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$\begingroup$ @questioner123 Yes, it's valid formula. You can check that by multiplying $ml$ into numerator and denominator of standard formula $$T_{0}=2\pi {\sqrt {\frac {\ell }{g}}}$$. And point-like mass has $I=m\ell^2$ moment of inertia. $\endgroup$– Agnius VasiliauskasCommented Jun 28, 2023 at 8:43
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$\begingroup$ From SHM you have $$\ddot{\theta} = - \left( \tfrac{2 \pi}{T} \right)^2 \theta$$ so find the equations of motion $$\ddot{\theta} = f(\theta)$$ and extract the period $T$ term from it. $\endgroup$– John AlexiouCommented Mar 19 at 14:12
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1 Answer
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How to drive the equation $~T=\ldots~$
you start with the position vectors to the pendulum center of mass $~\vec P_{\rm cm}~$ and the position vector to the disc $~\vec P_d~$
$$\vec P_{\rm cm}=\frac l2\, \begin{bmatrix} \sin(\varphi) \\ \cos(\varphi) \\ \end{bmatrix}\quad, \vec P_d=l\, \begin{bmatrix} \sin(\varphi) \\ \cos(\varphi) \\ \end{bmatrix} $$
from here the velocity of the CM and the velocity of the disc
$$\vec v_{\rm cm}=\frac{d}{dt}\,\vec P_{\rm cm}=\ldots\quad, \vec v_{d}=\frac{d}{dt}\,\vec P_d=\ldots$$
with the kinetic energy
$$T=\frac 12\left(m_p\,\vec v_{\rm cm}\cdot \vec v_{\rm cm}+m_d\vec v_{d}\cdot\,\vec v_{d}+I_{P}\dot\varphi^2\right)$$
the potenzial energy
$$U=m_p\,g\,(\vec P_{\rm cm})_y+m_d\,g\,(\vec P_{d})_y$$
and EL you obtain the equation of motion
$$\ddot\varphi+ \underbrace{\frac{l(m_p\,g+2\,m_d)}{2\,I_P}}_{\omega^2}\,\sin(\varphi)=0$$
for a small angle $~\varphi~$ $$\ddot\varphi+\omega^2\,\varphi=0$$
and with
$$\omega=\frac{2\,\pi}{T}$$
you obtain $~T~$
