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Figure 6.10.1 created by Rebecka Tumblin.
Section 6.10 The Rigid-Body Pendulum
A pendulum does not need to be a simple mass on a string. In fact, many objects that can be modeled in the rigid-body model can act as pendulums. For example, as you walk, your leg swings like a pendulum. Consider a pendulum made from a rigid-body object.
Use a torque analysis and your model of simple harmonic motion to analyze the system. The gravitational torque is
\begin{equation*}
\tau = -LMg\sin\theta(t) = I \alpha
\end{equation*}
Recall that the gravitational force acts at the center of mass of a rigid-body. Here, the distance from the pivot to the center of mass is \(L\text{.}\) Apply the small-angle approximation,
\begin{equation*}
\sin\theta(t) \approx \theta(t)
\end{equation*}
Note that this analysis will only provide an accurate predictive model when the angle \(\theta(t)\) is small. From this you can determine the equation of motion for the system
\begin{equation}
\frac{d^2}{dt^2}\theta(t)=-\frac{MgL}{I} \theta(t)\tag{6.10.1}
\end{equation}
The equation of motion is in the form of equation (6.5.1) and so you can directly determine the oscillation frequency and period of the system.
\begin{equation}
\omega=\sqrt{\frac{MgL}{I}}= \frac{2 \pi}{T}\tag{6.10.2}
\end{equation}
This equation is general for any rigid-body object so long as you specify the moment of inertia and properly determine the distance from the pivot to the center of mass.
